Operations in some graded Hopf categories

Volodymyr Lyubashenko

June 13, 2001

Institute of Mathematics, 3 Tereshchenkivska st., Kyiv-4, 01601 MSP, Ukraine

lub@imath.kiev.ua

1. Introduction

Crane and Frenkel proposed a notion of a Hopf category in [CF94]. It was motivated by Lusztig’s approach to quantum groups – his theory of canonical bases. In particular, Lusztig obtains braided deformations U_{q}𝔫_{+} of universal enveloping algebras U𝔫_{+} for some nilpotent Lie algebras 𝔫_{+} together with canonical bases of these braided Hopf algebras [Lus90, Lus91, Lus93]. The elements of the canonical basis are identified with isomorphism classes of simple perverse sheaves – certain objects of equivariant derived categories. They are contained in a semisimple abelian category of semisimple complexes. Conjectural properties of this category were collected into a system of axioms of a Hopf category, equipped with functors of multiplication and comultiplication, isomorphisms of associativity, coassociativity and coherence which satisfy four equations [CF94]. Crane and Frenkel gave an example of a Hopf category resembling the semisimple category encountered in Lusztig’s theory corresponding to one-dimensional Lie algebra 𝔫_{+} – nilpotent subalgebra of 𝔰𝔩(2). The mathematical framework and some further examples of Hopf categories were provided by Neuchl [Neu].

We discuss an example of a related notion – triangulated Hopf category – the whole equivariant derived category equipped with multiplication and comultiplication functors and structure isomorphisms. In particular, it is a monoidal category. The additive relations between operations proposed in [CF94] are replaced with distinguished triangles. The preliminary study of the subject can be found in [Lyu99a, Lyu99b].

1.1. Operations in a graded Hopf algebra

Let Q_{+} be a commutative monoid additively generated by elements of a finite set I. Denote R = ℤ[q, q^{−1}]. Let ℋ be a Q_{+}-graded braided Hopf R-algebra, for instance, the algebra U_{q}𝔫_{+} of Lusztig [Lus93]. The comultiplication in ℋ = ⊕_{v∈Q_{+}}ℋ_{v} can be written as

Δ = ∑_{u,v∈Q_{+}} Δ_{u,v}, Δ_{u,v} : ℋ_{u+v} → ℋ_{u} ⊗_{R} ℋ_{v}.

Similarly for iterated comultiplication Δ^{(l)} = (Δ^{(l−1)} ⊗ 1) ∘ Δ : ℋ → ℋ^{⊗l},

Δ^{(l)} = ∑_{v_{r}∈Q_{+}} Δ_{v_{1},...,v_{l}}^{(l)}, Δ_{v_{1},...,v_{l}}^{(l)} : ℋ_{v_{1}+···+v_{l}} → ℋ_{v_{1}} ⊗_{R} ⋯ ⊗_{R} ℋ_{v_{l}}.

The associativity, the coassociativity and the bialgebra axiom imply the equation

Δ^{(l)}(x^{1}) · . . . · Δ^{(l)}(x^{k}) = Δ^{(l)}(x^{1} · . . . · x^{k})  (1.1.1)

for arbitrary elements x^{m} ∈ ℋ. Note that multiplication in the left hand side uses the braiding. Apply equation (1.1.1) to homogeneous elements x^{m} of degree v^{m} and write down its homogeneous component of multidegree (v_{1}, …, v_{l}) ∈ Q_{+}^{l}:

∑_{\begin{matrix}∑_{r} v_{r}^{m}=v^{m} \\ ∑_{m} v_{r}^{m}=v_{r} \end{matrix}} Δ_{v_{1}^{1},...,v_{l}^{1}}^{(l)}(x^{1}) · · · · · Δ_{v_{1}^{k},...,v_{l}^{k}}^{(l)}(x^{k}) = Δ_{v_{1},...,v_{l}}^{(l)}(x^{1} · . . . · x^{k}).  (1.1.2)

Each summand in the left hand side can be viewed as an operation with k inputs and l outputs. These operations are not distinguished in algebra setup. However, in graded Hopf categories their explicit use seems advantageous.

2. The main ingredients

2.1. The basic data

Categories will be equivariant derived categories \frac{X}{G} := D_{G}^{b,c}(X), where X is a complex algebraic variety, equipped with the action of a complex algebraic group G, as defined by Bernstein and Lunts [BL94].

The functors will be compositions of functors of the three types (see [BL94]). Let ϕ : G → H be a group homomorphism, let X be a G-space, let Y be an H-space, and let f : X → Y be a ϕ-equivariant map. Then there are

— the inverse image functor \frac{f}{ϕ}^{∗} : \frac{Y}{H} → \frac{X}{G},

— if ϕ : G → H is surjective, K = Ker(ϕ), X is K-free, and Y = K\X, the direct image functor (in this case it is an equivalence) \frac{f}{ϕ}_{∗} : \frac{X}{G} → \frac{Y}{H}.

— if ϕ = 1 : G = H is the identity, the direct image functor with proper supports \frac{f}{1}_{!} : \frac{X}{G} → \frac{Y}{G}.

Quiver. Let (H, I) be a finite oriented graph with the set of vertices I, the set of edges H, the structure map H → I × I, h ↦ (h′, h′′), where h′ ∈ I is the source of h ∈ H, and h′′ ∈ I is the target of h ∈ H, such that h′ ≠ h′′.

Let V be a finite dimensional I-graded ℂ-vector space, (a function V : I → Ob ℂ -vect, i ↦ V (i)). Its automorphism group is

G_{V} = Aut_{I -grad -vect} V = ∏_{i∈I} GL(V (i)).

Define a linear space

E_{V} = ⊕_{h∈H} Hom_{ℂ}(V (h′), V (h′′)).

The union of all E_{V} is the class of representations of the quiver. The group G_{V} acts on E_{V} by (g.x)_{h} = g_{h′′}x_{h}g_{h′}^{−1}. The union of all G_{V} \E_{V} is the set of isomorphism classes of representations of the quiver. We consider the collection of equivariant derived categories \frac{E_{V}}{G_{V}} as our Hopf category.

Filtrations. To introduce operations we need to consider decompositions of V

𝒱 : V^{1} ⊕ V^{2} ⊕ ⋯ ⊕ V^{k} = V

into I-graded subspaces. Associate with it a filtration of V

0 = V^{(0)} ⊂ V^{(1)} ⊂ ⋯ ⊂ V^{(k)} = V,  V^{(m)} = V^{1} ⊕ ⋯ ⊕ V^{m}.  (2.1.1)

Associate with it the parabolic group P_{𝒱}

P_{𝒱} = {g ∈ G_{V} | ∀m g(V^{(m)}) ⊂ V^{(m)}}.

The unipotent radical of P_{𝒱} is denoted U_{𝒱}. The group

L_{𝒱} = {g ∈ G_{V} | ∀m g(V^{m}) ⊂ V^{m}} = ∏_{m=1}^{k} G_{V^{m}} ≃ P_{𝒱}/U_{𝒱}

is a Levi subgroup of P_{𝒱}.

Let F_{𝒱} be the linear subspace of E_{V} respecting the filtration:

F_{𝒱} = {x ∈ E_{V} | ∀m, h x_{h}(V^{(m)}(h′)) ⊂ V^{(m)}(h′′)}.

The group P_{𝒱} acts in F_{𝒱} as well as in its subspace f_{𝒱}, consisting of maps lowering the filtration:

f_{𝒱} = {x ∈ E_{V} | ∀m, h x_{h}(V^{(m)}(h′)) ⊂ V^{(m−1)}(h′′)}.

Operations. Let two decompositions of V into a direct sum be given:

𝒱 :  V^{1} ⊕ V^{2} ⊕ ⋯ ⊕ V^{k} ^{∼}→ V,  (2.1.2)
𝒲 : W_{1} ⊕ W_{2} ⊕ ⋯ ⊕ W_{l} ^{∼}→ V.  (2.1.3)

We denote by 𝒱 and 𝒲 also the associated filtrations (2.1.1) and

0 = W_{(0)} ⊂ W_{(1)} ⊂ ⋯ ⊂ W_{(l)} = V, W_{(r)} = W_{1} ⊕ ⋯ ⊕ W_{r}.

Let 𝒪 ⊂ G_{V} be a left P_{𝒲} -invariant and right P_{𝒱}-invariant locally closed subset. We associate with it an operation

� _{𝒲}^{𝒪;𝒱} = _{𝒪}_{W_{1} W_{2}  W_{l}}^{V^{1} V^{2}  V^{k}} = [figure: ^{V^{1} V^{2}  V^{k}}; 𝒪; W_{1}; W_{2}; W_{l}]=⋔_{𝒲}^{𝒱} ∘Ψ_{𝒲}^{𝒪;𝒱} .

The components of it are generalized multiplication and comultiplication functors defined below.

Graphical conventions. In graphical notations a straight line |X labeled by an I-graded space X denotes category \frac{E_{X}}{G_{X}}. A dashed line 𝒳 labeled by a filtration 𝒳 of an I-graded space X denotes category \frac{E_{X}}{P_{𝒳}}. If a horizontal line crosses a diagram intersecting transversally with several (solid or dashed) open intervals labeled by spaces Y_{1}, Y_{2}, … and filtrations 𝒳_{1}, 𝒳_{2} of spaces X_{1}, X_{2}, …, this line represents the category

\frac{E_{Y_{1}} × E_{Y_{2}} × ⋯ × E_{X_{1}} × E_{X_{2}} × ⋯}{G_{Y_{1}} × G_{Y_{2}} × ⋯ × P_{𝒳_{1}} × P_{𝒳_{2}} × ⋯}.

The order of factors repeats the order of intersection points on the horizontal line.

A vertex (or a horizontal row of vertices) represents a functor from the category, corresponding to horizontal line just above the vertex, to the category represented by a line just below the vertex.

Multiplication half. The multiplication half operation is

𝒪_{𝒲}^{V^{1} V^{2}  V^{k}} = Ψ_{𝒲}^{𝒪;𝒱} = (\frac{∏_{i=1}^{k} E_{V^{i}}}{∏_{i=1}^{k} G_{V^{i}}}  ^{ϕ^{∗}}→ \frac{𝒪 × F_{𝒱}}{P_{𝒲} × P_{𝒱}} ^{π_{∗}}→ \frac{𝒪 ×_{P_{𝒱}} F_{𝒱}}{P_{𝒲}} ^{α_{!}}→ \frac{E_{V}}{P_{𝒲}} ).

The scheme of multiplication is similar to that of Lusztig [Lus90, Lus91, Lus93]:

∏_{i=1}^{k} E_{V^{i}} ←^{ϕ} 𝒪 × F_{𝒱} ^{π}→ 𝒪 ×_{P_{𝒱}} F_{𝒱} ^{α}→ E_{V} ,

where ϕ(o, f) = κ(f) is the forgetful map, κ : F_{𝒱} → ∏_{i=1}^{k} E_{V^{i}} is the natural projection, π is the canonical projection, α(o, f) = o.ι(f) is induced from the action map, and ι : F_{𝒱} → E_{V} is the natural embedding.

Comultiplication half. The comultiplication half operation functor is

W_{1} _{W_{2}}  _{W_{l}}^{𝒲} =⋔_{𝒲}^{𝒱} = (\frac{E_{V}}{P_{𝒲}} ^{ι^{∗}}→ \frac{F_{𝒲}}{∏_{j=1}^{l} G_{W_{j}}}  ^{κ_{!}}→ \frac{∏_{j=1}^{l} E_{W_{j}}}{∏_{j=1}^{l} G_{W_{j}}}  ^{τ}→ \frac{∏_{j=1}^{l} E_{W_{j}}}{∏_{j=1}^{l} G_{W_{j}}} ),

where τ is the shift

τL = L[2 ∑_{r>s;h} dim W_{r}(h′) · dim W_{s}(h′′)].

The scheme of comultiplication is made of the natural embedding ι and the natural projection κ (as in Lusztig [Lus90, Lus91, Lus93]):

E_{V} ←^{ι} F_{𝒲} ^{κ}→ ∏_{j=1}^{l} E_{W_{j}}.

Braiding. Let (A_{m})_{m=1}^{k}, (B_{n})_{n=1}^{l} be sequences of I-graded vector spaces. We define the braiding functor as

c = = (\frac{∏ E_{A_{m}} × ∏ E_{B_{n}}}{∏ G_{A_{m}} × ∏ G_{B_{n}}} ⟶^{τ} \frac{∏ E_{A_{m}} × ∏ E_{B_{n}}}{∏ G_{A_{m}} × ∏ G_{B_{n}}} ⟶^{σ^{∗}} \frac{∏ E_{B_{n}} × ∏ E_{A_{m}}}{∏ G_{B_{n}} × ∏ G_{A_{m}}} ),

where σ is the permutation isomorphism of groups and modules and the functor τ is the shift

τ(L) = L[−2 ∑_{\begin{matrix}m,n \\ i∈I \end{matrix}} dim A_{m}(i) dim B_{n}(i) + 2 ∑_{\begin{matrix}m,n \\ h∈H \end{matrix}} dim A_{m}(h′) dim B_{n}(h′′)].

When 𝒜_{m}, ℬ_{n} are filtrations of I-graded vector spaces A_{m}, B_{n} we use the same τ and similar σ to define the braiding functor

c = = (\frac{∏ E_{A_{m}} × ∏ E_{B_{n}}}{∏ P_{𝒜_{m}} × ∏ P_{ℬ_{n}}} ⟶^{τ} \frac{∏ E_{A_{m}} × ∏ E_{B_{n}}}{∏ P_{𝒜_{m}} × ∏ P_{ℬ_{n}}} ⟶^{σ^{∗}} \frac{∏ E_{B_{n}} × ∏ E_{A_{m}}}{∏ P_{ℬ_{n}} × ∏ P_{𝒜_{m}}}).

2.1.1. Minimal parametrizing subsets. Minimal bi-invariant subsets 𝒪 ⊂ G_{V} are orbits of the action of P_{𝒲} × P_{𝒱} in G_{V}, (p, r).g = pgr^{−1}. They are in natural bijection with orbits of left action of G_{V} in the space of pairs of filtrations P_{𝒲} \G_{V} × G_{V} /P_{𝒱} ≃ G_{V} /P_{𝒲} × G_{V} /P_{𝒱}, (P_{𝒲} g, hP_{𝒱}) ↔ (g^{−1}P_{𝒲} , hP_{𝒱}). Indeed, multiplication

P_{𝒲} \G_{V} ×_{G_{V}} G_{V} /P_{𝒱} → P_{𝒲} \G_{V} /P_{𝒱}, (P_{𝒲} g, hP_{𝒱}) ↦ P_{𝒲} ghP_{𝒱}

is a bijection. By [BLM90] these orbits are in bijection with k × l-matrices (v_{r}^{m}), 1 ⩽ m ⩽ k, 1 ⩽ r ⩽ l with elements in Q_{+} = ℤ_{+}^{I}, such that ∑_{r} v_{r}^{m} = v^{m} is the dimension of V^{m} and ∑_{m} v_{r}^{m} = v_{r} is the dimension of W_{r}. Indeed, to filtration g^{−1}𝒲 of length l and filtration h𝒱 of length k Beilinson, Lusztig and MacPherson associate a filtration 𝒰 of the same space of length kl

0 = X_{0}^{1} ⊂ X_{1}^{1} ⊂ ⋯ ⊂ X_{l}^{1} = X_{0}^{2} ⊂ ⋯ ⊂ X_{l}^{2} = X_{0}^{3} ⊂ ⋯ ⊂ X_{l−1}^{k} ⊂ X_{l}^{k} = V,

where

X_{r}^{m} = hV^{(m−1)} + (g^{−1}W_{(r)} ∩ hV^{(m)}) = (hV^{(m−1)} + g^{−1}W_{(r)}) ∩ hV^{(m)}.  (2.1.4)

The matrix (v_{r}^{m}) consists of relative dimensions v_{r}^{m} = dim_{ℂ} X_{r}^{m}/X_{r−1}^{m}. For our purposes it suffices to take g = 1. We conclude that the orbits of P_{𝒲} × P_{𝒱} in G_{V} are in bijection with the summands in the left hand side of equation (1.1.2).

Distinguished triangles. The P_{𝒲} ×P_{𝒱}-invariant subsets of G_{V} are unions of orbits, thereby, they correspond to sums of several summands in (1.1.2). The additive relation (1.1.2) in algebra is replaced for our Hopf category by a system of functorial distinguished triangles

� ^{𝒪_{U}} → � ^{𝒪_{X}} → � ^{𝒪_{F}} →

given for any bi-invariant subset 𝒪_{X} ⊂ G_{V} and a bi-invariant closed subset 𝒪_{F} ⊂ 𝒪_{X} with 𝒪_{U} = 𝒪_{X} −𝒪_{F}. The following diagram made with given distinguished triangles is an octahedron

\begin{aligned}� ^{𝒪_{R}} & & & \overline{→} & � ^{𝒪_{W}} \\ d \\ = & � ^{𝒪_{Z}} & \underline{←} & ^{\underline{←} ^{1}} & = \\ d \\ � ^{𝒪_{Q}} & ^{^{1}↑} & ←  _{1} & \overline{→} & � ^{𝒪_{F}} & ↓ & \overline{→} \end{aligned}  \begin{aligned}� ^{𝒪_{R}} & \overline{→} & � ^{𝒪_{W}} \\ & = \\ d & � ^{𝒪_{U}} & \overline{→} & \overline{→} & d \\ & = \\ � ^{𝒪_{Q}} & ^{^{1}↑} & ←  _{1} & \underline{←} & � ^{𝒪_{F}} & ↓ & ^{\underline{←} ^{1}} \end{aligned}

for any pair of closed embeddings 𝒪_{F} ⊂ 𝒪_{Z} ⊂ 𝒪_{W}, where 𝒪_{U} = 𝒪_{W} − 𝒪_{F}, 𝒪_{Q} = 𝒪_{Z} − 𝒪_{F}, 𝒪_{R} = 𝒪_{W} − 𝒪_{Z}. This means commutativity of two squares formed by diagonal maps and of the four triangles marked “=”. This is proven in [Lyu, Proposition 3.11.1].

2.2. Coherence isomorphism

Assume given a vector space decomposition

V = ⊕_{m=1}^{k} ⊕_{r=1}^{l} V_{r}^{m}.

We shall denote it V_{−}^{=} = 𝒰 if the summands are ordered as follows

V_{1}^{1}, V_{2}^{1}, …, V_{l}^{1}, V_{1}^{2}, V_{2}^{2}, …, V_{l}^{2}, …, V_{1}^{k}, V_{2}^{k}, …, V_{l}^{k}.  (2.2.1)

The same decomposition with the order of summands

V_{1}^{1}, V_{1}^{2}, …, V_{1}^{k}, V_{2}^{1}, V_{2}^{2}, …, V_{2}^{k}, …, V_{l}^{1}, V_{l}^{2}, …, V_{l}^{k}

will be denoted V_{=}^{−} = 𝒳. We also have decompositions

V_{−}^{m} = 𝒱^{m} : V^{m} = ⊕_{r=1}^{l}V_{r}^{m},  V^{−} = 𝒱 : V = ⊕_{m=1}^{k}V^{m},
V_{r}^{−} = 𝒲_{r} : W_{r} = ⊕_{m=1}^{k}V_{r}^{m}, V_{−} = 𝒲 : V = ⊕_{r=1}^{l}W_{r},

which produce filtrations V_{(r)}^{m} of V^{m}, V^{(m)} of V, V_{r}^{(m)} of W_{r} and W_{(r)} of V. We associate with these filtrations the parabolic groups P_{𝒱^{m}} ⊂ G_{V^{m}}, P_{𝒱} ⊂ G_{V}, P_{𝒲_{r}} ⊂ G_{W_{r}} and P_{𝒲} ⊂ G_{𝒲} .

We claim that for any collection of indices and for any collection of bi-invariant locally closed subsets (𝒪′_{1}, …, 𝒪′_{k}, 𝒪′′_{1}, …, 𝒪′′_{l}), which may occur in the following diagram, there exists a bi-invariant locally closed subset 𝒪 and a coherence isomorphism

[figure: 𝒪′_{1}; 𝒪′_{k}; l; l; l; σ_{k,l}; 𝒪′′_{1}; k; k; 𝒪′′_{l}; k] ≡ [figure: 𝒴; 𝒱; 𝒪′_{1}; 𝒪′_{k}; 𝒰; 𝒳; 𝒲; 𝒪′′_{1}; 𝒪′′_{l}; 𝒵]  ^{Coher} →  ^{𝒪} ^{},

explicitly described below. Here σ_{k,l} = (s_{k,l})_{+}^{∼} is the braid, corresponding to the permutation s_{k,l} of the set {1, 2, …, kl},

s_{k,l}(1 + t + nl) = 1 + n + tk for 0 ⩽ t < l, 0 ⩽ n < k,  (2.2.2)

under the standard splitting S_{kl} → B_{kl}, which maps the elementary transpositions to the generators of the braid group. The subset 𝒪 is computed as follows.

Bi-invariance of initial parametrising subsets means that

G_{V^{m}} ⊃ 𝒪′_{m} ∈ P_{V_{−}^{m}} -sets- P_{𝒴^{m}},
G_{W_{r}} ⊃ 𝒪′′_{r} ∈ P_{𝒵_{r}} -sets- P_{V_{r}^{−}},

where 𝒴^{m} is the incoming filtration of V^{m}, and 𝒵_{r} is the outgoing filtration of W_{r}. The subsets

P_{𝒱} ⊃ \overline{𝒪} =^{def} U_{𝒱} · ∏_{m} 𝒪′_{m} = ∏_{m} 𝒪′_{m} · U_{𝒱} ∈ P_{V_{−}^{=}} -sets- P_{𝒴},
P_{𝒱} ⊃ \underline{𝒪} =^{def} U_{𝒲} · ∏_{r} 𝒪′′_{r} = ∏_{r} 𝒪′′_{r} · U_{𝒲} ∈ P_{𝒵} -sets- P_{V_{=}^{−}}

are locally closed. The subset

G_{V} ⊃ 𝒪 =^{def} \underline{𝒪} · \overline{𝒪} ∈ P_{𝒵} -sets- P_{𝒴}, 𝒪 = \underline{𝒪}P_{𝒰} ×_{P_{𝒰}} \overline{𝒪} = \underline{𝒪} ×_{P_{𝒲} ∩P_{𝒱}} \overline{𝒪}

is also locally closed. Indeed, \underline{𝒪} × \overline{𝒪} is locally closed in P_{𝒲} × P_{𝒱}, hence, \underline{𝒪} ×_{P_{𝒲} ∩P_{𝒱}} \overline{𝒪} is locally closed in P_{𝒲} ×_{P_{𝒲} ∩P_{𝒱}} P_{𝒱} ≃ P_{𝒲} · P_{𝒱}. The subspace C = P_{𝒲} · P_{𝒱} of G_{V} is locally closed, since it is an orbit of P_{𝒲} × P_{𝒱} in G_{V}, and the number of such orbits is finite by Section 2.1.1. Indeed, C is embedded in its closure as

C = \overline{C} − ⋃\{\overline{𝒪_{a}} | 𝒪_{a} ⊂ \overline{C} is an orbit of P_{𝒲} × P_{𝒱}, 𝒪_{a} ≠ C\}

due to dimension considerations.

Both associativity isomorphism and coassociativity isomorphism of [Lyu99b] are particular cases of the general coherence isomorphism.

In the particular case, when the upper row of operations consists of comultiplications (operations with single input), and the lower row of operations consists of multiplications (operations with single output), we have necessarily 𝒪′_{m} = G_{V^{m}} and 𝒪′′_{r} = G_{W_{r}}. In such cases we omit the parametrizing set, since it is unique. Graphical notation here is the following:

[figure: _{V^{1}  V^{2}  V^{k}}; _{V_{1}^{1}  V_{l}^{1}  V_{1}^{2}  V_{l}^{2}  V_{1}^{k}  V_{l}^{k}}; V_{2}^{1}; V_{2}^{2}; V_{2}^{k}; _{V_{1}^{2}  V_{2}^{2}  V_{l}^{2}}; V_{1}^{1}; V_{1}^{k}; V_{2}^{1}; V_{2}^{k}; V_{l}^{1}; V_{l}^{k}; W^{1}  W^{2}  W^{l}]  ^{Coher} → _{𝒪}_{W_{1}  W_{2}  W_{l}}^{V^{1}  V^{2}  V^{k}} .

By general formulas we find here \overline{𝒪} = P_{𝒱}, \underline{𝒪} = P_{𝒲} , hence, 𝒪 = P_{𝒱}P_{𝒲} . We may set g = h = 1 in definition (2.1.4) of the filtration

X_{r}^{m} = V^{(m−1)} + (W_{(r)} ∩ V^{(m)}) = (⨁_{n=1}^{m−1} ⨁_{s=1}^{l} V_{s}^{n}) ⊕ (⨁_{s=1}^{r} V_{s}^{m}).

This filtration coincides with V_{−}^{=} = 𝒰 obtained from (2.2.1). The computed orbit 𝒪 = P_{𝒱}P_{𝒲} corresponds via bijection of Section 2.1.1 to the term in the left hand side of (1.1.2) indexed by (v_{r}^{m} = dim V_{r}^{m}).

An impression might be created, that to a matrix (v_{r}^{m}) correspond many operations � _{𝒲}^{𝒪;𝒱} , when 𝒱, 𝒲 and 𝒪 vary. However, up to natural equivalences and isomorphisms they coincide.

The general coherence isomorphism is built as the composition

[figure: 𝒴; 𝒱; 𝒪′_{1}; 𝒪′_{k}; 𝒰; 𝒳; 𝒲; 𝒪′′_{1}; 𝒪′′_{l}; 𝒵] = [figure: 𝒪′_{1}; 𝒪′_{k}; 𝒱^{1}  𝒱^{k}; 𝒪′′_{1}; 𝒪′′_{l}; 𝒵^{1}  𝒵^{l}]  ^{coher} →

[figure: 𝒪′_{1}; 𝒱^{1}; 𝒪′_{k}; 𝒱^{k}; \underline{𝒪} P_{𝒰}; 𝒵; 𝒵_{1}; 𝒵_{l}] assoc coass →  [figure: 𝒪; 𝒵]  =  ^{𝒪}  ^{}.

The three components of the coherence isomorphism are defined next. All cells in the following diagrams represent isomorphisms, defined in Section 4.1.

\frac{∏ E_{Y^{m}_{s}}}{∏ L_{𝒴^{m}}}  \overline{→^{ϕ^{∗}}} \frac{∏ 𝒪′_{m}×F_{𝒴^{m}}}{∏ P_{𝒱^{m}} ×P_{𝒴^{m}}} →^{π_{∗}} \frac{∏ 𝒪′_{m}×_{P_{𝒴^{m}}} F_{𝒴^{m}}}{∏ P_{𝒱^{m}}}  \overline{^{α_{!}} →} \frac{∏ E_{V^{m}}}{∏ P_{𝒱^{m}}}

\frac{𝒪×F_{𝒴}}{P_{𝒵} ×P_{𝒴}} ↓^{ϕ^{∗}}  \overline{→^{π^{∗}}} \frac{\underline{𝒪} P_{𝒰} ×\overline{𝒪} ×F_{𝒴}}{P_{𝒵} ×P_{𝒰} ×P_{𝒴}} ↓^{ϕ^{∗}}  \overline{→^{π_{∗}}} \frac{\underline{𝒪} P_{𝒰} ×\overline{𝒪} ×_{P_{𝒴}} F_{𝒴}}{P_{𝒵} ×P_{𝒰}} ↓^{ϕ^{∗}}  \overline{→^{1×α_{!}}} \frac{\underline{𝒪} P_{𝒰} ×F_{𝒱}}{P_{𝒵} ×P_{𝒰}} ↓^{ϕ^{∗}}

\frac{𝒪×F_{𝒴}}{P_{𝒵} ×P_{𝒴}} ↓^{π_{∗}}  \overline{^{π_{∗}} →}^{^{Id}^{=⇀ ⇁}}  \frac{𝒪×_{P_{𝒴}} F_{𝒴}}{P_{𝒵}} ↓^{π_{∗}}  \overline{→^{β_{!}}} \frac{\underline{𝒪} P_{𝒰} ×_{P_{𝒰}} F_{𝒱}}{P_{𝒵}} ↓^{π_{∗}}

assoc

\frac{E_{V}}{P_{𝒵}} ↓^{α_{!}}^{^{α_{!}}  \overline{→}}
\frac{E_{V}}{P_{𝒵}}  \overline{^{ι^{∗}}→} \frac{F_{𝒲}}{∏_{r} P_{𝒵r}}  \overline{^{κ_{!}}→} \frac{∏_{r} E_{Wr}}{∏_{r} P_{𝒵r}}

\frac{F_{𝒵}}{L_{𝒵}} ↓^{ι^{∗}}  \overline{^{κ_{!}}→}^{^{ι^{∗}}  \overline{→}}  \frac{∏_{r} F_{𝒵r}}{∏_{r} L_{𝒵r}} ↓^{ι^{∗}}

coass  \frac{∏_{n,r} E_{Z^{n}_{r}}}{L_{𝒵}} ↓^{κ_{!}}^{^{κ_{!}}  \overline{→}}

The isomorphism coher is presented in Figure 1, where the numbers

A = ∑_{m<n;r>s} ∑_{i∈I} dim V_{r}^{m}(i) · dim V_{s}^{n}(i),

B = ∑_{m>n;r>s} ∑_{h∈H} dim V_{r}^{m}(h′) · dim V_{s}^{n}(h′′)

are nothing else but dimensions of the spaces

U_{𝒲} /(U_{𝒲} ∩ P_{𝒱}) ≃ ⊕_{m<n;r>s} Hom(V_{r}^{m}, V_{s}^{n}),
f_{𝒲} ∩ f_{𝒱} = ⊕_{h∈H;m>n;r>s} Hom_{ℂ}(V_{r}^{m}(h′), V_{s}^{n}(h′′)) ⊂ F_{𝒲} ∩ F_{𝒱}.

A v) type isomorphism T^{−2B} → σ_{!}σ^{∗} comes from affine linear P_{𝒲} ∩ P_{𝒱}-bundle F_{𝒲} ∩ F_{𝒱} → (F_{𝒲} ∩ F_{𝒱})/(f_{𝒲} ∩ f_{𝒱}).

A V) type isomorphism T^{−2A} → t_{!}t^{∗} is obtained from a sequence of affine linear ∏ P_{𝒲_{r}} - bundles

U_{𝒲} /(U_{𝒲} ∩ P_{𝒱}) ▷ … ▷ U_{𝒲} /[U_{𝒲}^{(k+1)}(U_{𝒲} ∩ P_{𝒱})] ▷ U_{𝒲} /[U_{𝒲}^{(k)}(U_{𝒲} ∩ P_{𝒱})] ▷ … pt

constructed via Young tableau (B.2.2) in Section B.2.8. The group ∏ P_{𝒲_{r}} acts by conjugation. Pull-back of this sequence along ∏ 𝒪′′_{r} × F_{𝒲_{r}} | ∏ P_{𝒵_{r}} × P_{𝒲_{r}} → pt| ∏ P_{𝒲_{r}} gives a sequence of affine linear ∏ P_{𝒵_{r}} × P_{𝒲_{r}} -bundles

∏ 𝒪′′_{r} × U_{𝒲} /(U_{𝒲} ∩ P_{𝒱}) × F_{𝒲_{r}} ▷ … ▷ ∏ 𝒪′′_{r} × U_{𝒲} /[U_{𝒲}^{(k+1)}(U_{𝒲} ∩ P_{𝒱})] × F_{𝒲_{r}}

▷ ∏ 𝒪′′_{r} × U_{𝒲} /[U_{𝒲}^{(k)}(U_{𝒲} ∩ P_{𝒱})] × F_{𝒲_{r}} ▷ … ▷ ∏ 𝒪′′_{r} × F_{𝒲_{r}} .

\frac{∏ E_{V^{m}}}{∏ P_{𝒱^{m}}}  \overline{^{∏ ι^{∗}}  →} \frac{∏ F_{𝒱^{m}}}{∏ G_{V^{m}_{r}}}  \overline{^{∏ κ_{!}}  →} \frac{∏ E_{V^{m}_{r}}}{∏ G_{V^{m}_{r}}}
\frac{∏ F_{𝒱^{m}}}{∏ P_{𝒱^{m}}}  \overline{_{_{∏ κ_{!}}  →}}^{Res}  \overline{→}^{^{ι^{∗}}  \overline{→}}  \frac{∏ E_{V^{m}_{r}}}{∏ P_{𝒱^{m}}} =_{Id}  ⇀ ⇁^{Res=i^{∗}} s  \overline{→} \frac{∏ E_{V^{m}_{r}}}{∏ P_{𝒱^{m}}} \underline{←}^{p^{∗}}
\frac{\underline{𝒪} /(U_{𝒲} ∩P_{𝒱} )×∏ F_{𝒲r}}{∏ P_{𝒵r} ×P_{𝒲r}} \underline{←}^{p^{∗}}  \frac{∏ 𝒪′′_{r}×F_{𝒲r}}{∏ P_{𝒵r} ×P_{𝒲r}} ↓^{ϕ^{∗}}
\frac{\underline{𝒪} ×(F_{𝒲} ∩F_{𝒱} )/(f_{𝒲} ∩f_{𝒱} )}{(∏ P_{𝒵r} )×(P_{𝒲} ∩P_{𝒱} )} ↓^{ϕ^{∗}}  \overline{^{r_{!}} →} \frac{\underline{𝒪} ×∏ F_{𝒲r}}{(∏ P_{𝒵r} )×(P_{𝒲} ∩P_{𝒱} )}  \overline{→^{q_{∗}}} \underline{←}^{p^{∗}} \underline{←}^{q^{∗}}  \frac{\underline{𝒪} /(U_{𝒲} ∩P_{𝒱} )×∏ F_{𝒲r}}{∏ P_{𝒵r} ×P_{𝒲r}}^{\begin{matrix}F & Id \\ & ⇃⇂ \end{matrix} ‖}  \overline{_{_{t_{!}}→}} \underline{←}^{t^{∗}}  \frac{∏ 𝒪′′_{r}×F_{𝒲r}}{∏ P_{𝒵r} ×P_{𝒲r}} \begin{aligned}V & & [−2A] \\ & ↓ \end{aligned}
\frac{\underline{𝒪} P_{𝒰} ×F_{𝒱}}{P_{𝒵} ×P_{𝒰}} ↓^{ϕ^{∗}}  \overline{^{i^{∗}}→} \frac{\underline{𝒪} ×(F_{𝒲} ∩F_{𝒱} )}{(∏ P_{𝒵r} )×(P_{𝒲} ∩P_{𝒱} )} ↓^{σ^{∗}}  \frac{\underline{𝒪} ×_{P_{𝒲} ∩P_{𝒱}} (F_{𝒲} ∩F_{𝒱} )/(f_{𝒲} ∩f_{𝒱} )}{∏ P_{𝒵r}}  \overline{^{r_{!}} →}^{^{q_{∗}}  \overline{→}}  \frac{\underline{𝒪} ×_{P_{𝒲} ∩P_{𝒱}} ∏ F_{𝒲r}}{∏ P_{𝒵r}} ↓^{q_{∗}}  \overline{→^{t_{!}}}^{^{q_{∗}}  \overline{→}}  \frac{∏ 𝒪′′_{r}×_{P_{𝒲r}} F_{𝒲r}}{∏ P_{𝒵r}} ↓^{π_{∗}}

F^{−1}

\frac{\underline{𝒪} P_{𝒰} ×_{P_{𝒰}} F_{𝒱}}{P_{𝒵}} ↓^{π_{∗}}  \overline{^{ι^{∗}}→} \frac{\underline{𝒪} ×_{P_{𝒲} ∩P_{𝒱}} (F_{𝒲} ∩F_{𝒱} )}{∏ P_{𝒵r}} ↓^{q_{∗}}  \overline{→^{σ_{!}}} \underline{←}^{σ^{∗}} \frac{\underline{𝒪} ×_{P_{𝒲} ∩P_{𝒱}} (F_{𝒲} ∩F_{𝒱} )/(f_{𝒲} ∩f_{𝒱} )}{∏ P_{𝒵r}} \begin{aligned}v & & [−2B] \\ & ↓ \end{aligned}  \overline{^{r_{!}} →} \frac{\underline{𝒪} ×_{P_{𝒲} ∩P_{𝒱}} ∏ F_{𝒲r}}{∏ P_{𝒵r}} ↓^{[−2B]}  \overline{→^{t_{!}}} \frac{∏ 𝒪′′_{r}×_{P_{𝒲r}} F_{𝒲r}}{∏ P_{𝒵r}} ↓^{[−2B]}
\frac{E_{V}}{P_{𝒵}} ↓^{α_{!}}  \overline{^{ι^{∗}}  →} \frac{F_{𝒲}}{∏ P_{𝒵r}} ↓^{α_{!}}  \overline{^{κ_{!}}  →} \frac{∏ E_{Wr}}{∏ P_{𝒵r}} ↓^{α_{!}}^{^{α_{!}}  \overline{→}}

Figure 1: Coherence isomorphism coher 9 Multiplication map ∏ 𝒪′′_{r} × U_{𝒲} → \underline{𝒪} is an isomorphism of ∏ P_{𝒵_{r}} × P_{𝒲_{r}} -spaces, hence, we may replace one with another. Finally, we take quotient by ∏ P_{𝒵_{r}} and get a sequence of affine linear ∏ P_{𝒲_{r}} -bundles

(∏ P_{𝒵_{r}})\\underline{𝒪}/(U_{𝒲} ∩ P_{𝒱}) × F_{𝒲_{r}} ▷ … ▷ (∏ P_{𝒵_{r}})\\underline{𝒪}/[U_{𝒲}^{(k+1)}(U_{𝒲} ∩ P_{𝒱})] × F_{𝒲_{r}}

▷ (∏ P_{𝒵_{r}})\\underline{𝒪}/[U_{𝒲}^{(k)}(U_{𝒲} ∩ P_{𝒱})] × F_{𝒲_{r}} ▷ … ▷ ∏ P_{𝒵_{r}} \𝒪′′_{r} × F_{𝒲_{r}} .

2.2.1. Lemma. The group P_{𝒲} ∩ P_{𝒰} = P_{𝒲} ∩ P_{𝒱} is a semidirect product of its normal subgroup U_{𝒲} ∩ P_{𝒰} = U_{𝒲} ∩ P_{𝒱} and diagonally embedded subgroup ∏_{r} P_{𝒲_{r}} .

Proof. For an element g ∈ G_{V} the following conditions are equivalent:

g ∈ P_{𝒱} ⟺ ∀μ, ρ (g − 1)(V_{ρ}^{μ}) ⊂ ⊕_{m,r}{V_{r}^{m} | m ⩽ μ},
g ∈ P_{𝒰} ⟺ ∀μ, ρ (g − 1)(V_{ρ}^{μ}) ⊂ ⊕_{m,r}{V_{r}^{m} | m < μ, or (m = μ, r ⩽ ρ)},
g ∈ P_{𝒲} ⟺ ∀μ, ρ (g − 1)(V_{ρ}^{μ}) ⊂ ⊕_{m,r}{V_{r}^{m} | r ⩽ ρ},
g ∈ U_{𝒲} ⟺ ∀μ, ρ (g − 1)(V_{ρ}^{μ}) ⊂ ⊕_{m,r}{V_{r}^{m} | r < ρ},
g ∈ P_{𝒲} ∩ P_{𝒰} ⟺ ∀μ, ρ (g − 1)(V_{ρ}^{μ}) ⊂ ⊕_{m,r}{V_{r}^{m} | m ⩽ μ, r ⩽ ρ} ⟺ g ∈ P_{𝒲} ∩ P_{𝒱},
g ∈ U_{𝒲} ∩ P_{𝒰} ⟺ ∀μ, ρ (g − 1)(V_{ρ}^{μ}) ⊂ ⊕_{m,r}{V_{r}^{m} | m ⩽ μ, r < ρ} ⟺ g ∈ U_{𝒲} ∩ P_{𝒱},
g ∈ ∏_{r} P_{𝒲_{r}} ⟺ ∀μ, ρ (g − 1)(V_{ρ}^{μ}) ⊂ ⊕_{m,r}{V_{r}^{m} | m ⩽ μ, r = ρ}

This implies the lemma.

Diagram in Fig. 2 shows how three isomorphisms combine into full coherence isomorphism Coher. Its complete description including types of all cells is given in diagram on page 12.

3. Coherence equations

3.1. General description

The introduced coherence isomorphisms have to satisfy certain coherence equations. Let us describe these equations.

Assume that k, n are positive integers, (a_{1}, …, a_{k}), (b_{1}, …, b_{n}) are sequences of positive integers, and (c_{1}^{1}, …, c_{1}^{a_{1}}), …, (c_{k}^{1}, …, c_{k}^{a_{k}}), (d_{1}^{1}, …, d_{1}^{b_{1}}), …, (d_{n}^{1}, …, d_{n}^{b_{n}}) are sequences of positive integers. Set

m = ∑_{s=1}^{k} a_{s}, l = ∑_{r=1}^{n} b_{r},

c_{s} = ∑_{t=1}^{a_{s}} c_{s}^{t}, for 1 ⩽ s ⩽ k, and c = ∑_{s=1}^{k} c_{s},

d_{r} = ∑_{p=1}^{b_{r}} d_{r}^{p}, for 1 ⩽ r ⩽ n, and d = ∑_{r=1}^{n} d_{r}.

These integers are numbers of incoming or outgoing edges for vertices (operations � _{𝒲}^{𝒪;𝒱} ) in the following diagram made of composite functors as well as isomorphisms between them. Braiding

\frac{∏_{m,s} E_{Y^{m}_{s}}}{∏_{m} L_{𝒴^{m}}} →^{ϕ^{∗}} \frac{∏_{m} 𝒪′_{m}×F_{𝒴^{m}}}{∏_{m} P_{𝒱^{m}} ×P_{𝒴^{m}}} →^{π_{∗}} \frac{∏_{m} 𝒪′_{m}×_{P_{𝒴^{m}}} F_{𝒴^{m}}}{∏_{m} P_{𝒱^{m}}}  \overline{^{α_{!}}→} \frac{∏_{m} E_{V^{m}}}{∏_{m} P_{𝒱^{m}}}  \overline{\underrightarrow{∏ ι^{∗}}} \frac{∏_{m} F_{𝒱^{m}}}{∏_{m} L_{𝒱^{m}}} \overline{^{∏ κ_{!}} →} \frac{∏_{m,r} E_{V^{m}_{r}}}{∏_{m} L_{𝒱^{m}}}
\frac{∏_{r} 𝒪′′_{r}×F_{𝒲r}}{∏_{r} P_{𝒵r} ×P_{𝒲r}} ↓^{ϕ^{∗}}
\frac{𝒪×F_{𝒴}}{P_{𝒵} ×P_{𝒴}} ↓^{ϕ^{∗}} \overline{→^{π^{∗}}} \frac{\underline{𝒪} P_{𝒰} ×\overline{𝒪} ×F_{𝒴}}{P_{𝒵} ×P_{𝒰} ×P_{𝒴}} ↓^{ϕ^{∗}} \overline{^{π_{∗}}→} \frac{\underline{𝒪} P_{𝒰} ×\overline{𝒪} ×_{P_{𝒴}} F_{𝒴}}{P_{𝒵} ×P_{𝒰}} ↓^{ϕ^{∗}} \overline{→^{1×α_{!}}} \frac{\underline{𝒪} P_{𝒰} ×F_{𝒱}}{P_{𝒵} ×P_{𝒰}} ↓^{ϕ^{∗}} \frac{∏_{r} 𝒪′′_{r}×F_{𝒲r}}{∏_{r} P_{𝒵r} ×P_{𝒲r}} ↓^{[−2A]}

coher

\frac{∏_{r} 𝒪′′_{r}×_{P_{𝒲r}} F_{𝒲r}}{∏_{r} P_{𝒵r}} ↓^{π_{∗}}
\frac{𝒪×F_{𝒴}}{P_{𝒵} ×P_{𝒴}} ↓^{π_{∗}}  \overline{^{π_{∗}} →}^{^{Id}^{=⇀ ⇁}}  \frac{𝒪×_{P_{𝒴}} F_{𝒴}}{P_{𝒵}} ↓^{π_{∗}}  \overline{→^{β_{!}}} \frac{\underline{𝒪} P_{𝒰} ×_{P_{𝒰}} F_{𝒱}}{P_{𝒵}} ↓^{π_{∗}}  \frac{∏_{r} 𝒪′′_{r}×_{P_{𝒲r}} F_{𝒲r}}{∏_{r} P_{𝒵r}} ↓^{[−2B]}
\frac{E_{V}}{P_{𝒵}} ↓^{α_{!}}  \overline{→^{ι^{∗}}}^{^{α_{!}}  \overline{→}}  \frac{F_{𝒲}}{∏_{r} P_{𝒵r}}  \overline{^{κ_{!}} →} \frac{∏_{r} E_{Wr}}{∏_{r} P_{𝒵r}} ↓^{α_{!}}
\frac{F_{𝒵}}{L_{𝒵}} ↓^{ι^{∗}}  \overline{^{κ_{!}} →}^{^{ι^{∗}}  \overline{→}}  \frac{∏_{r} F_{𝒵r}}{∏_{r} L_{𝒵r}} ↓^{ι^{∗}}
\frac{∏_{n,r} E_{Z^{n}_{r}}}{L_{𝒵}} ↓^{κ_{!}}^{^{κ_{!}}  \overline{→}}

Figure 2: Scheme of the composite coherence isomorphism Coher 11

\frac{∏_{m,s} E_{Y^{m}_{s}}}{∏_{m,s} G_{Y^{m}_{s}}} →^{ϕ^{∗}} \frac{∏_{m} 𝒪′_{m}×F_{𝒴^{m}}}{∏ P_{𝒱^{m}} ×P_{𝒴^{m}}}  →^{π_{∗}} \frac{∏ 𝒪′_{m}×_{P_{𝒴^{m}}} F_{𝒴^{m}}}{∏_{m} P_{𝒱^{m}}}  \overline{^{α_{!}} →} \frac{∏ E_{V^{m}}}{∏ P_{𝒱^{m}}}  \overline{^{∏ ι^{∗}}  →} \frac{∏ F_{𝒱^{m}}}{∏ G_{V^{m}_{r}}}  \overline{^{∏ κ_{!}}  →} \frac{∏ E_{V^{m}_{r}}}{∏ G_{V^{m}_{r}}}
a^{−1}  d^{−1}
\frac{∏ F_{𝒱^{m}}}{∏ P_{𝒱^{m}}}  \overline{_{_{∏ κ_{!}}  →}}^{Res}  \overline{→}^{^{ι^{∗}}  \overline{→}}  \frac{∏ E_{V^{m}_{r}}}{∏ P_{𝒱^{m}}} =_{Id}  ⇀ ⇁^{Res=i^{∗}} s  → \frac{∏ E_{V^{m}_{r}}}{∏ P_{𝒱^{m}}} \underline{←}^{p^{∗}}

a

a  F  d  a  d  \frac{\underline{𝒪} /(U_{𝒲} ∩P_{𝒱} )×∏ F_{𝒲r}}{∏ P_{𝒵r} ×P_{𝒲r}} \underline{←}^{p^{∗}} a \frac{∏ 𝒪′′_{r}×F_{𝒲r}}{∏ P_{𝒵r} ×P_{𝒲r}} ↓^{ϕ^{∗}}

\frac{\underline{𝒪} ×(F_{𝒲} ∩F_{𝒱} )/(f_{𝒲} ∩f_{𝒱} )}{(∏ P_{𝒵r} )×(P_{𝒲} ∩P_{𝒱} )} ↓^{ϕ^{∗}}  \overline{^{r_{!}} →} \frac{\underline{𝒪} ×∏ F_{𝒲r}}{(∏ P_{𝒵r} )×(P_{𝒲} ∩P_{𝒱} )}  \overline{→^{q_{∗}}} \underline{←}^{p^{∗}} \underline{←}^{q^{∗}}  \frac{\underline{𝒪} /(U_{𝒲} ∩P_{𝒱} )×∏ F_{𝒲r}}{∏ P_{𝒵r} ×P_{𝒲r}}^{\begin{matrix}F & Id \\ & ⇃⇂ \end{matrix} ‖}  \overline{_{_{t_{!}}→}} \underline{←}^{t^{∗}}  \frac{∏ 𝒪′′_{r}×F_{𝒲r}}{∏ P_{𝒵r} ×P_{𝒲r}} \begin{aligned}V & & [−2A] \\ & ↓ \end{aligned}
e  e
\frac{𝒪×F_{𝒴}}{P_{𝒵} ×P_{𝒴}} ↓^{ϕ^{∗}}  \overline{→^{π^{∗}}} \frac{\underline{𝒪} P_{𝒰} ×\overline{𝒪} ×F_{𝒴}}{P_{𝒵} ×P_{𝒰} ×P_{𝒴}} ↓^{ϕ^{∗}}  \overline{→^{π_{∗}}} \frac{\underline{𝒪} P_{𝒰} ×\overline{𝒪} ×_{P_{𝒴}} F_{𝒴}}{P_{𝒵} ×P_{𝒰}} ↓^{ϕ^{∗}}  \overline{→^{1×α_{!}}} \frac{\underline{𝒪} P_{𝒰} ×F_{𝒱}}{P_{𝒵} ×P_{𝒰}} ↓^{ϕ^{∗}}  \overline{^{i^{∗}}→} \frac{\underline{𝒪} ×(F_{𝒲} ∩F_{𝒱} )}{(∏ P_{𝒵r} )×(P_{𝒲} ∩P_{𝒱} )} ↓^{σ^{∗}}  f \frac{\underline{𝒪} ×_{P_{𝒲} ∩P_{𝒱}} (F_{𝒲} ∩F_{𝒱} )/(f_{𝒲} ∩f_{𝒱} )}{∏ P_{𝒵r}}  \overline{^{r_{!}} →}^{^{q_{∗}}  \overline{→}}  \frac{\underline{𝒪} ×_{P_{𝒲} ∩P_{𝒱}} ∏ F_{𝒲r}}{∏ P_{𝒵r}} \begin{aligned}b & & q_{∗} \\ & ↓ \end{aligned}  \overline{→^{t_{!}}}^{^{q_{∗}}  \overline{→}}  \frac{∏ 𝒪′′_{r}×_{P_{𝒲r}} F_{𝒲r}}{∏ P_{𝒵r}} ↓^{π_{∗}}
b  e  F^{−1}  =  =
\frac{𝒪×F_{𝒴}}{P_{𝒵} ×P_{𝒴}} ↓^{F^{−1} π_{∗}}  \overline{^{π_{∗}} →}^{^{Id}^{=⇀ ⇁}}  \frac{𝒪×_{P_{𝒴}} F_{𝒴}}{P_{𝒵}} ↓^{π_{∗}}  \overline{→^{β_{!}}} \frac{\underline{𝒪} P_{𝒰} ×_{P_{𝒰}} F_{𝒱}}{P_{𝒵}} ↓^{π_{∗}}  \overline{^{ι^{∗}}→} \frac{\underline{𝒪} ×_{P_{𝒲} ∩P_{𝒱}} (F_{𝒲} ∩F_{𝒱} )}{∏ P_{𝒵r}} ↓^{q_{∗}}  \overline{→^{σ_{!}}} \underline{←}^{σ^{∗}} \frac{\underline{𝒪} ×_{P_{𝒲} ∩P_{𝒱}} (F_{𝒲} ∩F_{𝒱} )/(f_{𝒲} ∩f_{𝒱} )}{∏ P_{𝒵r}} \begin{aligned}v & & [−2B] \\ & ↓ \end{aligned}  \overline{^{r_{!}} →} \frac{\underline{𝒪} ×_{P_{𝒲} ∩P_{𝒱}} ∏ F_{𝒲r}}{∏ P_{𝒵r}} ↓^{[−2B]}  \overline{→^{t_{!}}} \frac{∏ 𝒪′′_{r}×_{P_{𝒲r}} F_{𝒲r}}{∏ P_{𝒵r}} ↓^{[−2B]}
d^{−1}  c
\frac{E_{V}}{P_{𝒵}} \begin{aligned}c & & α_{!} \\ & ↓ \end{aligned} \overline{^{ι^{∗}}  →}^{^{α_{!}}  \overline{→}}  \frac{F_{𝒲}}{∏_{r} P_{𝒵r}} ↓^{α_{!}  c^{−1}}  \overline{^{κ_{!}}  →} \frac{∏_{r} E_{Wr}}{∏_{r} P_{𝒵r}} ↓^{α_{!}}^{^{α_{!}}  \overline{→}}

d

\frac{F_{𝒵}}{L_{𝒵}} ↓^{ι^{∗}}  \overline{^{κ_{!}}  →}^{^{ι^{∗}} a  \overline{→}}  \frac{∏_{r} F_{𝒵r}}{∏_{r} L_{𝒵r}} ↓^{ι^{∗}}

c

Full  isomorphism  Coher  \frac{∏_{n,r} E_{Z^{n}_{r}}}{L_{𝒵}} ↓^{κ_{!}}^{^{κ_{!}}  \overline{→}}

12 functors correspond to braids which are liftings of permutations of (2.2.2) type. The functors and isomorphisms in this diagram are already constructed.

[figure: c_{1}^{1}; n; ^{1}𝒪^{1}; …; ^{a_{1}}𝒪^{1}; c_{1}^{a_{1}}; n; c_{k}^{1}; n; ^{1}𝒪^{k}; …; ^{a_{k}} 𝒪^{k}; c_{k}^{a_{k}}; n; ^{σ_{a_{1},n}  …  σ_{a_{k},n}}; a_{1}; b_{1}; 𝒪_{1}^{1}; …; 𝒪_{n}^{1}; a_{1}; b_{n}; a_{k}; b_{1}; 𝒪_{1}^{k}; …; 𝒪_{n}^{k}; a_{k}; b_{n}; σ_{k,l}; k; d_{1}^{1}; _{1}𝒪_{1}; …; _{b_{1}}𝒪_{1}; k; d_{1}^{b_{1}}; …; k; d_{n}^{1}; _{1}𝒪_{n}; …; _{bn}𝒪_{n}; k; d_{n}^{bn}; c_{1}^{1}; n; ^{1}𝒪^{1}; …; ^{a_{1}}𝒪^{1}; c_{1}^{a_{1}}; n; …; c_{k}^{1}; n; ^{1}𝒪^{k}; …; ^{a_{k}} 𝒪^{k}; c_{k}^{a_{k}}; n; σ_{m,n}; a_{1}; b_{1}; 𝒪_{1}^{1}; …; 𝒪_{1}^{k}; a_{k}; b_{1}; a_{1}; b_{n}; 𝒪_{n}^{1}; …; 𝒪_{n}^{k}; a_{k}; b_{n}; ^{σ_{k,b_{1}}  …  σ_{k,b_{n}}}; k; d_{1}^{1}; _{1}𝒪_{1}; …; _{b_{1}}𝒪_{1}; k; d_{1}^{b_{1}}; k; d_{n}^{1}; _{1}𝒪_{n}; …; _{bn}𝒪_{n}; k; d_{n}^{bn}; ↓^{≀}]

⟶^{Coher}

⟶^{Coher}

[figure: c_{1}; l; 𝒪^{1}; …; 𝒪^{k}; c_{k}; l]

σ_{k,l}

[figure: k; d_{1}^{1}; _{1}𝒪_{1}; …; _{bn}𝒪_{n}; k; d_{n}^{bn}]

𝒪 ^{}^{c}_{d} ↓^{Coher}

[figure: c_{1}^{1}; n; ^{1}𝒪^{1}; …; ^{a_{k}} 𝒪^{k}; c_{k}^{a_{k}}; n; σ_{m,n}; m; d_{1}; 𝒪_{1}; …; 𝒪_{n}; m; d_{n}]^{↑^{Coher}}

The Hopf category property requires that the two composite isomorphisms were equal, so that the diagram is commutative. Here the left vertical isomorphism comes from the permutation isomorphism of spaces and groups, which corresponds to an isotopy of the tangle diagram. The isotopy consists of pulling down the middle row of vertices along the strings of σ_{k,l} closer to the lower row.

The same equation can be rewritten less formally as follows.

[figure: _{1}𝒪_{1}  _{a_{1}}𝒪_{1}  _{1}𝒪_{k}  _{a_{k}} 𝒪_{k}; 𝒪_{1}^{1}; 𝒪_{n}^{1}; 𝒪_{1}^{k}; 𝒪_{n}^{k}; _{1}𝒪_{1}; _{b_{1}}𝒪_{1}; _{1}𝒪_{n}; _{bn}𝒪_{n}; \overline{→^{Coher}}; 𝒪^{1}; 𝒪^{k}; _{_{1}𝒪_{1}  _{b_{1}}𝒪_{1}  _{1}𝒪_{n}  _{bn}𝒪_{n}}]

[figure: _{^{1}𝒪^{1}  ^{a_{1}}𝒪^{1}  ^{1}𝒪^{k}  ^{a_{k}} 𝒪^{k}}; 𝒪_{1}  𝒪_{n}; ↓^{Coher}; \overline{→^{Coher}}; 𝒪; ↓^{Coher}]

The left top corner functor is obviously isomorphic to both functors on the left in the previous diagram. So we replace the mentioned two functors with the one above.

To prove this equation we use presentation of Coher as composition of three isomorphisms coher, assoc and coass. We divide the above square into several polygons, and try to prove commutativity of each of them separately. To divide the square we introduce new isomorphisms, which are similar to already used coher, assoc and coass, however, do not formally follows from them. Squares marked with = commute for trivial reasons. Main Theorem. The following relations, marked with !=! in diagram on the next page, hold true: coassociativity–coassociativity, associativity–associativity and coherence–coassociativity.

Concrete expressions for all functors and morphisms in this diagram are provided below. Proof of the theorem will appear elsewhere. Main Missing Theorem. The coherence–associativity relation, marked with ?=? in diagram on the following page also holds?

The general case of isomorphisms between direct sums reduces to the case of subspaces of a common fixed space. So let a vector space V be given, together with its subspaces _{p}^{t}X^{s}, ^{t}V_{r}^{s}, jY_{r}^{s}, _{j}^{i}Z_{r}, such that all sums

∑_{p,t,s} _{p}^{t}X^{s} = ∑_{t,s,r}^{t}V_{r}^{s} = ∑_{s,r,j}_{j}Y_{r}^{s} = ∑_{r,j,i} _{j}^{i}Z_{r} = V

are direct sums. Introduce also subspaces

V_{r}^{s} = ∑_{t}^{t}V_{r}^{s},  ^{t}V_{r} = ∑_{s}^{t}V_{r}^{s},  ^{t}V^{s} = ∑_{r}^{t}V_{r}^{s},
tV = ∑_{s,r}^{t}V_{r}^{s},  V^{s} = ∑_{r,t}^{t}V_{r}^{s},  V_{r} = ∑_{t,s}^{t}V_{r}^{s}  (3.1.1)

[figure: \overline{^{coher}→}; \overline{^{coass}→}; \overline{^{assoc}→}]

=  ! =!  =

[figure: ↓^{coher}; \overline{^{coher}→}; ↓^{coher}; ↓^{coher}; \overline{=^{assoc}→}; ↓^{coher}]

? =?

[figure: ↓^{assoc}; \overline{^{coher}→}; ← =^{assoc}; ^{^{coass}↑}; assoc coass →; ^{coher} →]

=  ! =!

[figure]

↓^{coass}

coher→

[figure]

↓^{coass}

assoc coass  →
\frac{←}{assoc coass}

assoc coass

↓

and similarly for other letters X, Y, Z. We require the following subspaces to coincide pairwise:

tX^{s} = ^{t}V^{s},  V_{r}^{s} = Y_{r}^{s},  _{j}Y_{r} = _{j}Z_{r}.

This implies, of course, the equations

tX = ^{t}V, X^{s} = V^{s} = Y^{s}, V_{r} = Y_{r} = Z_{r}, _{j}Y = _{j}Z.

Let us refer to the functors in diagram as Φ_{ij}, 1 ⩽ i, j ⩽ 4; i is the row and j is the column number. An example of Φ_{11} is given below.

Φ_{11} = [figure: _{1}^{1}X^{1} _{2}^{1}X^{1} _{1}^{2}X^{1} _{2}^{2}X^{1} _{1}^{1}X^{2} _{2}^{1}X^{2} _{1}^{2}X^{2} _{2}^{2}X^{2}; _{_{1}𝒪_{1}  ^{1}V_{−}^{1}}  _{_{2}𝒪_{1}  ^{2}V_{−}^{1}}  _{^{1}V_{−}^{2} _{1}𝒪_{2}}  _{^{2}V_{−}^{2} _{2}𝒪_{2}}; ^{1}V_{2}^{1}; ^{1}V_{1}^{1}; ^{2}V_{1}^{1}; ^{2}V_{2}^{1}; ^{1}V_{2}^{2}; ^{1}V_{1}^{2}; ^{2}V_{1}^{2}; ^{2}V_{2}^{2}; 𝒪_{1}^{1}  _{−}Y_{1}^{1}; 𝒪_{2}^{1}; _{−}Y_{2}^{1}; _{−}Y_{1}^{2}; 𝒪_{1}^{2}; \begin{aligned}& 𝒪_{2}^{2} \\ _{−}Y_{2}^{2} \end{aligned}; \begin{aligned}& _{2}Y_{1}^{1} \\ _{1}Y_{1}^{1} \end{aligned}  _{1}Y_{1}^{2}  _{2}Y_{1}^{2}  \begin{aligned}& _{2}Y_{2}^{1} \\ _{1}Y_{2}^{1} \end{aligned}  _{1}Y_{2}^{2}  _{2}Y_{2}^{2}; _{1}𝒪_{1}; _{2}𝒪_{1}; _{1}𝒪_{2}; _{2}𝒪_{2}; ^{_{1}^{−}Z_{1}  _{2}^{−}Z_{1}  _{1}^{−}Z_{2}  _{2}^{−}Z_{2}}; _{1}^{1}Z_{1} _{1}^{2}Z_{1} _{2}^{1}Z_{1} _{2}^{2}Z_{1} _{1}^{1}Z_{2} _{1}^{2}Z_{2} _{2}^{1}Z_{2} _{2}^{2}Z_{2}]

We use the lexicographical order of subsequences of (_{p}^{t}X^{s}), (^{t}V_{r}^{s}), (_{j}Y_{r}^{s}), (_{j}^{i}Z_{r}) or (3.1.1) or similar for other letters. The subsequence is fixed by fixing 2, 1 or 0 indices. It is denoted by the corresponding letter X, V, Y or Z with these fixed indices, where the remaining indices are replaced with symbols −, = or ≡. These symbols indicate places at which indices are compared to establish the lexicographical order. We compare two triples of indices first at place indicated by ≡ (if present). If the values coincide or are not specified, we compare at place indicated by = (if present). If these coincide or are not specified, we compare values at −. For instance,

_{−}^{t}X^{s} = (_{1}^{t}X^{s}, _{2}^{t}X^{s}, …),
−Y_{=}^{s} = (_{1}Y_{1}^{s},_{2}Y_{1}^{s}, …,_{1}Y_{2}^{s},_{2}Y_{2}^{s}, …),
Y_{r}^{−} = (Y_{r}^{1}, Y_{r}^{2}, …),
_{=}^{−}Z_{≡} = (_{1}^{1}Z_{1}, _{1}^{2}Z_{1}, …, _{2}^{1}Z_{1}, _{2}^{2}Z_{1}, …, _{1}^{1}Z_{2}, _{1}^{2}Z_{2}, …, _{2}^{1}Z_{2}, _{2}^{2}Z_{2}, …)
−Z_{=} = (_{1}Z_{1},_{2}Z_{1}, …,_{1}Z_{2},_{2}Z_{2}, …),
Y^{−} = (Y^{1}, Y^{2}, …).

From so ordered sequence we produce an increasing filtration of the space V = X = Y = Z. The filtration is denoted by the same symbols as the sequence. The m-th term of the filtration is the sum of first m elements of the sequence.

3.2. Detailed description

The locally closed subsets parametrising operations are the following:

G_{^{t}V^{s}} ⊃ ^{t}𝒪^{s} ∈ P_{^{t}V_{−}^{s}} -sets- P_{_{−}^{t}X^{s}},
G_{Y_{r}^{s}} ⊃ 𝒪_{r}^{s} ∈ P_{_{−}Y_{r}^{s}} -sets- P_{^{−}V_{r}^{s}},
G_{_{j}Y_{r}} ⊃ _{j}𝒪_{r} ∈ P_{_{j}^{−}Z_{r}} -sets- P_{_{j}Y_{r}^{−}}.

All parametrising subsets obtained from these will be also locally closed as shown in Section 2.2. We have to deal with the following functors.

Φ_{11} = \frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  ^{ϕ^{∗}}→ \frac{∏^{t}𝒪^{s}×F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}×P_{^{t}_{−}X^{s}}}  ^{π_{∗}}→ ^{\frac{∏^{t}𝒪^{s}×_{P_{^{t}_{−}X^{s}}} F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}}}  ^{α_{!}}→ \frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}  ^{ι^{∗}}→ \frac{∏ F_{^{t}V^{s}_{−}}}{∏ L_{^{t}V^{s}_{−}}}  ^{κ_{!}}→ \frac{∏ E_{^{t}V^{s}_{r}}}{∏ G_{^{t}V^{s}_{r}}}
ϕ^{∗}→ \frac{∏ 𝒪_{r}^{s}×F_{^{−}V^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}×P_{^{−}V^{s}_{r}}}  ^{π_{∗}}→ \frac{∏ 𝒪_{r}^{s}×_{P_{^{−}V^{s}_{r}}} F_{^{−}V^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}  ^{α_{!}}→ \frac{∏ E_{Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}  ^{ι^{∗}}→ \frac{∏ F_{_{−}Y^{s}_{r}}}{∏ L_{_{−}Y^{s}_{r}}}  ^{κ_{!}}→ \frac{∏ E_{_{j} Y^{s}_{r}}}{∏ G_{_{j} Y^{s}_{r}}}
ϕ^{∗}→ \frac{∏_{j}𝒪_{r}×F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr} ×P_{_{j} Y^{−}_{r}}}  ^{π_{∗}}→ \frac{∏_{j}𝒪_{r}×_{P_{_{j} Y^{−}_{r}}} F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr}}  ^{α_{!}}→ \frac{∏ E_{_{j} Zr}}{∏ P_{^{−}_{j}Zr}}  ^{ι^{∗}}→ \frac{∏ F_{^{−}_{j}Zr}}{∏ L_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} ^{shift} → \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} .

Φ_{12} = \frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  ^{ϕ^{∗}}→ \frac{∏^{t}𝒪^{s}×F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}×P_{^{t}_{−}X^{s}}}  ^{π_{∗}}→ ^{\frac{∏^{t}𝒪^{s}×_{P_{^{t}_{−}X^{s}}} F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}}}  ^{α_{!}}→ \frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}
ϕ^{∗}→ \frac{∏ 𝒪_{−}^{s}×F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}V^{s}_{−}}}  ^{π_{∗}}→ \frac{∏ 𝒪_{−}^{s}×_{P_{^{=}V^{s}_{−}}} F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}}  ^{α_{!}}→ \frac{∏ E_{Y^{s}}}{∏ P_{_{−}Y^{s}_{=}}}
ι^{∗}→ \frac{∏ F_{V^{s}_{−}}}{∏ P_{_{−}Y^{s}_{r}}}  ^{κ_{!}}→ \frac{∏ E_{Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}  ^{ι^{∗}}→ \frac{∏ F_{_{−}Y^{s}_{r}}}{∏ L_{_{−}Y^{s}_{r}}}  ^{κ_{!}}→ \frac{∏ E_{_{j} Y^{s}_{r}}}{∏ G_{_{j} Y^{s}_{r}}}
ϕ^{∗}→ \frac{∏_{j}𝒪_{r}×F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr} ×P_{_{j} Y^{−}_{r}}}  ^{π_{∗}}→ \frac{∏_{j}𝒪_{r}×_{P_{_{j} Y^{−}_{r}}} F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr}}  ^{α_{!}}→ \frac{∏ E_{_{j} Zr}}{∏ P_{^{−}_{j}Zr}}  ^{ι^{∗}}→ \frac{∏ F_{^{−}_{j}Zr}}{∏ L_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} ^{shift} → \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}}

Here

P_{V_{−}^{s}} ⊃ \underline{𝒪}_{−}^{s} =^{def} U_{V_{−}^{s}} · ∏_{r} 𝒪_{r}^{s} = ∏_{r} 𝒪_{r}^{s} · U_{V_{−}^{s}} ∈ P_{_{−}Y_{=}^{s}} -sets- P_{^{−}V_{=}^{s}},
G_{V^{s}} ⊃ 𝒪_{−}^{s} =^{def} \underline{𝒪}_{−}^{s} · P_{^{=}V_{−}^{s}} ∈ P_{_{−}Y_{=}^{s}} -sets- P_{^{=}V_{−}^{s}},  𝒪_{−}^{s} = \underline{𝒪}_{−}^{s} ×_{P_{V^{s}_{−}}∩P_{^{−}V^{s}}} P_{^{=}V_{−}^{s}}.

Φ_{13} = \frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  ^{ϕ^{∗}}→ \frac{∏^{t}𝒪^{s}×F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}×P_{^{t}_{−}X^{s}}}  ^{π_{∗}}→ ^{\frac{∏^{t}𝒪^{s}×_{P_{^{t}_{−}X^{s}}} F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}}}  ^{α_{!}}→ \frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}
ϕ^{∗}→ \frac{∏ 𝒪_{−}^{s}×F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}V^{s}_{−}}}  ^{π_{∗}}→ \frac{∏ 𝒪_{−}^{s}×_{P_{^{=}V^{s}_{−}}} F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}}  ^{α_{!}}→ \frac{∏ E_{Y^{s}}}{∏ P_{_{−}Y^{s}_{=}}}  ^{ι^{∗}}→ \frac{∏ F_{_{−}Y^{s}_{=}}}{∏ L_{_{−}Y^{s}_{r}}}  ^{κ_{!}}→ \frac{∏ E_{_{j} Y^{s}_{r}}}{∏ G_{_{j} Y^{s}_{r}}}
ϕ^{∗}→ \frac{∏_{j}𝒪_{r}×F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr} ×P_{_{j} Y^{−}_{r}}}  ^{π_{∗}}→ \frac{∏_{j}𝒪_{r}×_{P_{_{j} Y^{−}_{r}}} F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr}}  ^{α_{!}}→ \frac{∏ E_{_{j} Zr}}{∏ P_{^{−}_{j}Zr}}  ^{ι^{∗}}→ \frac{∏ F_{^{−}_{j}Zr}}{∏ L_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} ^{shift} → \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}}

The isomorphism coass : Φ_{12} → Φ_{13} is essentially given by

\frac{∏ E_{Y^{s}}}{∏ P_{_{−}Y^{s}_{=}}}  \overline{^{ι^{∗}}→} \frac{∏ F_{Y^{s}_{−}}}{∏ P_{_{−}Y^{s}_{r}}}  \overline{^{κ_{!}}→} \frac{∏ E_{Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}

\frac{∏ F_{_{−}Y^{s}_{=}}}{∏ L_{_{−}Y^{s}_{r}}} ↓^{ι^{∗}}  \overline{^{κ_{!}}→}^{^{ι^{∗}}  \overline{→}}  \frac{∏ F_{_{−}Y^{s}_{r}}}{∏ L_{_{−}Y^{s}_{r}}} ↓^{ι^{∗}}

\frac{∏ E_{_{j} Y^{s}_{r}}}{∏ G_{_{j} Y^{s}_{r}}} ↓^{κ_{!}}^{^{κ_{!}}  \overline{→}}

Φ_{14} = \frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  ^{ϕ^{∗}}→ \frac{∏ 𝒪^{s}×F_{^{=}_{−}X^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}_{−}X^{s}}}  ^{π_{∗}}→ ^{\frac{∏ 𝒪^{s}×_{P_{^{=}_{−}X^{s}}} F_{^{=}_{−}X^{s}}}{∏ P_{_{−}Y^{s}_{=}}}}  ^{α_{!}}→ \frac{∏ E_{Y^{s}}}{∏ P_{_{−}Y^{s}_{=}}}  ^{ι^{∗}}→ \frac{∏ F_{_{−}Y^{s}_{=}}}{∏ L_{_{−}Y^{s}_{r}}}  ^{κ_{!}}→ \frac{∏ E_{_{j} Y^{s}_{r}}}{∏ G_{_{j} Y^{s}_{r}}}
ϕ^{∗}→ \frac{∏_{j}𝒪_{r}×F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr} ×P_{_{j} Y^{−}_{r}}}  ^{π_{∗}}→ \frac{∏_{j}𝒪_{r}×_{P_{_{j} Y^{−}_{r}}} F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr}}  ^{α_{!}}→ \frac{∏ E_{_{j} Zr}}{∏ P_{^{−}_{j}Zr}}  ^{ι^{∗}}→ \frac{∏ F_{^{−}_{j}Zr}}{∏ L_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} ^{shift} → \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}}

Here

P_{^{−}V^{s}} ⊃ ^{−}𝒪^{s} =^{def} U_{^{−}V^{s}} · ∏_{t}^{t}𝒪^{s} = ∏_{t}^{t}𝒪^{s} · U_{^{−}V^{s}} ∈ P_{^{=}V_{−}^{s}} -sets- P_{_{−}^{=}X^{s}},
G_{Y^{s}} ⊃ 𝒪^{s} =^{def} \underline{𝒪}_{−}^{s} ·^{−}𝒪^{s} = 𝒪_{−}^{s} ·^{−}𝒪^{s} ∈ P_{_{−}Y_{=}^{s}} -sets- P_{_{−}^{=}X^{s}},
𝒪^{s} = 𝒪_{−}^{s} ×_{P_{^{=}V^{s}_{−}}} ^{−}𝒪^{s} = \underline{𝒪}_{−}^{s} ×_{P_{V^{s}_{−}}∩P_{^{−}V^{s}}} ^{−}𝒪^{s}.

The associativity isomorphism assoc : Φ_{13} → Φ_{14} (as well as assoc : Φ_{23} → Φ_{24} and the associativity part of the isomorphism assoc : Φ_{33} → Φ_{24}) is essentially given by

\frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  \overline{^{ϕ^{∗}} →} \frac{∏^{t}𝒪^{s}×F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}×P_{^{t}_{−}X^{s}}}  \overline{^{π_{∗}} →} ^{\frac{∏^{t}𝒪^{s}×_{P_{^{t}_{−}X^{s}}} F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}}}  \overline{^{α_{!}} →} \frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}

\frac{∏ 𝒪^{s}×F_{^{=}_{−}X^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}_{−}X^{s}}} ↓^{ϕ^{∗}}  →^{π^{∗}} \frac{∏ 𝒪_{−}^{s}×^{−}𝒪^{s}×F_{^{=}_{−}X^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}V^{s}_{−}}×P_{^{=}_{−}X^{s}}} ↓^{ϕ^{∗}}  →^{π_{∗}} ^{\frac{∏ 𝒪_{−}^{s}×^{−}𝒪^{s}×_{P_{^{=}_{−}X^{s}}} F_{^{=}_{−}X^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}V^{s}_{−}}}} ↓^{ϕ^{∗}}  \overline{→^{1×α_{!}}} \frac{∏ 𝒪_{−}^{s}×F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}V^{s}_{−}}} ↓^{ϕ^{∗}}

\frac{∏ 𝒪^{s}×F_{^{=}_{−}X^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}_{−}X^{s}}} ↓^{π_{∗}}  \overline{^{π_{∗}} →}^{^{Id}^{=⇀ ⇁}}  ^{\frac{∏ 𝒪^{s}×_{P_{^{=}_{−}X^{s}}} F_{^{=}_{−}X^{s}}}{∏ P_{_{−}Y^{s}_{=}}}} ↓^{π_{∗}}  \overline{→^{β_{!}}} ^{\frac{∏ 𝒪_{−}^{s}×_{P_{^{=}V^{s}_{−}}} F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}}} ↓^{π_{∗}}

\frac{∏ E_{Y^{s}}}{∏ P_{_{−}Y^{s}_{=}}} ↓^{α_{!}}^{^{α_{!}}  \overline{→}}

Φ_{21} = \frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  ^{ϕ^{∗}}→ \frac{∏^{t}𝒪^{s}×F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}×P_{^{t}_{−}X^{s}}}  ^{π_{∗}}→ ^{\frac{∏^{t}𝒪^{s}×_{P_{^{t}_{−}X^{s}}} F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}}}  ^{α_{!}}→ \frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}  ^{ι^{∗}}→ \frac{∏ F_{^{t}V^{s}_{−}}}{∏ L_{^{t}V^{s}_{−}}}  ^{κ_{!}}→ \frac{∏ E_{^{t}V^{s}_{r}}}{∏ G_{^{t}V^{s}_{r}}}
ϕ^{∗}→ \frac{∏ 𝒪_{r}^{s}×F_{^{−}V^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}×P_{^{−}V^{s}_{r}}}  ^{π_{∗}}→ \frac{∏ 𝒪_{r}^{s}×_{P_{^{−}V^{s}_{r}}} F_{^{−}V^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}  ^{α_{!}}→ \frac{∏ E_{Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}
ϕ^{∗}→ \frac{∏_{−}𝒪_{r}×F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr} ×P_{_{−}Y^{=}_{r}}}  ^{π_{∗}}→ \frac{∏_{−}𝒪_{r}×_{P_{_{−}Y^{=}_{r}}} F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr}}  ^{α_{!}}→ \frac{∏ E_{Zr}}{∏ P_{^{−} =Zr}}
ι^{∗}→ \frac{∏ F_{_{−}Zr}}{∏ P_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{_{j} Zr}}{∏ P_{^{−}_{j}Zr}}  ^{ι^{∗}}→ \frac{∏ F_{^{−}_{j}Zr}}{∏ L_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} ^{shift} → \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} .

Here

P_{_{−}Y_{r}} ⊃ _{−}\underline{𝒪}_{r} =^{def} U_{_{−}Y_{r}} · ∏_{j}_{j}𝒪_{r} = ∏_{j}_{j}𝒪_{r} · U_{_{−}Y_{r}} ∈ P_{_{=}^{−}Z_{r}} -sets- P_{_{=}Y_{r}^{−}},
G_{Y_{r}} ⊃ _{−}𝒪_{r} =^{def} _{−}\underline{𝒪}_{r} · P_{_{−}Y_{r}^{=}} ∈ P_{_{=}^{−}Z_{r}} -sets- P_{_{−}Y_{r}^{=}},  _{−}𝒪_{r} = _{−}\underline{𝒪}_{r} ×_{P_{_{−}Yr} ∩P_{Y^{−}_{r}}} P_{_{−}Y_{r}^{=}}.

Φ_{22} = \frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  ^{ϕ^{∗}}→ \frac{∏^{t}𝒪^{s}×F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}×P_{^{t}_{−}X^{s}}}  ^{π_{∗}}→ ^{\frac{∏^{t}𝒪^{s}×_{P_{^{t}_{−}X^{s}}} F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}}}  ^{α_{!}}→ \frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}
ϕ^{∗}→ \frac{∏ 𝒪_{−}^{s}×F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}V^{s}_{−}}}  ^{π_{∗}}→ \frac{∏ 𝒪_{−}^{s}×_{P_{^{=}V^{s}_{−}}} F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}}  ^{α_{!}}→ \frac{∏ E_{Y^{s}}}{∏ P_{_{−}Y^{s}_{=}}}
ι^{∗}→ \frac{∏ F_{V^{s}_{−}}}{∏ P_{_{−}Y^{s}_{r}}}  ^{κ_{!}}→ \frac{∏ E_{Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}  ^{ϕ^{∗}}→ \frac{∏_{−}𝒪_{r}×F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr} ×P_{_{−}Y^{=}_{r}}}  ^{π_{∗}}→ \frac{∏_{−}𝒪_{r}×_{P_{_{−}Y^{=}_{r}}} F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr}}  ^{α_{!}}→ \frac{∏ E_{Zr}}{∏ P_{^{−} =Zr}}
ι^{∗}→ \frac{∏ F_{_{−}Zr}}{∏ P_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{_{j} Zr}}{∏ P_{^{−}_{j}Zr}}  ^{ι^{∗}}→ \frac{∏ F_{^{−}_{j}Zr}}{∏ L_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} ^{shift} → \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} .

Φ_{23} = \frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  ^{ϕ^{∗}}→ \frac{∏^{t}𝒪^{s}×F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}×P_{^{t}_{−}X^{s}}}  ^{π_{∗}}→ ^{\frac{∏^{t}𝒪^{s}×_{P_{^{t}_{−}X^{s}}} F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}}}  ^{α_{!}}→ \frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}
ϕ^{∗}→ \frac{∏ 𝒪_{−}^{s}×F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}V^{s}_{−}}}  ^{π_{∗}}→ \frac{∏ 𝒪_{−}^{s}×_{P_{^{=}V^{s}_{−}}} F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}}  ^{α_{!}}→ \frac{∏ E_{Y^{s}}}{∏ P_{_{−}Y^{s}_{=}}}
ϕ^{∗}→ _{\frac{_{−}𝒪_{=}×F_{Y^{−}}}{P_{^{−} =Z≡}×P_{_{−}Y^{≡}_{=}}}}  ^{π_{∗}}→ _{\frac{_{−}𝒪_{=}×_{P_{_{−}Y^{≡}_{=}}} F_{Y^{−}}}{P_{^{−} =Z≡}}}  ^{α_{!}}→ _{\frac{E_{Z}}{P_{^{−} =Z≡}}}
ι^{∗}→ _{\frac{F_{_{−}Z=}}{∏ P_{^{−}_{j}Zr}}}  ^{κ_{!}}→ \frac{∏ E_{_{j} Zr}}{∏ P_{^{−}_{j}Zr}}  ^{ι^{∗}}→ \frac{∏ F_{^{−}_{j}Zr}}{∏ L_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} ^{shift} → \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}}

Here

P_{_{−}Y_{=}} ⊃ _{−}\underline{𝒪}_{=} =^{def} U_{_{−}Y_{=}} · ∏_{j,r}_{j}𝒪_{r} = ∏_{j,r}_{j}𝒪_{r} · U_{_{−}Y_{=}} ∈ P_{_{=}^{−}Z_{≡}} -sets- P_{_{=}Y_{≡}^{−}},
G_{Y} ⊃ _{−}𝒪_{=} =^{def} _{−}\underline{𝒪}_{=} · P_{_{−}Y_{=}^{≡}} ∈ P_{_{=}^{−}Z_{≡}} -sets- P_{_{−}Y_{=}^{≡}},  _{−}𝒪_{=} = _{−}\underline{𝒪}_{=} ×_{P_{_{−}Y=}∩P_{Y^{−}}} P_{_{−}Y_{=}^{≡}}.

Φ_{24} = \frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  ^{ϕ^{∗}}→ \frac{∏ 𝒪^{s}×F_{^{=}_{−}X^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}_{−}X^{s}}}  ^{π_{∗}}→ ^{\frac{∏ 𝒪^{s}×_{P_{^{=}_{−}X^{s}}} F_{^{=}_{−}X^{s}}}{∏ P_{_{−}Y^{s}_{=}}}}  ^{α_{!}}→ \frac{∏ E_{Y^{s}}}{∏ P_{_{−}Y^{s}_{=}}}
ϕ^{∗}→ _{\frac{_{−}𝒪_{=}×F_{Y^{−}}}{P_{^{−} =Z≡}×P_{_{−}Y^{≡}_{=}}}}  ^{π_{∗}}→ _{\frac{_{−}𝒪_{=}×_{P_{_{−}Y^{≡}_{=}}} F_{Y^{−}}}{P_{^{−} =Z≡}}}  ^{α_{!}}→ _{\frac{E_{Z}}{P_{^{−} =Z≡}}}
ι^{∗}→ _{\frac{F_{_{−}Z=}}{∏ P_{^{−}_{j}Zr}}}  ^{κ_{!}}→ \frac{∏ E_{_{j} Zr}}{∏ P_{^{−}_{j}Zr}}  ^{ι^{∗}}→ \frac{∏ F_{^{−}_{j}Zr}}{∏ L_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} ^{shift} → \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}}

Φ_{31} = \frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  ^{ϕ^{∗}}→ \frac{∏^{t}𝒪^{s}×F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}×P_{^{t}_{−}X^{s}}}  ^{π_{∗}}→ ^{\frac{∏^{t}𝒪^{s}×_{P_{^{t}_{−}X^{s}}} F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}}}  ^{α_{!}}→ \frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}
ι^{∗}→ \frac{∏ F_{^{t}V^{s}_{−}}}{∏ L_{^{t}V^{s}_{−}}}  ^{κ_{!}}→ \frac{∏ E_{^{t}V^{s}_{r}}}{∏ G_{^{t}V^{s}_{r}}}  ^{ϕ^{∗}}→ \frac{∏ 𝒪_{r}×F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr} ×P_{^{−}V^{=}_{r}}}  ^{π_{∗}}→ \frac{∏ 𝒪_{r}×_{P_{^{−}V^{=}_{r}}} F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr}}  ^{α_{!}}→ \frac{∏ E_{Zr}}{∏ P_{^{−} =Zr}}
ι^{∗}→ \frac{∏ F_{_{−}Zr}}{∏ P_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{_{j} Zr}}{∏ P_{^{−}_{j}Zr}}  ^{ι^{∗}}→ \frac{∏ F_{^{−}_{j}Zr}}{∏ L_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} ^{shift} → \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} .

Here

P_{V_{r}^{−}} ⊃ 𝒪_{r}^{−} =^{def} U_{V_{r}^{−}} · ∏_{s} 𝒪_{r}^{s} = ∏_{s} 𝒪_{r}^{s} · U_{V_{r}^{−}} ∈ P_{_{−}Y_{r}^{=}} -sets- P_{^{−}V_{r}^{=}},
G_{Y_{r}} ⊃ 𝒪_{r} =^{def} _{−}\underline{𝒪}_{r} · 𝒪_{r}^{−} = _{−}𝒪_{r} · 𝒪_{r}^{−} ∈ P_{_{=}^{−}Z_{r}} -sets- P_{^{−}V_{r}^{=}},
𝒪_{r} = _{−}𝒪_{r} ×_{P_{_{−}Y^{=}_{r}}} 𝒪_{r}^{−} = _{−}\underline{𝒪}_{r} ×_{P_{_{−}Yr} ∩P_{Y^{−}_{r}}} 𝒪_{r}^{−}.

The associativity isomorphism assoc : Φ_{21} → Φ_{31} is essentially the following.

\frac{∏ E_{^{t}V^{s}_{r}}}{∏ G_{^{t}V^{s}_{r}}}  \overline{^{ϕ^{∗}} →} \frac{∏ 𝒪_{r}^{s}×F_{^{−}V^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}×P_{^{−}V^{s}_{r}}}  \overline{^{π_{∗}} →} ^{\frac{∏ 𝒪_{r}^{s}×_{P_{^{−}V^{s}_{r}}} F_{^{−}V^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}}  \overline{^{α_{!}} →} \frac{∏ E_{Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}

\frac{∏ 𝒪_{r}×F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr} ×P_{^{−}V^{=}_{r}}} ↓^{ϕ^{∗}}  →^{π^{∗}} \frac{∏_{−}𝒪_{r}×𝒪_{r}^{−}×F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr} ×P_{_{−}Y^{=}_{r}}×P_{^{−}V^{=}_{r}}} ↓^{ϕ^{∗}}  →^{π_{∗}} \frac{∏_{−}𝒪_{r}×𝒪_{r}^{−}×_{P_{^{−}V^{=}_{r}}} F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr} ×P_{_{−}Y^{=}_{r}}} ↓^{ϕ^{∗}}  \overline{→^{1×α_{!}}} \frac{∏_{−}𝒪_{r}×F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr} ×P_{_{−}Y^{=}_{r}}} ↓^{ϕ^{∗}}

\frac{∏ 𝒪_{r}×F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr} ×P_{^{−}V^{=}_{r}}} ↓^{π_{∗}}  \overline{^{π_{∗}} →}^{^{Id}^{=⇀ ⇁}}  \frac{∏ 𝒪_{r}×_{P_{^{−}V^{=}_{r}}} F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr}} ↓^{π_{∗}}  \overline{→^{β_{!}}} \frac{∏_{−}𝒪_{r}×_{P_{_{−}Y^{=}_{r}}} F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr}} ↓^{π_{∗}}

\frac{∏ E_{Zr}}{∏ P_{^{−} =Zr}} ↓^{α_{!}}^{^{α_{!}}  \overline{→}}

Φ_{32} = \frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  ^{ϕ^{∗}}→ \frac{∏^{t}𝒪^{s}×F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}×P_{^{t}_{−}X^{s}}}  ^{π_{∗}}→ ^{\frac{∏^{t}𝒪^{s}×_{P_{^{t}_{−}X^{s}}} F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}}}  ^{α_{!}}→ \frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}
ϕ^{∗}→ _{\frac{𝒪_{−}×F_{^{−}V^{=}}}{P_{^{−} =Z≡}×P_{^{=}V^{≡}_{−}}}}  ^{π_{∗}}→ \frac{𝒪_{−}×_{P_{^{=}V^{≡}_{−}}} F_{^{−}V^{=}}}{P_{^{−} =Z≡}}  ^{α_{!}}→ _{\frac{E_{V}}{P_{^{−} =Z≡}}}  ^{ι^{∗}}→ _{\frac{F_{Z_{−}}}{∏ P_{^{−} =Zr}}}  ^{κ_{!}}→ \frac{∏ E_{Zr}}{∏ P_{^{−} =Zr}}
ι^{∗}→ \frac{∏ F_{_{−}Zr}}{∏ P_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{_{j} Zr}}{∏ P_{^{−}_{j}Zr}}  ^{ι^{∗}}→ \frac{∏ F_{^{−}_{j}Zr}}{∏ L_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} ^{shift} → \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} .

Here

P_{V_{−}} ⊃ \underline{𝒪}_{−} =^{def} U_{V_{−}} · ∏_{r} 𝒪_{r} = ∏_{r} 𝒪_{r} · U_{V_{−}} ∈ P_{_{=}^{−}Z_{≡}} -sets- P_{^{−}V_{≡}^{=}},
G_{V} ⊃ 𝒪_{−} =^{def} \underline{𝒪}_{−} · P_{^{=}V_{−}^{≡}} ∈ P_{_{=}^{−}Z_{≡}} -sets- P_{^{=}V_{−}^{≡}},  𝒪_{−} = \underline{𝒪}_{−} ×_{P_{V_{−}}∩P_{^{−}V^{=}}} P_{^{=}V_{−}^{≡}}.

Φ_{33} = \frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  ^{ϕ^{∗}}→ \frac{∏^{t}𝒪^{s}×F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}×P_{^{t}_{−}X^{s}}}  ^{π_{∗}}→ ^{\frac{∏^{t}𝒪^{s}×_{P_{^{t}_{−}X^{s}}} F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}}}  ^{α_{!}}→ \frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}
ϕ^{∗}→ \frac{∏ 𝒪_{−}^{s}×F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}V^{s}_{−}}}  ^{π_{∗}}→ \frac{∏ 𝒪_{−}^{s}×_{P_{^{=}V^{s}_{−}}} F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}}  ^{α_{!}}→ \frac{∏ E_{Y^{s}}}{∏ P_{_{−}Y^{s}_{=}}}
ϕ^{∗}→ _{\frac{_{−}𝒪_{=}×F_{Y^{−}}}{P_{^{−} =Z≡}×P_{_{−}Y^{≡}_{=}}}}  ^{π_{∗}}→ _{\frac{_{−}𝒪_{=}×_{P_{_{−}Y^{≡}_{=}}} F_{Y^{−}}}{P_{^{−} =Z≡}}}  ^{α_{!}}→ _{\frac{E_{V}}{P_{^{−} =Z≡}}}  ^{ι^{∗}}→ _{\frac{F_{Z_{−}}}{∏ P_{^{−} =Zr}}}  ^{κ_{!}}→ \frac{∏ E_{Zr}}{∏ P_{^{−} =Zr}}
ι^{∗}→ \frac{∏ F_{_{−}Zr}}{∏ P_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{_{j} Zr}}{∏ P_{^{−}_{j}Zr}}  ^{ι^{∗}}→ \frac{∏ F_{^{−}_{j}Zr}}{∏ L_{^{−}_{j}Zr}}  ^{κ_{!}}→ \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} ^{shift} → \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} .

Denote

P_{V^{−}} ⊃ 𝒪_{−}^{=} =^{def} U_{V^{−}} · ∏_{s} 𝒪_{−}^{s} = ∏_{s} 𝒪_{−}^{s} · U_{V^{−}} ∈ P_{_{−}Y_{=}^{≡}} -sets- P_{^{=}V_{−}^{≡}}.

3.2.1. Lemma. The following equations hold

𝒪_{−} = _{−}\underline{𝒪}_{=} · 𝒪_{−}^{=} = _{−}𝒪_{=} · 𝒪_{−}^{=} ⊂ G_{V} , 𝒪_{−} = _{−}𝒪_{=} ×_{P_{_{−}Y^{≡}_{=}}} 𝒪_{−}^{=}.

Proof. Clearly, _{−}\underline{𝒪}_{=} · 𝒪_{−}^{=} = _{−}𝒪_{=} · 𝒪_{−}^{=}. This is the image of the multiplication map

mult : _{−}\underline{𝒪}_{=} · P_{_{−}Y_{=}^{≡}} ×_{P_{_{−}Y^{≡}_{=}}} 𝒪_{−}^{=} → G_{Y} .  (3.2.1)

Let us prove that this map is injective.

Suppose that two elements are mapped to the same element p_{1}p_{2}p_{3} = p̄_{1}p̄_{2}p̄_{3} in G_{Y}, where p_{1}, p̄_{1} ∈ _{−}𝒪_{=} ⊂ P_{_{−}Y_{=}}, p_{2}, p̄_{2} ∈ P_{_{−}Y_{=}^{≡}} ⊂ P_{Y_{−}^{=}}, p_{3}, p̄_{3} ∈ 𝒪_{−}^{=} ⊂ P_{Y_{−}^{=}}. Hence, p_{2}p_{3}, p̄_{2}p̄_{3} ∈ P_{Y_{−}^{=}}. Then the element

r := p̄_{1}^{−1}p_{1} = p̄_{2}p̄_{3}(p_{2}p_{3})^{−1}

belongs simultaneously to P_{_{−}Y_{=}} and to P_{Y_{−}^{=}}.

Notice that

P_{_{−}Y_{=}} ∩ P_{Y_{−}^{=}} ⊂ P_{_{−}Y_{=}^{≡}}.

Indeed, for any p ∈ P_{_{−}Y_{=}} we have for all possible ι, σ, ρ

p(_{ι}Y_{ρ}^{σ}) ⊂ ⊕_{j,s,r}{_{j}Y_{r}^{s} | r < ρ or (r = ρ, j ⩽ ι)},

and for any p ∈ P_{Y_{−}^{=}} we have

p(_{ι}Y_{ρ}^{σ}) ⊂ ⊕_{j,s,r}{_{j}Y_{r}^{s} | s < σ or (s = σ, r ⩽ ρ)}.

Thus, for any p ∈ P_{_{−}Y_{=}} ∩ P_{Y_{−}^{=}} we have

p(_{ι}Y_{ρ}^{σ}) ⊂ ⊕_{j,s,r}{_{j}Y_{r}^{s} | s < σ or (s = σ, r < ρ) or (s = σ, r = ρ, j ⩽ ι)},

which means that p ∈ P_{_{−}Y_{=}^{≡}}.

Since p̄_{2}p̄_{3} = rp_{2}p_{3}, p̄_{1} = p_{1}r^{−1} and r ∈ P_{_{−}Y_{=}^{≡}}, we get in _{−}\underline{𝒪}_{=} · P_{_{−}Y_{=}^{≡}} ×_{P_{_{−}Y^{≡}_{=}}} 𝒪_{−}^{=}

(p̄_{1}p̄_{2}, p̄_{3}) = (p̄_{1}, p̄_{2}p̄_{3}) = (p_{1}r^{−1}, rp_{2}p_{3}) = (p_{1}, p_{2}p_{3}) = (p_{1}p_{2}, p_{3}).

Therefore, map (3.2.1) is injective.

Now we wish to prove that the image of (3.2.1) coincides with 𝒪_{−}, that is,

−\underline{𝒪}_{=} · 𝒪_{−}^{=} = \underline{𝒪}_{−} · P_{^{=}V_{−}^{≡}}.

Recall the definitions of these subsets:

−\underline{𝒪}_{=} = ∏_{j,r}_{j}𝒪_{r} · U_{_{−}Y_{=}}, 𝒪_{−}^{=} = ∏_{s} 𝒪_{−}^{s} · U_{V^{−}} = ∏_{s} U_{V_{−}^{s}} · ∏_{s,r} 𝒪_{r}^{s} · ∏_{s} P_{^{=}V_{−}^{s}} · U_{V^{−}},

\underline{𝒪}_{−} = ∏_{r} 𝒪_{r} · U_{V_{−}} = ∏_{r}_{−}\underline{𝒪}_{r} · 𝒪_{r}^{−} · U_{V_{−}} = ∏_{r}_{−}\underline{𝒪}_{r} · U_{V_{−}} · ∏_{r} 𝒪_{r}^{−}

= ∏_{j,r}_{j}𝒪_{r} · ∏_{r} U_{_{−}Y_{r}} · U_{V_{−}} · ∏_{s,r} 𝒪_{r}^{s} · ∏_{r} U_{V_{r}^{−}}.

We have to check that

U_{_{−}Y_{=}} · ∏_{s} U_{V_{−}^{s}} · ∏_{s,r} 𝒪_{r}^{s} · ∏_{s} P_{^{=}V_{−}^{s}} · U_{V^{−}} = ∏_{r} U_{_{−}Y_{r}} · U_{V_{−}} · ∏_{s,r} 𝒪_{r}^{s} · ∏_{r} U_{V_{r}^{−}} · P_{^{=}V_{−}^{≡}}.

This equation is implied by the following facts

∏_{s} U_{Y_{−}^{s}} ⊂ U_{_{−}Y_{=}},  U_{_{−}Y_{=}} = ∏_{r} U_{_{−}Y_{r}} · U_{Y_{−}},  ∏_{r} U_{V_{r}^{−}} ⊂ U_{^{=}V_{−}^{≡}},  ∏_{s} P_{^{=}V_{−}^{s}} · U_{V^{−}} = P_{^{=}V_{−}^{≡}},

which are easy to verify.

Now we may describe the associativity isomorphism assoc : Φ_{33} → Φ_{32} as well as the associativity part of the isomorphism assoc : Φ_{33} → Φ_{42} as the following.

\frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}  \overline{^{ϕ^{∗}} →} \frac{∏ 𝒪_{−}^{s}×F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}V^{s}_{−}}}  \overline{^{π_{∗}} →} ^{\frac{∏ 𝒪_{−}^{s}×_{P_{^{=}V^{s}_{−}}} F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}}}  \overline{^{α_{!}} →} \frac{∏ E_{Y^{s}}}{∏ P_{_{−}Y^{s}_{=}}}

\frac{𝒪_{−}×F_{^{−}V^{=}}}{P_{^{−} =Z≡}×P_{^{=}V^{≡}_{−}}} ↓^{ϕ^{∗}}  →^{π^{∗}} \frac{_{−}𝒪_{=}×𝒪_{−}^{=}×F_{^{−}V^{=}}}{P_{^{−} =Z≡}×P_{_{−}Y^{≡}_{=}}×P_{^{=}V^{≡}_{−}}} ↓^{ϕ^{∗}}  →^{π_{∗}} \frac{_{−}𝒪_{=}×𝒪_{−}^{=}×_{P_{^{=}V^{≡}_{−}}} F_{^{−}V^{=}}}{P_{^{−} =Z≡}×P_{_{−}Y^{≡}_{=}}} ↓^{ϕ^{∗}}  \overline{→^{1×α_{!}}} \frac{_{−}𝒪_{=}×F_{Y^{−}}}{P_{^{−} =Z≡}×P_{_{−}Y^{≡}_{=}}} ↓^{ϕ^{∗}}

\frac{𝒪_{−}×F_{^{−}V^{=}}}{P_{^{−} =Z≡}×P_{^{=}V^{≡}_{−}}} ↓^{π_{∗}}  \overline{^{π_{∗}} →}^{^{Id}^{=⇀ ⇁}}  \frac{𝒪_{−}×_{P_{^{=}V^{≡}_{−}}} F_{^{−}V^{=}}}{P_{^{−} =Z≡}} ↓^{π_{∗}}  \overline{→^{β_{!}}} \frac{_{−}𝒪_{=}×_{P_{_{−}Y^{≡}_{=}}} F_{Y^{−}}}{P_{^{−} =Z≡}} ↓^{π_{∗}}

\frac{E_{V}}{P_{^{−} =Z≡}} ↓^{α_{!}}^{^{α_{!}}  \overline{→}}

The isomorphism coass : Φ_{33} → Φ_{23} is essentially given by

\frac{E_{V}}{P_{^{−} =Z≡}}  \overline{^{ι^{∗}}→} \frac{F_{Z_{−}}}{∏ P_{^{−} =Zr}}  \overline{^{κ_{!}}→} \frac{∏ E_{Zr}}{∏ P_{^{−} =Zr}}

\frac{F_{_{−}Z=}}{∏ P_{^{−}_{j}Zr}} ↓^{ι^{∗}}  \overline{^{κ_{!}}→}^{^{ι^{∗}}  \overline{→}}  \frac{∏ F_{_{−}Zr}}{∏ P_{^{−}_{j}Zr}} ↓^{ι^{∗}}

\frac{∏ E_{_{j} Zr}}{∏ P_{^{−}_{j}Zr}} ↓^{κ_{!}}^{^{κ_{!}}  \overline{→}}

Φ_{41} = \frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  ^{ϕ^{∗}}→ \frac{∏^{t}𝒪^{s}×F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}×P_{^{t}_{−}X^{s}}}  ^{π_{∗}}→ ^{\frac{∏^{t}𝒪^{s}×_{P_{^{t}_{−}X^{s}}} F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}}}  ^{α_{!}}→ \frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}
ι^{∗}→ \frac{∏ F_{^{t}V^{s}_{−}}}{∏ L_{^{t}V^{s}_{−}}}  ^{κ_{!}}→ \frac{∏ E_{^{t}V^{s}_{r}}}{∏ G_{^{t}V^{s}_{r}}}  ^{ϕ^{∗}}→ \frac{∏ 𝒪_{r}×F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr} ×P_{^{−}V^{=}_{r}}}  ^{π_{∗}}→ \frac{∏ 𝒪_{r}×_{P_{^{−}V^{=}_{r}}} F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr}}  ^{α_{!}}→ \frac{∏ E_{Zr}}{∏ P_{^{−} =Zr}}
ι^{∗}→ \frac{∏ F_{^{−} =Zr}}{∏ L_{^{−} =Zr}} ^{κ_{!}}→ \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} ^{shift} → \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} .

Φ_{42} = \frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  ^{ϕ^{∗}}→ \frac{∏^{t}𝒪^{s}×F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}×P_{^{t}_{−}X^{s}}}  ^{π_{∗}}→ ^{\frac{∏^{t}𝒪^{s}×_{P_{^{t}_{−}X^{s}}} F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}}}  ^{α_{!}}→ \frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}
ϕ^{∗}→ _{\frac{𝒪_{−}×F_{^{−}V^{=}}}{P_{^{−} =Z≡}×P_{^{=}V^{≡}_{−}}}}  ^{π_{∗}}→ \frac{𝒪_{−}×_{P_{^{=}V^{≡}_{−}}} F_{^{−}V^{=}}}{P_{^{−} =Z≡}}  ^{α_{!}}→ _{\frac{E_{V}}{P_{^{−} =Z≡}}}
ι^{∗}→ \frac{F_{Z_{−}}}{∏ P_{^{−} =Zr}}  ^{κ_{!}}→ \frac{∏ E_{Zr}}{∏ P_{^{−} =Zr}}  ^{ι^{∗}}→ \frac{∏ F_{^{−} =Zr}}{∏ L_{^{−} =Zr}}  ^{κ_{!}}→ \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} ^{shift} → \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} .

Φ_{44} = \frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  ^{ϕ^{∗}}→ \frac{𝒪×F_{^{=}_{−}X^{≡}}}{P_{^{−} =Z≡}×P_{^{=}_{−}X^{≡}}}  ^{π_{∗}}→ \frac{𝒪×_{P_{^{=}_{−}X^{≡}}} F_{^{=}_{−}X^{≡}}}{P_{^{−} =Z≡}}  ^{α_{!}}→ _{\frac{E_{V}}{P_{^{−} =Z≡}}}
ι^{∗}→ \frac{F_{^{−} =Z≡}}{L_{^{−} =Z≡}} ^{κ_{!}}→ \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} ^{shift} → \frac{∏ E_{^{i}_{j} Zr}}{∏ G_{^{i}_{j} Zr}} .

We introduce

P_{Y^{−}} ⊃ 𝒪^{−} =^{def} U_{Y^{−}} · ∏_{s} 𝒪^{s} = ∏_{s} 𝒪^{s} · U_{Y^{−}} ∈ P_{_{−}Y_{=}^{≡}} -sets- P_{_{−}^{=}X^{≡}},
G_{V} ⊃ 𝒪 =^{def} _{−}\underline{𝒪}_{=} · 𝒪^{−} = _{−}𝒪_{=} · 𝒪^{−} ∈ P_{_{=}^{−}Z_{≡}} -sets- P_{_{−}^{=}X^{≡}},
𝒪 = _{−}𝒪_{=} ×_{P_{_{−}Y^{≡}_{=}}} 𝒪^{−} = _{−}\underline{𝒪}_{=} ×_{P_{_{−}Y=}∩P_{Y^{−}}} 𝒪^{−}.

Denote also

P_{^{−}V^{=}} ⊃ ^{−}𝒪^{=} =^{def} U_{V^{−}} · ∏_{s}^{−}𝒪^{s} = ∏_{s}^{−}𝒪^{s} · U_{V^{−}}

= U_{^{−}V^{=}} · ∏_{t,s}^{t}𝒪^{s} = ∏_{t,s}^{t}𝒪^{s} · U_{^{−}V^{=}} ∈ P_{^{=}V_{−}^{≡}} -sets- P_{_{−}^{=}X^{≡}}.

3.2.2. Lemma. The following equations hold

𝒪^{−} = 𝒪_{−}^{=} ·^{−}𝒪^{=} = 𝒪_{−}^{=} ×_{P_{^{=}V^{≡}_{−}}} ^{−}𝒪^{=}.

Proof. We verify first the equation 𝒪^{−} = 𝒪_{−}^{=} · ^{−}𝒪^{=}. The definitions of these subsets are as follows:

𝒪^{−} = ∏_{s} 𝒪^{s} · U_{Y^{−}} = ∏_{s} 𝒪_{−}^{s} ·^{−}𝒪^{s} · U_{Y^{−}} = ∏_{s} 𝒪_{−}^{s} · U_{Y^{−}} · ∏_{s}^{−}𝒪^{s}

= ∏_{s} 𝒪_{−}^{s} · U_{Y^{−}} · ∏_{s} U_{^{−}V^{s}} · ∏_{t,s}^{t}𝒪^{s},

𝒪_{−}^{=} = ∏_{s} 𝒪_{−}^{s} · U_{V^{−}}, ^{−}𝒪^{=} = U_{^{−}V^{=}} · ∏_{t,s}^{t}𝒪^{s}.

The required equation is implied by the following facts

U_{V^{−}} ⊂ U_{^{−}V^{=}} = U_{V^{−}} · ∏_{s} U_{^{−}V^{s}}.

Now we prove the second equation. Recall that P_{V_{−}^{=}} ⊃ 𝒪_{−}^{=} ∈ P_{_{−}Y_{=}^{≡}} -sets- P_{^{=}V_{−}^{≡}} and P_{^{−}V^{=}} ⊃ −𝒪^{=} ∈ P_{^{=}V_{−}^{≡}} -sets- P_{_{−}^{=}X^{≡}}. The multiplication map mult : 𝒪_{−}^{=} ×_{P_{^{=}V^{≡}_{−}}} ^{−}𝒪^{=} → 𝒪_{−}^{=} · ^{−}𝒪^{=} is obviously surjective. We have to prove its injectivity. Suppose that two elements (g_{1}, h_{1}) and (g_{2}, h_{2}) are mapped to the same element g_{1}h_{1} = g_{2}h_{2} in G_{V}, where g_{1}, g_{2} ∈ 𝒪_{−}^{=}, h_{1}, h_{2} ∈ ^{−}𝒪^{=}. Then the element r := g_{2}^{−1}g_{1} = h_{2}h_{1}^{−1} belongs simultaneously to P_{V_{−}^{=}} and to P_{^{−}V^{=}}. Notice that

P_{V_{−}^{=}} ∩ P_{^{−}V^{=}} ⊂ P_{^{=}V_{−}^{≡}}.

Indeed, for any p ∈ P_{V_{−}^{=}} ∩ P_{^{−}V^{=}} we have for all possible τ, σ, ρ

p(^{τ}V_{ρ}^{σ}) ⊂ ⊕_{t,s,r}{^{t}V_{r}^{s} | s < σ or (s = σ, t ⩽ τ, r ⩽ ρ)}.

This implies also that

p(^{τ}V_{ρ}^{σ}) ⊂ ⊕_{t,s,r}{^{t}V_{r}^{s} | s < σ or (s = σ, t < τ) or (s = σ, t = τ, r ⩽ ρ)},

which means that p ∈ P_{^{=}V_{−}^{≡}}. Therefore, in 𝒪_{−}^{=} ×_{P_{^{=}V^{≡}_{−}}} ^{−}𝒪^{=} we have

(g_{1}, h_{1}) = (g_{2}r, r^{−1}h_{2}) = (g_{2}, h_{2}),

which is the desired injectivity.

3.2.3. Lemma. The following equations hold

𝒪 = \underline{𝒪}_{−} ·^{−}𝒪^{=} = 𝒪_{−} ·^{−}𝒪^{=} ⊂ G_{V} , 𝒪 = 𝒪_{−} ×_{P_{^{=}V^{≡}_{−}}} ^{−}𝒪^{=}.

Proof. The definition and Lemmas 3.2.2 and 3.2.1 imply the equations

𝒪 = _{−}𝒪_{=} · 𝒪^{−} = _{−}𝒪_{=} · 𝒪_{−}^{=} ·^{−}𝒪^{=} = 𝒪_{−} ·^{−}𝒪^{=}.

The equation \underline{𝒪}_{−} ·^{−}𝒪^{=} = 𝒪_{−} ·^{−}𝒪^{=} is obvious. The equation 𝒪_{−} ×_{P_{^{=}V^{≡}_{−}}} ^{−}𝒪^{=} = 𝒪_{−} ·^{−}𝒪^{=} follows from the existence of the associativity part of the isomorphism assoc : Φ_{42} → Φ_{44} presented below.

Now we may describe the associativity part of the isomorphism assoc : Φ_{42} → Φ_{44} as follows.

\frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  \overline{^{ϕ^{∗}} →} \frac{∏^{t}𝒪^{s}×F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}×P_{^{t}_{−}X^{s}}}  \overline{^{π_{∗}} →} ^{\frac{∏^{t}𝒪^{s}×_{P_{^{t}_{−}X^{s}}} F_{^{t}_{−}X^{s}}}{∏ P_{^{t}V^{s}_{−}}}}  \overline{^{α_{!}} →} \frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}

\frac{𝒪×F_{^{=}_{−}X^{≡}}}{P_{^{−} =Z≡}×P_{^{=}_{−}X^{≡}}} ↓^{ϕ^{∗}}  →^{π^{∗}} \frac{𝒪_{−}×^{−}𝒪^{=}×F_{^{=}_{−}X^{≡}}}{P_{^{−} =Z≡}×P_{^{=}V^{≡}_{−}}×P_{^{=}_{−}X^{≡}}} ↓^{ϕ^{∗}}  →^{π_{∗}} \frac{𝒪_{−}×^{−}𝒪^{=}×_{P_{^{=}_{−}X^{≡}}} F_{^{=}_{−}X^{≡}}}{P_{^{−} =Z≡}×P_{^{=}V^{≡}_{−}}} ↓^{ϕ^{∗}}  \overline{→^{1×α_{!}}} \frac{𝒪_{−}×F_{^{−}V^{=}}}{P_{^{−} =Z≡}×P_{^{=}V^{≡}_{−}}} ↓^{ϕ^{∗}}

\frac{𝒪×F_{^{=}_{−}X^{≡}}}{P_{^{−} =Z≡}×P_{^{=}_{−}X^{≡}}} ↓^{π_{∗}}  \overline{^{π_{∗}} →}^{^{Id}^{=⇀ ⇁}}  \frac{𝒪×_{P_{^{=}_{−}X^{≡}}} F_{^{=}_{−}X^{≡}}}{P_{^{−} =Z≡}} ↓^{π_{∗}}  \overline{→^{β_{!}}} \frac{𝒪_{−}×_{P_{^{=}V^{≡}_{−}}} F_{^{−}V^{=}}}{P_{^{−} =Z≡}} ↓^{π_{∗}}

\frac{E_{Z}}{P_{^{−} =Z≡}} ↓^{α_{!}}^{^{α_{!}}  \overline{→}}

The associativity part of the isomorphism assoc : Φ_{24} → Φ_{44} is essentially the following.

\frac{∏ E_{^{t}_{p}X^{s}}}{∏ G_{^{t}_{p}X^{s}}}  \overline{^{ϕ^{∗}} →} \frac{∏ 𝒪^{s}×F_{^{=}_{−}X^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}_{−}X^{s}}}  \overline{^{π_{∗}} →} ^{\frac{∏ 𝒪^{s}×_{P_{^{=}_{−}X^{s}}} F_{^{=}_{−}X^{s}}}{∏ P_{_{−}Y^{s}_{=}}}}  \overline{^{α_{!}} →} \frac{∏ E_{Y^{s}}}{∏ P_{_{−}Y^{s}_{=}}}

\frac{𝒪×F_{^{=}_{−}X^{≡}}}{P_{^{−} =Z≡}×P_{^{=}_{−}X^{≡}}} ↓^{ϕ^{∗}}  →^{π^{∗}} \frac{_{−}𝒪_{=}×𝒪^{−}×F_{^{=}_{−}X^{≡}}}{P_{^{−} =Z≡}×P_{_{−}Y^{≡}_{=}}×P_{^{=}_{−}X^{≡}}} ↓^{ϕ^{∗}}  →^{π_{∗}} \frac{_{−}𝒪_{=}×𝒪^{−}×_{P_{^{=}_{−}X^{≡}}} F_{^{=}_{−}X^{≡}}}{P_{^{−} =Z≡}×P_{_{−}Y^{≡}_{=}}} ↓^{ϕ^{∗}}  →^{1×α_{!}} \frac{_{−}𝒪_{=}×F_{Y^{−}}}{P_{^{−} =Z≡}×P_{_{−}Y^{≡}_{=}}} ↓^{ϕ^{∗}}

\frac{𝒪×F_{^{=}_{−}X^{≡}}}{P_{^{−} =Z≡}×P_{^{=}_{−}X^{≡}}} ↓^{π_{∗}}  \overline{^{π_{∗}} →}^{^{Id}^{=⇀ ⇁}}  \frac{𝒪×_{P_{^{=}_{−}X^{≡}}} F_{^{=}_{−}X^{≡}}}{P_{^{−} =Z≡}} ↓^{π_{∗}}  \overline{→^{β_{!}}} \frac{_{−}𝒪_{=}×_{P_{_{−}Y^{≡}_{=}}} F_{Y^{−}}}{P_{^{−} =Z≡}} ↓^{π_{∗}}

\frac{E_{Z}}{P_{^{−} =Z≡}} ↓^{α_{!}}^{^{α_{!}}  \overline{→}}

For the morphism coher : Φ_{12} → Φ_{22} we use notation

A_{12} = dim ∏_{r} U_{_{−}Y_{r}}/(U_{_{−}Y_{r}} ∩ P_{Y_{r}^{−}}), B_{12} = dim ∏_{r} f_{_{−}Y_{r}} ∩ f_{Y_{r}^{−}}.

Notice that P_{_{−}Y_{r}} ∩P_{Y_{r}^{−}} = P_{_{−}Y_{r}} ∩P_{_{−}Y_{r}^{=}} ⊂ P_{_{−}Y_{r}^{=}} and U_{_{−}Y_{r}} ∩P_{Y_{r}^{−}} = U_{_{−}Y_{r}} ∩P_{_{−}Y_{r}^{=}} by Lemma 2.2.1.

\frac{∏ E_{Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}  \overline{^{∏ ι^{∗}}  →} \frac{∏ F_{_{−}Y^{s}_{r}}}{∏ G_{_{j} Y^{s}_{r}}}  \overline{^{∏ κ_{!}}  →} \frac{∏ E_{_{j} Y^{s}_{r}}}{∏ G_{_{j} Y^{s}_{r}}}

\frac{∏ F_{_{−}Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}  \overline{_{_{∏ κ_{!}}  →}}^{Res}  \overline{→}^{^{ι^{∗}}  \overline{→}}  \frac{∏ E_{_{j} Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}} =_{Id}  ⇀ ⇁^{Res=i^{∗}} s  \overline{→} \frac{∏ E_{_{j} Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}} \underline{←}^{p^{∗}}

\frac{∏_{−}\underline{𝒪} _{r}/(U_{_{−}Yr} ∩P_{Y^{−}_{r}})×∏ F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr} ×P_{_{j} Y^{−}_{r}}} \underline{←}^{p^{∗}}  \frac{∏_{j}𝒪_{r}×F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr} ×P_{_{j} Y^{−}_{r}}} ↓^{ϕ^{∗}}

\frac{∏_{−}\underline{𝒪} _{r}×(F_{_{−}Yr} ∩F_{Y^{−}_{r}})/(f_{_{−}Yr} ∩f_{Y^{−}_{r}})}{(∏ P_{^{−}_{j}Zr})×(P_{_{−}Yr} ∩P_{Y^{−}_{r}})} ↓^{ϕ^{∗}}  \overline{^{r_{!}}  →} \frac{∏_{−}\underline{𝒪} _{r}×∏ F_{_{j} Y^{−}_{r}}}{(∏ P_{^{−}_{j}Zr})×(P_{_{−}Yr} ∩P_{Y^{−}_{r}})}  \overline{→^{q_{∗}}} \underline{←}^{p^{∗}} \underline{←}^{q^{∗}}  \frac{∏_{−}\underline{𝒪} _{r}/(U_{_{−}Yr} ∩P_{Y^{−}_{r}})×∏ F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr} ×P_{_{j} Y^{−}_{r}}}^{\begin{matrix}F & Id \\ & ⇃⇂ \end{matrix} ‖}  \overline{_{_{t_{!}} →}} \underline{←}^{t^{∗}}  \frac{∏_{j}𝒪_{r}×F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr} ×P_{_{j} Y^{−}_{r}}} \begin{aligned}V & & [−2A_{12}] \\ & ↓ \end{aligned}

\frac{∏_{−}\underline{𝒪} _{r}P_{_{−}Y^{=}_{r}}×F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr} ×P_{_{−}Y^{=}_{r}}} ↓^{ϕ^{∗}}  \overline{^{i^{∗}} →} \frac{∏_{−}\underline{𝒪} _{r}×(F_{_{−}Yr} ∩F_{Y^{−}_{r}})}{(∏ P_{^{−}_{j}Zr})×(P_{_{−}Yr} ∩P_{Y^{−}_{r}})} ↓^{σ^{∗}}  \frac{∏_{−}\underline{𝒪} _{r}×_{P_{_{−}Yr} ∩P_{Y^{−}_{r}}} (F_{_{−}Yr} ∩F_{Y^{−}_{r}})/(f_{_{−}Yr} ∩f_{Y^{−}_{r}})}{∏ P_{^{−}_{j}Zr}}  \overline{^{r_{!}} →}^{^{q_{∗}}  \overline{→}}  \frac{∏_{−}\underline{𝒪} _{r}×_{P_{_{−}Yr} ∩P_{Y^{−}_{r}}} ∏ F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr}} ↓^{q_{∗}}  \overline{→^{t_{!}}}^{^{q_{∗}}  \overline{→}}  \frac{∏_{j}𝒪_{r}×_{P_{_{j} Y^{−}_{r}}} F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr}} ↓^{π_{∗}}

F^{−1}

\frac{∏_{−}\underline{𝒪} _{r}P_{_{−}Y^{=}_{r}}×_{P_{_{−}Y^{=}_{r}}} F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr}} ↓^{π_{∗}}  \overline{^{ι^{∗}}→} \frac{∏_{−}\underline{𝒪} _{r}×_{P_{_{−}Yr} ∩P_{Y^{−}_{r}}} (F_{_{−}Yr} ∩F_{Y^{−}_{r}})}{∏ P_{^{−}_{j}Zr}} ↓^{q_{∗}}  \overline{→^{σ_{!}}} \underline{←}^{σ^{∗}} \frac{∏_{−}\underline{𝒪} _{r}×_{P_{_{−}Yr} ∩P_{Y^{−}_{r}}} (F_{_{−}Yr} ∩F_{Y^{−}_{r}})/(f_{_{−}Yr} ∩f_{Y^{−}_{r}})}{∏ P_{^{−}_{j}Zr}} \begin{aligned}v & & [−2B_{12}] \\ & ↓ \end{aligned}  \overline{^{r_{!}} →} \frac{∏_{−}\underline{𝒪} _{r}×_{P_{_{−}Yr} ∩P_{Y^{−}_{r}}} ∏ F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr}} ↓^{[−2B_{12}]}  \overline{→^{t_{!}}} \frac{∏_{j}𝒪_{r}×_{P_{_{j} Y^{−}_{r}}} F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr}} ↓^{[−2B_{12}]}

\frac{∏ E_{Zr}}{∏ P_{^{−} =Zr}} ↓^{α_{!}}  \overline{^{ι^{∗}}  →} \frac{∏ F_{_{−}Yr}}{∏ P_{^{−}_{j}Zr}} ↓^{α_{!}}  \overline{^{κ_{!}}  →} \frac{∏ E_{_{j} Zr}}{∏ P_{^{−}_{j}Zr}} ↓^{α_{!}}^{^{α_{!}}  \overline{→}}

25

\frac{∏ E_{Y^{s}}}{∏ P_{_{−}Y^{s}_{=}}}  \overline{^{∏ ι^{∗}}  →} \frac{∏ F_{_{−}Y^{s}_{=}}}{∏ G_{_{j} Y^{s}_{r}}}  \overline{^{∏ κ_{!}}  →} \frac{∏ E_{_{j} Y^{s}_{r}}}{∏ G_{_{j} Y^{s}_{r}}}

\frac{∏ F_{_{−}Y^{s}_{=}}}{∏ P_{_{−}Y^{s}_{=}}}  \overline{_{_{∏ κ_{!}}  →}}^{Res}  \overline{→}^{^{ι^{∗}}  \overline{→}}  \frac{∏ E_{_{j} Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{=}}} =_{Id}  ⇀ ⇁^{Res=i^{∗}} s  \overline{→} \frac{∏ E_{_{j} Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{=}}} \underline{←}^{p^{∗}}

\frac{_{−}\underline{𝒪} _{=}/(U_{_{−}Y=}∩P_{Y^{−}})×∏ F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr} ×P_{_{j} Y^{−}_{r}}} \underline{←}^{p^{∗}}  \frac{∏_{j}𝒪_{r}×F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr} ×P_{_{j} Y^{−}_{r}}} ↓^{ϕ^{∗}}

\frac{_{−}\underline{𝒪} _{=}×(F_{_{−}Y=}∩F_{Y^{−}})/(f_{_{−}Y=}∩f_{Y^{−}})}{(∏ P_{^{−}_{j}Zr})×(P_{_{−}Y=}∩P_{Y^{−}})} ↓^{ϕ^{∗}}  \overline{^{r_{!}}  →} \frac{_{−}\underline{𝒪} _{=}×∏ F_{_{j} Y^{−}_{r}}}{(∏ P_{^{−}_{j}Zr})×(P_{_{−}Y=}∩P_{Y^{−}})}  \overline{→^{q_{∗}}} \underline{←}^{p^{∗}} \underline{←}^{q^{∗}}  \frac{_{−}\underline{𝒪} _{=}/(U_{_{−}Y=}∩P_{Y^{−}})×∏ F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr} ×P_{_{j} Y^{−}_{r}}}^{\begin{matrix}F & Id \\ & ⇃⇂ \end{matrix} ‖}  \overline{_{_{t_{!}} →}} \underline{←}^{t^{∗}}  \frac{∏_{j}𝒪_{r}×F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr} ×P_{_{j} Y^{−}_{r}}} \begin{aligned}V & & [−2A_{13}] \\ & ↓ \end{aligned}

\frac{_{−}\underline{𝒪} _{=}P_{_{−}Y^{≡}_{=}}×F_{Y^{−}}}{P_{^{−} =Z≡}×P_{_{−}Y^{≡}_{=}}} ↓^{ϕ^{∗}}  \overline{^{i^{∗}} →} \frac{_{−}\underline{𝒪} _{=}×(F_{_{−}Y=}∩F_{Y^{−}})}{(∏ P_{^{−}_{j}Zr})×(P_{_{−}Y=}∩P_{Y^{−}})} ↓^{σ^{∗}}  \frac{_{−}\underline{𝒪} _{=}×_{P_{_{−}Y=} ∩P_{Y^{−}}} (F_{_{−}Y=}∩F_{Y^{−}})/(f_{_{−}Y=}∩f_{Y^{−}})}{∏ P_{^{−}_{j}Zr}}  \overline{^{r_{!}} →}^{^{q_{∗}}  \overline{→}}  \frac{_{−}\underline{𝒪} _{=}×_{P_{_{−}Y=} ∩P_{Y^{−}}} ∏ F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr}} ↓^{q_{∗}}  \overline{→^{t_{!}}}^{^{q_{∗}}  \overline{→}}  \frac{∏_{j}𝒪_{r}×_{P_{_{j} Y^{−}_{r}}} F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr}} ↓^{π_{∗}}

F^{−1}

\frac{_{−}\underline{𝒪} _{=}P_{_{−}Y^{≡}_{=}}×_{P_{_{−}Y^{≡}_{=}}} F_{Y^{−}}}{P_{^{−} =Z≡}} ↓^{π_{∗}}  \overline{^{ι^{∗}}→} \frac{_{−}\underline{𝒪} _{=}×_{P_{_{−}Y=} ∩P_{Y^{−}}} (F_{_{−}Y=}∩F_{Y^{−}})}{∏ P_{^{−}_{j}Zr}} ↓^{q_{∗}}  \overline{→^{σ_{!}}} \underline{←}^{σ^{∗}} \frac{_{−}\underline{𝒪} _{=}×_{P_{_{−}Y=} ∩P_{Y^{−}}} (F_{_{−}Y=}∩F_{Y^{−}})/(f_{_{−}Y=}∩f_{Y^{−}})}{∏ P_{^{−}_{j}Zr}} \begin{aligned}v & & [−2B_{13}] \\ & ↓ \end{aligned}  \overline{^{r_{!}} →} \frac{_{−}\underline{𝒪} _{=}×_{P_{_{−}Y=} ∩P_{Y^{−}}} ∏ F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr}} ↓^{[−2B_{13}]}  \overline{→^{t_{!}}} \frac{∏_{j}𝒪_{r}×_{P_{_{j} Y^{−}_{r}}} F_{_{j} Y^{−}_{r}}}{∏ P_{^{−}_{j}Zr}} ↓^{[−2B_{13}]}

\frac{E_{Y}}{P_{^{−} =Z≡}} ↓^{α_{!}}  \overline{^{ι^{∗}}  →} \frac{F_{_{−}Y=}}{∏ P_{^{−}_{j}Zr}} ↓^{α_{!}}  \overline{^{κ_{!}}  →} \frac{∏ E_{_{j} Zr}}{∏ P_{^{−}_{j}Zr}} ↓^{α_{!}}^{^{α_{!}}  \overline{→}}

26 For the morphism coher : Φ_{13} → Φ_{23} we use notation

A_{13} = dim U_{_{−}Y_{=}}/(U_{_{−}Y_{=}} ∩ P_{Y^{−}}), B_{13} = dim f_{_{−}Y_{=}} ∩ f_{Y^{−}}.

Notice that P_{_{−}Y_{=}} ∩P_{Y^{−}} = P_{_{−}Y_{=}} ∩P_{_{−}Y_{=}^{≡}} ⊂ P_{_{−}Y_{=}^{≡}} and U_{_{−}Y_{=}} ∩P_{Y^{−}} = U_{_{−}Y_{=}} ∩P_{_{−}Y_{=}^{≡}} by Lemma 2.2.1. For the morphism coher : Φ_{21} → Φ_{22} we use notation

A_{21} = dim ∏_{s} U_{V_{−}^{s}}/(U_{V_{−}^{s}} ∩ P_{^{−}V^{s}}), B_{21} = dim ∏_{s} f_{V_{−}^{s}} ∩ f_{^{−}V^{s}}.

Notice that P_{V_{−}^{s}} ∩ P_{^{−}V^{s}} = P_{V_{−}^{s}} ∩ P_{^{=}V_{−}^{s}} ⊂ P_{^{=}V_{−}^{s}} and U_{V_{−}^{s}} ∩ P_{^{−}V^{s}} = U_{V_{−}^{s}} ∩ P_{^{=}V_{−}^{s}} by Lemma 2.2.1. For the morphism coher : Φ_{31} → Φ_{32} we use notation

A_{31} = dim U_{V_{−}}/(U_{V_{−}} ∩ P_{^{−}V^{=}}), B_{31} = dim f_{V_{−}} ∩ f_{^{−}V^{=}}.

Notice that P_{V_{−}} ∩ P_{^{−}V^{=}} = P_{V_{−}} ∩ P_{^{=}V_{−}^{≡}} ⊂ P_{^{=}V_{−}^{≡}} and U_{V_{−}} ∩ P_{^{−}V^{=}} = U_{V_{−}} ∩ P_{^{=}V_{−}^{≡}} by Lemma 2.2.1. For the morphism coher : Φ_{22} → Φ_{33} we use notation

A_{22} = dim U_{Y_{−}}/(U_{Y_{−}} ∩ P_{Y^{−}}), B_{22} = dim f_{Y_{−}} ∩ f_{Y^{−}}.

3.2.4. Lemma. The group P_{_{−}Y_{≡}^{=}} ∩ P_{Y^{−}} = P_{Y_{−}} ∩ P_{_{−}Y_{=}^{≡}} is a semidirect product of its normal subgroup U_{Y_{−}} ∩ P_{Y^{−}} and diagonally embedded subgroup ∏_{r} P_{_{−}Y_{r}^{=}}.

Proof. For an element g ∈ G_{V} the following conditions are equivalent:

g ∈ U_{Y_{−}} ∩ P_{Y^{−}} ⟺ (g − 1)(_{ι}Y_{ρ}^{σ}) ⊂ ⊕_{j,s,r}{_{j}Y_{r}^{s} | s ⩽ σ, r < ρ},
g ∈ ∏_{r} P_{_{−}Y_{r}^{=}} ⟺ (g − 1)(_{ι}Y_{ρ}^{σ}) ⊂ ⊕_{j,s,r}{_{j}Y_{r}^{s} | (s < σ, r = ρ) or (j ⩽ ι, s = σ, r = ρ)} (3.2.2)

(for all ι, σ, ρ).

Semisplit exact sequence obtained in Lemma 2.2.1 gives the lower row in the following diagram.

\begin{aligned}1 & \overline{→} & U_{Y_{−}} ∩ P_{Y^{−}} & \overline{→} & X & \overline{→} & ∏_{r} P_{_{−}Y_{r}^{=}} & \overline{→} & 1 \\ 1 & \overline{→} & U_{Y_{−}} ∩ P_{Y^{−}} & ‖ & → & P_{Y_{−}} ∩ P_{Y^{−}} & ^{↓ ∩} & \overline{→} & ∏_{r} P_{Y_{r}^{−}} & ^{↓ ∩} & \overline{→} & 1 \end{aligned}

The subgroup X in the upper row is determined as a preimage. By construction X is a semidirect product of its normal subgroup U_{Y_{−}} ∩ P_{Y^{−}} and diagonally embedded subgroup ∏_{r} P_{_{−}Y_{r}^{=}}.

An element g ∈ G_{V} is in X iff for all ι, σ, ρ

(g − 1)(_{ι}Y_{ρ}^{σ}) ⊂ ⊕_{j,s,r}{_{j}Y_{r}^{s} | (s < σ, r < ρ) or (s = σ, r < ρ) or (s < σ, r = ρ) or (j ⩽ ι, s = σ, r = ρ)}. (3.2.3)

An element g ∈ G_{V} is in P_{_{−}Y_{≡}^{=}} iff for all ι, σ, ρ

(g − 1)(_{ι}Y_{ρ}^{σ}) ⊂ ⊕_{j,s,r}{_{j}Y_{r}^{s} | r < ρ or (s < σ, r = ρ) or (j ⩽ ι, s = σ, r = ρ)}.

Hence, X = P_{_{−}Y_{≡}^{=}} ∩ P_{Y^{−}}. Symmetrically, an element g ∈ G_{V} is in P_{_{−}Y_{=}^{≡}} iff

(g − 1)(_{ι}Y_{ρ}^{σ}) ⊂ ⊕_{j,s,r}{_{j}Y_{r}^{s} | s < σ or (s = σ, r < ρ) or (j ⩽ ι, s = σ, r = ρ)}.  (3.2.4)

Hence, X = P_{Y_{−}} ∩ P_{_{−}Y_{=}^{≡}}.

\frac{∏ E_{^{t}V^{s}}}{∏ P_{^{t}V^{s}_{−}}}  \overline{^{∏ ι^{∗}}  →} \frac{∏ F_{^{t}V^{s}_{−}}}{∏ G_{^{t}V^{s}_{r}}}  \overline{^{∏ κ_{!}}  →} \frac{∏ E_{^{t}V^{s}_{r}}}{∏ G_{^{t}V^{s}_{r}}}

\frac{∏ F_{^{t}V^{s}_{−}}}{∏ P_{^{t}V^{s}_{−}}}  \overline{_{_{∏ κ_{!}}  →}}^{Res}  \overline{→}^{^{ι^{∗}}  \overline{→}}  \frac{∏ E_{^{t}V^{s}_{r}}}{∏ P_{^{t}V^{s}_{−}}} =_{Id}  ⇀ ⇁^{Res=i^{∗}} s  \overline{→} \frac{∏ E_{^{t}V^{s}_{r}}}{∏ P_{^{t}V^{s}_{−}}} \underline{←}^{p^{∗}}

\frac{∏ \underline{𝒪} _{−}^{s}/(U_{V^{s}_{−}}∩P_{^{−}V^{s}})×∏ F_{^{−}V^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}×P_{^{−}V^{s}_{r}}} \underline{←}^{p^{∗}}  \frac{∏ 𝒪_{r}^{s}×F_{^{−}V^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}×P_{^{−}V^{s}_{r}}} ↓^{ϕ^{∗}}

\frac{∏ \underline{𝒪} _{−}^{s}×(F_{V^{s}_{−}}∩F_{^{−}V^{s}})/(f_{V^{s}_{−}}∩f_{^{−}V^{s}})}{(∏ P_{_{−}Y^{s}_{r}})×(P_{V^{s}_{−}}∩P_{^{−}V^{s}})} ↓^{ϕ^{∗}}  \overline{^{r_{!}}  →} \frac{∏ \underline{𝒪} _{−}^{s}×∏ F_{^{−}V^{s}_{r}}}{(∏ P_{_{−}Y^{s}_{r}})×(P_{V^{s}_{−}}∩P_{^{−}V^{s}})}  \overline{→^{q_{∗}}} \underline{←}^{p^{∗}} \underline{←}^{q^{∗}}  \frac{∏ \underline{𝒪} _{−}^{s}/(U_{V^{s}_{−}}∩P_{^{−}V^{s}})×∏ F_{^{−}V^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}×P_{^{−}V^{s}_{r}}}^{\begin{matrix}F & Id \\ & ⇃⇂ \end{matrix} ‖}  \overline{_{_{t_{!}}→}} \underline{←}^{t^{∗}}  \frac{∏ 𝒪_{r}^{s}×F_{^{−}V^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}×P_{^{−}V^{s}_{r}}} \begin{aligned}V & & [−2A_{21}] \\ & ↓ \end{aligned}

\frac{∏ \underline{𝒪} _{−}^{s}P_{^{=}V^{s}_{−}}×F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}×P_{^{=}V^{s}_{−}}} ↓^{ϕ^{∗}}  \overline{^{i^{∗}} →} \frac{∏ \underline{𝒪} _{−}^{s}×(F_{V^{s}_{−}}∩F_{^{−}V^{s}})}{(∏ P_{_{−}Y^{s}_{r}})×(P_{V^{s}_{−}}∩P_{^{−}V^{s}})} ↓^{σ^{∗}}  ^{\frac{∏ \underline{𝒪} _{−}^{s}×_{P_{V^{s}_{−}} ∩P_{^{−}V^{s}}} (F_{V^{s}_{−}}∩F_{^{−}V^{s}})/(f_{V^{s}_{−}}∩f_{^{−}V^{s}})}{∏ P_{_{−}Y^{s}_{r}}}}  \overline{^{r_{!}} →}^{^{q_{∗}}  \overline{→}}  ^{\frac{∏ \underline{𝒪} _{−}^{s}×_{P_{V^{s}_{−}} ∩P_{^{−}V^{s}}} ∏ F_{^{−}V^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}} ↓^{q_{∗}}  \overline{→^{t_{!}}}^{^{q_{∗}}  \overline{→}}  ^{\frac{∏ 𝒪_{r}^{s}×_{P_{^{−}V^{s}_{r}}} F_{^{−}V^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}} ↓^{π_{∗}}

F^{−1}

\frac{∏ \underline{𝒪} _{−}^{s}P_{^{=}V^{s}_{−}}×_{P_{^{=}V^{s}_{−}}} F_{^{−}V^{s}}}{∏ P_{_{−}Y^{s}_{=}}} ↓^{π_{∗}}  \overline{^{ι^{∗}}→} ^{\frac{∏ \underline{𝒪} _{−}^{s}×_{P_{V^{s}_{−}} ∩P_{^{−}V^{s}}} (F_{V^{s}_{−}}∩F_{^{−}V^{s}})}{∏ P_{_{−}Y^{s}_{r}}}} ↓^{q_{∗}}  \overline{→^{σ_{!}}} \underline{←}^{σ^{∗}} ^{\frac{∏ \underline{𝒪} _{−}^{s}×_{P_{V^{s}_{−}} ∩P_{^{−}V^{s}}} (F_{V^{s}_{−}}∩F_{^{−}V^{s}})/(f_{V^{s}_{−}}∩f_{^{−}V^{s}})}{∏ P_{_{−}Y^{s}_{r}}}} \begin{aligned}v & & [−2B_{21}] \\ & ↓ \end{aligned}  \overline{^{r_{!}} →} ^{\frac{∏ \underline{𝒪} _{−}^{s}×_{P_{V^{s}_{−}} ∩P_{^{−}V^{s}}} ∏ F_{^{−}V^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}} ↓^{[−2B_{21}]}  \overline{→^{t_{!}}} ^{\frac{∏ 𝒪_{r}^{s}×_{P_{^{−}V^{s}_{r}}} F_{^{−}V^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}} ↓^{[−2B_{21}]}

\frac{∏ E_{Y^{s}}}{∏ P_{_{−}Y^{s}_{=}}} ↓^{α_{!}}  \overline{^{ι^{∗}}  →} \frac{∏ F_{V^{s}_{−}}}{∏ P_{_{−}Y^{s}_{r}}} ↓^{α_{!}}  \overline{^{κ_{!}}  →} \frac{∏ E_{Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}} ↓^{α_{!}}^{^{α_{!}}  \overline{→}}

28

\frac{∏ F_{^{t}V^{s}_{−}}}{∏ P_{^{t}V^{s}_{−}}}  \overline{_{_{∏ κ_{!}}  →}}^{Res}  \overline{→}^{^{ι^{∗}}  \overline{→}}  \frac{∏ E_{^{t}V^{s}_{r}}}{∏ P_{^{t}V^{s}_{−}}} =_{Id}  ⇀ ⇁^{Res=i^{∗}} s  \overline{→} \frac{∏ E_{^{t}V^{s}_{r}}}{∏ P_{^{t}V^{s}_{−}}} \underline{←}^{p^{∗}}

\frac{\underline{𝒪} _{−}/(U_{V_{−}}∩P_{^{−}V^{=}})×∏ F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr} ×P_{^{−}V^{=}_{r}}} \underline{←}^{p^{∗}}  \frac{∏ 𝒪_{r}×F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr} ×P_{^{−}V^{=}_{r}}} ↓^{ϕ^{∗}}

\frac{\underline{𝒪} _{−}×(F_{V_{−}}∩F_{^{−}V^{=}})/(f_{V_{−}}∩f_{^{−}V^{=}})}{(∏ P_{^{−} =Zr})×(P_{V_{−}}∩P_{^{−}V^{=}})} ↓^{ϕ^{∗}}  \overline{^{r_{!}}  →} \frac{\underline{𝒪} _{−}×∏ F_{^{−}V^{=}_{r}}}{(∏ P_{^{−} =Zr})×(P_{V_{−}}∩P_{^{−}V^{=}})}  \overline{→^{q_{∗}}} \underline{←}^{p^{∗}} \underline{←}^{q^{∗}}  \frac{\underline{𝒪} _{−}/(U_{V_{−}}∩P_{^{−}V^{=}})×∏ F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr} ×P_{^{−}V^{=}_{r}}}^{\begin{matrix}F & Id \\ & ⇃⇂ \end{matrix} ‖}  \overline{_{_{t_{!}}→}} \underline{←}^{t^{∗}}  \frac{∏ 𝒪_{r}×F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr} ×P_{^{−}V^{=}_{r}}} \begin{aligned}V & & [−2A_{31}] \\ & ↓ \end{aligned}

\frac{\underline{𝒪} _{−}P_{^{=}V^{≡}_{−}}×F_{^{−}V^{=}}}{P_{^{−} =Z≡}×P_{^{=}V^{≡}_{−}}} ↓^{ϕ^{∗}}  \overline{^{i^{∗}} →} \frac{\underline{𝒪} _{−}×(F_{V_{−}}∩F_{^{−}V^{=}})}{(∏ P_{^{−} =Zr})×(P_{V_{−}}∩P_{^{−}V^{=}})} ↓^{σ^{∗}}  \frac{\underline{𝒪} _{−}×_{P_{V_{−}} ∩P_{^{−}V^{=}}} (F_{V_{−}}∩F_{^{−}V^{=}})/(f_{V_{−}}∩f_{^{−}V^{=}})}{∏ P_{^{−} =Zr}}  \overline{^{r_{!}} →}^{^{q_{∗}}  \overline{→}}  \frac{\underline{𝒪} _{−}×_{P_{V_{−}} ∩P_{^{−}V^{=}}} ∏ F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr}} ↓^{q_{∗}}  \overline{→^{t_{!}}}^{^{q_{∗}}  \overline{→}}  \frac{∏ 𝒪_{r}×_{P_{^{−}V^{=}_{r}}} F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr}} ↓^{π_{∗}}

F^{−1}

\frac{\underline{𝒪} _{−}P_{^{=}V^{≡}_{−}}×_{P_{^{=}V^{≡}_{−}}} F_{^{−}V^{=}}}{P_{^{−} =Z≡}} ↓^{π_{∗}}  \overline{^{ι^{∗}}→} \frac{\underline{𝒪} _{−}×_{P_{V_{−}} ∩P_{^{−}V^{=}}} (F_{V_{−}}∩F_{^{−}V^{=}})}{∏ P_{^{−} =Zr}} ↓^{q_{∗}}  \overline{→^{σ_{!}}} \underline{←}^{σ^{∗}} \frac{\underline{𝒪} _{−}×_{P_{V_{−}} ∩P_{^{−}V^{=}}} (F_{V_{−}}∩F_{^{−}V^{=}})/(f_{V_{−}}∩f_{^{−}V^{=}})}{∏ P_{^{−} =Zr}} \begin{aligned}v & & [−2B_{31}] \\ & ↓ \end{aligned}  \overline{^{r_{!}} →} \frac{\underline{𝒪} _{−}×_{P_{V_{−}} ∩P_{^{−}V^{=}}} ∏ F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr}} ↓^{[−2B_{31}]}  \overline{→^{t_{!}}} \frac{∏ 𝒪_{r}×_{P_{^{−}V^{=}_{r}}} F_{^{−}V^{=}_{r}}}{∏ P_{^{−} =Zr}} ↓^{[−2B_{31}]}

\frac{E_{V}}{P_{^{−} =Z≡}} ↓^{α_{!}}  \overline{^{ι^{∗}}  →} \frac{F_{V_{−}}}{∏ P_{^{−} =Zr}} ↓^{α_{!}}  \overline{^{κ_{!}}  →} \frac{∏ E_{Zr}}{∏ P_{^{−} =Zr}} ↓^{α_{!}}^{^{α_{!}}  \overline{→}}

29 3.2.5. Lemma. We have

P_{Y_{−}} ⊃ (∏_{−}𝒪_{r})U_{Y_{−}} = _{−}\underline{𝒪}_{=} · P_{_{−}Y_{≡}^{=}} ∈ P_{_{=}^{−}Z_{≡}} -sets- P_{_{−}Y_{≡}^{=}},
−𝒪_{=} = (∏_{−}𝒪_{r})U_{Y_{−}}P_{_{−}Y_{=}^{≡}} ≃ (∏_{−}𝒪_{r})U_{Y_{−}} ×_{P_{Y_{−}}∩P_{_{−}Y^{≡}_{=}}} P_{_{−}Y_{=}^{≡}}.  (3.2.5)

Proof. In the first equation both sides are equal to (∏_{−}\underline{𝒪}_{r}P_{_{−}Y_{r}^{=}})U_{Y_{−}} as follows from

(∏_{−}\underline{𝒪}_{r})U_{Y_{−}} = ∏_{r}(∏_{j}_{j}𝒪_{r} · U_{_{−}Y_{r}})U_{Y_{−}} = ∏_{j,r}_{j}𝒪_{r} · U_{_{−}Y_{=}} = _{−}\underline{𝒪}_{=}.  (3.2.6)

Hence, we find

(∏_{−}𝒪_{r})U_{Y_{−}}P_{_{−}Y_{=}^{≡}} = (∏_{−}\underline{𝒪}_{r}P_{_{−}Y_{r}^{=}})U_{Y_{−}}P_{_{−}Y_{=}^{≡}} = _{−}\underline{𝒪}_{=}(∏ P_{_{−}Y_{r}^{=}})P_{_{−}Y_{=}^{≡}} = _{−}\underline{𝒪}_{=}P_{_{−}Y_{=}^{≡}} = _{−}𝒪_{=}

since ∏ P_{_{−}Y_{r}^{=}} ⊂ P_{_{−}Y_{=}^{≡}} by (3.2.2) and (3.2.4).

The isomorphism in (3.2.5) follows from the fact that (∏_{−}𝒪_{r})U_{Y_{−}} ⊂ P_{Y_{−}}.

To prove the existence of some of the pull-back squares in diagram on page 32 we use the following lemmas.

3.2.6. Lemma. There is a pull-back square

\begin{aligned}∏_{s,r} E_{Y_{r}^{s}} & \underline{←} & ∏_{r} F_{Y_{r}^{−}} \\ ∏_{s} F_{Y_{−}^{s}} & ↑ & \underline{←} & (F_{Y_{−}} ∩ F_{Y^{−}})/(f_{Y_{−}} ∩ f_{Y^{−}}) & ↑ \end{aligned}

Proof. This square is the sum over all arrows h ∈ H of the pull-back squares

⊕_{s=σ;r=ρ} Hom \underline{←}  ⊕_{s⩽σ;r=ρ} Hom

⊕_{s=σ;r⩽ρ} Hom ↑  ← ⊕_{s⩽σ;r⩽ρ} Hom / ⊕_{s<σ;r<ρ} Hom ↑

made of direct sums of spaces Hom = Hom_{ℂ}(Y_{ρ}^{σ}(h′), Y_{r}^{s}(h′′)).

3.2.7. Lemma. There is a pull-back square

\begin{aligned}_{−}𝒪_{=} ×_{P_{_{−}Y^{≡}_{=}}} F_{Y^{−}} & ←^{ι^{⊃}} & (_{−}\underline{𝒪}_{=}P_{_{−}Y_{≡}^{=}}) ×_{P_{Y_{−}}∩P_{_{−}Y^{≡}_{=}}} (F_{Y_{−}} ∩ F_{Y^{−}}) \\ E_{Y} & ↓^{α} & ←  ^{ι  ^{⊃}} & F_{Y_{−}} & ↓^{α} \end{aligned}

Proof. By (3.2.5) we have

−𝒪_{=} ×_{P_{_{−}Y^{≡}_{=}}} F_{Y^{−}} ≃ (_{−}\underline{𝒪}_{=}P_{_{−}Y_{≡}^{=}}) ×_{P_{Y_{−}}∩P_{_{−}Y^{≡}_{=}}} F_{Y^{−}}.

This identification gives naturally the upper row embedding in the square. The action map α in the right column makes sense, since _{−}\underline{𝒪}_{=}P_{_{−}Y_{≡}^{=}} ⊂ P_{Y_{−}} by Lemma 3.2.5. Commutativity of the square is immediate.

Let us prove the pull-back property. Let g ∈ _{−}\underline{𝒪}_{=} · P_{_{−}Y_{≡}^{=}} and let f ∈ F_{Y^{−}}. Assume that g.f ∈ F_{Y_{−}}. Then g^{−1} ∈ P_{Y_{−}}, f = g^{−1}.(g.f) ∈ F_{Y_{−}}, hence, f ∈ F_{Y_{−}} ∩ F_{Y^{−}}.

3.2.8. Lemma. There exists a mapping

ι : P_{_{−}Y_{=}^{≡}} ×_{P_{Y_{−}}∩P_{_{−}Y^{≡}_{=}}} ((_{−}\underline{𝒪}_{=}P_{_{−}Y_{≡}^{=}}) × (F_{Y_{−}} ∩ F_{Y^{−}}))

→ (_{−}\underline{𝒪}_{=}P_{_{−}Y_{≡}^{=}}) ×_{P_{Y_{−}}∩P_{_{−}Y^{≡}_{=}}} P_{_{−}Y_{=}^{≡}} × F_{Y^{−}} ≃ _{−}𝒪_{=} × F_{Y^{−}},

ι : (p, g, f) ↦ (g, p^{−1}, p.f) ↦ (gp^{−1}, p.f),

where p ∈ P_{_{−}Y_{=}^{≡}}, g ∈ _{−}\underline{𝒪}_{=}P_{_{−}Y_{≡}^{=}}, f ∈ F_{Y_{−}} ∩ F_{Y^{−}}. It is an embedding and it makes

\begin{aligned}_{−}𝒪_{=} × F_{Y^{−}} & ← ^{ι^{⊃}} & P_{_{−}Y_{=}^{≡}} ×_{P_{Y_{−}}∩P_{_{−}Y^{≡}_{=}}} ((_{−}\underline{𝒪}_{=}P_{_{−}Y_{≡}^{=}}) × (F_{Y_{−}} ∩ F_{Y^{−}})) \\ _{−}𝒪_{=} ×_{P_{_{−}Y^{≡}_{=}}} F_{Y^{−}} & ↓^{π} & ←^{ι ^{⊃}} & (_{−}\underline{𝒪}_{=}P_{_{−}Y_{≡}^{=}}) ×_{P_{Y_{−}}∩P_{_{−}Y^{≡}_{=}}} (F_{Y_{−}} ∩ F_{Y^{−}}) & ↓^{π} \end{aligned} ; (3.2.7)

into a pull-back square.

Proof. For any r ∈ P_{Y_{−}} ∩ P_{_{−}Y_{=}^{≡}} the constructed mapping takes a point (pr^{−1}, gr^{−1}, r.f), equivalent to (p, g, f), into (gr^{−1}rp^{−1}, pr^{−1}r.f) = (gp^{−1}, p.f). Hence, the mapping ι exists. It is an equivariant mapping of free P_{_{−}Y_{=}^{≡}}-spaces.

Commutativity of square (3.2.7) is clear. Notice that the vertical mappings are quotient by P_{_{−}Y_{=}^{≡}} maps. Therefore, there is a quotient by P_{_{−}Y_{=}^{≡}} map \overline{ι} of the constructed mapping ι. It also closes up the square, making it into a pull-back square by [BL94, Lemma 2.1.1]. Since the right π is surjective, there exists only one map in the lower row, which makes the square commutative. Therefore, \overline{ι} = ι in the lower row. Since in this pull-back square the lower ι is an embedding, the same holds for the upper ι.

4. Elementary coherence relations

4.1. Standard isomorphisms

Let 𝔇 be a 2-subcategory of the 2-category of categories, whose objects are equivariant derived categories \frac{X}{G}, 1-morphisms are compositions of functors of the following 3 types:

• inverse image functor \frac{f}{ϕ}^{∗};

• direct image functor \frac{f}{ϕ}_{∗}, where f|ϕ is a quotient map;

• direct image functor with proper supports \frac{f}{1}_{!},

and 2-morphisms are pastings of isomorphisms and their inverses of the following 6 types:

a) \frac{g}{ψ}^{∗} \frac{f}{ϕ}^{∗} ^{∼}→ (\frac{f}{ϕ} \frac{g}{ψ})^{∗};

b) \frac{f}{ϕ}_{∗} \frac{g}{ψ}_{∗} ^{∼}→ (\frac{f}{ϕ} \frac{g}{ψ})_{∗};

c) \frac{f}{1}_{!} \frac{g}{1}_{!} ^{∼}→ (\frac{f}{1} \frac{g}{1})_{!};

\frac{∏ E_{Y^{s}}}{∏ P_{_{−}Y^{s}_{=}}}  ^{∏ ι^{∗}}  → \frac{∏ F_{Y^{s}_{−}}}{∏ P_{_{−}Y^{s}_{r}}}  \overline{^{∏ κ_{!}}  →} \frac{∏ E_{Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{r}}}

\frac{∏ F_{Y^{s}_{−}}}{∏ P_{_{−}Y^{s}_{=}}}  \overline{_{_{∏ κ_{!}}  →}}^{Res}  \overline{→}^{^{ι^{∗}}  \overline{→}}  \frac{∏ E_{Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{=}}} =_{Id}  ⇀ ⇁^{Res=i^{∗}} s  \overline{→} \frac{∏ E_{Y^{s}_{r}}}{∏ P_{_{−}Y^{s}_{=}}} \underline{←}^{p^{∗}}

\frac{(_{−}\underline{𝒪} _{=}P_{_{−}Y^{=}_{≡}})/(U_{Y_{−}}∩P_{Y^{−}})×∏ F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr} ×P_{_{−}Y^{=}_{r}}} \underline{←}^{p^{∗}}  \frac{∏_{−}𝒪_{r}×F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr} ×P_{_{−}Y^{=}_{r}}} ↓^{ϕ^{∗}}

\frac{(_{−}\underline{𝒪} _{=}P_{_{−}Y^{=}_{≡}})×(F_{Y_{−}}∩F_{Y^{−}})/(f_{Y_{−}}∩f_{Y^{−}})}{(∏ P_{^{−} =Zr})×(P_{Y_{−}}∩P_{_{−}Y^{≡}_{=}})} ↓^{ϕ^{∗}}  \overline{^{r_{!}}  →} \frac{(_{−}\underline{𝒪} _{=}P_{_{−}Y^{=}_{≡}})×∏ F_{Y^{−}_{r}}}{(∏ P_{^{−} =Zr})×(P_{Y_{−}}∩P_{_{−}Y^{≡}_{=}})}  \overline{→^{q_{∗}}} \underline{←}^{p^{∗}} \underline{←}^{q^{∗}}  \frac{(_{−}\underline{𝒪} _{=}P_{_{−}Y^{=}_{≡}})/(U_{Y_{−}}∩P_{Y^{−}})×∏ F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr} ×P_{_{−}Y^{=}_{r}}}^{\begin{matrix}F & Id \\ & ⇃⇂ \end{matrix} ‖}  \overline{^{t_{!}}→} \underline{←}^{t^{∗}}  \frac{∏_{−}𝒪_{r}×F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr} ×P_{_{−}Y^{=}_{r}}} \begin{aligned}V & & [−2A_{22}] \\ & ↓ \end{aligned}

\frac{_{−}𝒪_{=}×F_{Y^{−}}}{P_{^{−} =Z≡}×P_{_{−}Y^{≡}_{=}}} ↓^{ϕ^{∗}}  \overline{^{i^{∗}} →} \frac{(_{−}\underline{𝒪} _{=}P_{_{−}Y^{=}_{≡}})×(F_{Y_{−}}∩F_{Y^{−}})}{(∏ P_{^{−} =Zr})×(P_{Y_{−}}∩P_{_{−}Y^{≡}_{=}})} ↓^{σ^{∗}}  \frac{(_{−}\underline{𝒪} _{=}P_{_{−}Y^{=}_{≡}})×_{P_{Y_{−}} ∩P_{_{−}Y^{≡}_{=}}} (F_{Y_{−}}∩F_{Y^{−}})/(f_{Y_{−}}∩f_{Y^{−}})}{∏ P_{^{−} =Zr}}  \overline{^{r_{!}}→}^{^{q_{∗}}  \overline{→}}  \frac{(_{−}\underline{𝒪} _{=}P_{_{−}Y^{=}_{≡}})×_{P_{Y_{−}} ∩P_{_{−}Y^{≡}_{=}}} ∏ F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr}} ↓^{q_{∗}}  \overline{→^{t_{!}}}^{^{q_{∗}}  \overline{→}}  \frac{∏_{−}𝒪_{r}×_{P_{_{−}Y^{=}_{r}}} F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr}} ↓^{π_{∗}}

F^{−1}

\frac{_{−}𝒪_{=}×_{P_{_{−}Y^{≡}_{=}}} F_{Y^{−}}}{P_{^{−} =Z≡}} ↓^{π_{∗}}  \overline{→^{ι^{∗}}} \frac{(_{−}\underline{𝒪} _{=}P_{_{−}Y^{=}_{≡}})×_{P_{Y_{−}} ∩P_{_{−}Y^{≡}_{=}}} (F_{Y_{−}}∩F_{Y^{−}})}{∏ P_{^{−} =Zr}} ↓^{q_{∗}}  \overline{→^{σ_{!}}} \underline{←}^{σ^{∗}} \frac{(_{−}\underline{𝒪} _{=}P_{_{−}Y^{=}_{≡}})×_{P_{Y_{−}} ∩P_{_{−}Y^{≡}_{=}}} (F_{Y_{−}}∩F_{Y^{−}})/(f_{Y_{−}}∩f_{Y^{−}})}{∏ P_{^{−} =Zr}} \begin{aligned}v & & [−2B_{22}] \\ & ↓ \end{aligned}  \overline{^{r_{!}}→} \frac{(_{−}\underline{𝒪} _{=}P_{_{−}Y^{=}_{≡}})×_{P_{Y_{−}} ∩P_{_{−}Y^{≡}_{=}}} ∏ F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr}} ↓^{[−2B_{22}]}  \overline{→^{t_{!}}} \frac{∏_{−}𝒪_{r}×_{P_{_{−}Y^{=}_{r}}} F_{Y^{−}_{r}}}{∏ P_{^{−} =Zr}} ↓^{[−2B_{22}]}

\frac{E_{Y}}{P_{^{−} =Z≡}} ↓^{α_{!}}  \overline{^{ι^{∗}}  →} \frac{F_{Y_{−}}}{∏ P_{^{−} =Zr}} ↓^{α_{!}}  \overline{^{κ_{!}}  →} \frac{∏ E_{Zr}}{∏ P_{^{−} =Zr}} ↓^{α_{!}}^{^{α_{!}}  \overline{→}}

32 d) base change isomorphism

\begin{matrix}\frac{X}{G} & \overline{^{\frac{f}{1}_{!}} →} & \frac{Y}{G} \\ \frac{W}{H} & ↓^{\frac{h}{ϕ}^{∗}} & \overline{^{\frac{j}{1}_{!}} →} & ⟸ & \frac{Z}{H} & ↓^{\frac{g}{ϕ}^{∗}} \end{matrix}

where W = X ×_{Y} Z, and h, j are the projections;

e) the isomorphism of \frac{X}{G} ^{\frac{f}{1}_{!}}→ \frac{Y}{G} ^{\frac{g}{ϕ}_{∗}}→ \frac{Z}{H} with \frac{X}{G} ^{\frac{j}{ϕ}_{∗}}→ \frac{W}{H} ^{\frac{h}{1}_{!}}→ \frac{Z}{H}, where K = Ker(ϕ : G → H), W = K\X, h = K\f : W = K\X → K\Y = Z, and j is the quotient map; it is given by the pasting

\frac{X}{G} =⇀ ⇁ \frac{X}{G}  \overline{^{\frac{f}{1}_{!}} →} \frac{Y}{G}

^{ε^{−1}} ‖

\frac{W}{H} ↓^{\frac{j}{ϕ}_{∗}}  \overline{_{_{\frac{h}{1}_{!}} →}}^{\frac{j}{ϕ}^{∗}}  \overline{→} \frac{Z}{H}^{‖} =⇀ ⇁^{\frac{g}{ϕ}^{∗}}  \overline{→}^{^{η^{−1}} ‖} \frac{Z}{H} ↓^{\frac{g}{ϕ}_{∗}}

f) the isomorphism of \frac{X}{G} ^{\frac{f}{ϕ}_{∗}}→ \frac{Y}{H} ^{\frac{g}{ψ}^{∗}}→ \frac{Z}{B} with \frac{X}{G} ^{\frac{h}{ξ}^{∗}}→ \frac{W}{K} ^{\frac{j}{χ}_{∗}}→ \frac{Z}{B}, where W = X ×_{Y} Z, K = G_{ϕ} ×_{H} _{ψ}B and h, j, ξ, χ are the projections; it is given by the pasting

\frac{X}{G}  \overline{^{\frac{f}{ϕ}_{∗}}  →} \frac{Y}{H}

\frac{X}{G}^{⇃⇂^{⟸^{ε}} ‖} \underline{←}^{\frac{f}{ϕ}^{∗}}  \frac{Z}{B} ↓^{\frac{g}{ψ}^{∗}}

\frac{W}{K} ↓^{\frac{h}{ξ}^{∗}}  \overline{_{_{\frac{j}{χ}_{∗}}  →}} \frac{←}{⇑ ; ⇑} \underline{←}^{\frac{j}{χ}^{∗}}  \frac{Z}{B}^{⇃⇂^{⟸^{η}} ‖}

4.1.1. Lemma. Any 1-morphism \frac{X}{G} → \frac{Y}{H} in 𝔇 is isomorphic to a composition of the form

\frac{X}{G} ^{\frac{f}{ϕ}^{∗}}→ \frac{Z}{B} ^{\frac{g}{ψ}_{∗}}→ \frac{W}{H} ^{\frac{h}{1}_{!}}→ \frac{Y}{H} .  (4.1.1)

Proof. Isomorphisms d)–f) allow to place all functors in a given composition in the correct order: inverse image functors are followed by direct image functors and then by direct image functors with proper supports. Isomorphisms a)–c) reduce the sequence of functors of similar type to only one functor.

4.1.2. Theorem. The category 𝔇 is a 2-groupoid, that is, for any 1-morphisms F and G with the common source and target the set Hom(F, G) either is empty or has exactly one element (and all 2-morphisms are invertible).

The coherence isomorphism uses 3 more types of elementary isomorphisms.

F) A generalization of type f) isomorphism: the isomorphism of \frac{X}{G} ^{\frac{f}{ϕ}_{∗}}→ \frac{Y}{H} ^{\frac{g}{ψ}^{∗}}→ \frac{Z}{B} with

\frac{X}{G} ^{\frac{h}{ξ}^{∗}}→ \frac{W}{K} ^{\frac{j}{χ}_{∗}}→ \frac{Z}{B}, whenever the following square commutes

\begin{matrix}X|G & \overline{^{f|ϕ} →} & Y |H \\ W |K & ^{^{h|ξ}↑} & \overline{^{j|χ} →} & Z|B & ^{↑^{g|ψ}} \end{matrix}

the isomorphism is given by the pasting

\frac{X}{G}  \overline{^{\frac{f}{ϕ}_{∗}}  →} \frac{Y}{H}

\frac{X}{G}^{⇃⇂^{⟸^{ε}} ‖} \underline{←}^{\frac{f}{ϕ}^{∗}}  \frac{Z}{B} ↓^{\frac{g}{ψ}^{∗}}

\frac{W}{K} ↓^{\frac{h}{ξ}^{∗}}  \overline{_{_{\frac{j}{χ}_{∗}}  →}} \frac{←}{⇑ ; ⇑} \underline{←}^{\frac{j}{χ}^{∗}}  \frac{Z}{B}^{⇃⇂^{⟸^{η}} ‖}

s) Whenever a linear complex algebraic group P is a semidirect product of its subgroups U and L, such that U is normal in P and the manifold U is an affine space, and the action of U in a P -variety E is trivial, there is an isomorphism

\frac{E}{P} =⇀ ⇁^{i^{∗}}  \overline{→}^{‖}  \frac{E}{P}^{^{p^{∗}}  \overline{→}}^{\frac{E}{L}}

constructed in Appendix A.1 as a corollary of Proposition A.1.1 for Y = E.

v) Whenever h : E → B is a complex affine linear G-bundle (h and linear combination maps are G-equivariant), then there is an isomorphism

\frac{B}{G}

\frac{E}{G} \begin{matrix}⇐ \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \end{matrix} \underline{←}^{h^{∗}}

\frac{B}{G} ↓^{T^{−2 rk_{ℂ} h}}^{^{h_{!}}  \overline{→}}

which is the inverse of isomorphism λ, constructed in Theorem B.2.5.

A more general V) type isomorphism is a composition of such. Let

E_{1} ^{h_{1}}→ E_{2} ^{h_{2}}→ … ^{h_{n−1}} → E_{n} ^{h_{n}}→ B

be a sequence of affine linear G-bundles h_{i} of ranks d_{i}. Let h denote the composition h_{n} ∘ ⋯ ∘ h_{2} ∘ h_{1}, and let s^{k} = ∑_{i=1}^{k} d_{i}, d = s^{n}. With this sequence of bundles we associate an isomorphism of V) type, inverse to the following isomorphism Λ:

\begin{aligned}\overline{h^{∗}} & \frac{B}{G} \\ \frac{E_{2}}{G} & ← ^{h_{2}^{∗}} & \frac{E_{3}}{G} & ← ^{h_{3}^{∗}} & … & \frac{E_{n−1}}{G} & ←^{h_{n−1}^{∗}} & \frac{E_{n}}{G} & =^{λ_{n}} ⇒ & \underline{←}^{h_{n}^{∗}} \\ \frac{E_{1}}{G} & ↓ & =^{λ_{1}} ⇒ & \underline{←}^{h_{1}^{∗}} & \frac{E_{3}}{G} & ↓^{⟹^{λ_{2}} [−2d_{2}]} & ^{^{h_{2!}}  \overline{→}} & ^{^{h_{3!}}  \overline{→}} & \frac{E_{n−1}}{G} & ↓^{[−2d_{n−2}]} & = & \frac{E_{n}}{G} & ↓^{⟹^{λ_{n−1}} [−2d_{n−1}]} & ^{^{h_{n−1!}}  \overline{→}} & \frac{B}{G} & ↓^{[−2d_{n}]} & ^{^{h_{n!}}  \overline{→}} \\ & = \\ \frac{E_{2}}{G} & ↓^{[−2d_{1}]} & \overline{_{_{h_{2!}} →}} & ^{^{h_{1!}}  \overline{→}} & \frac{E_{3}}{G} & ↓^{[−2d_{1}]} & \overline{_{_{h_{3!}} →}} & … & \frac{E_{n−1}}{G} & ↓^{[−2s^{n−3}]} & \overline{_{_{h_{n−1!}} →}} & \frac{E_{n}}{G} & ↓^{[−2s^{n−2}]} & = \\ \overline{^{h_{!}}  →} & \frac{B}{G} & ↓^{[−2s^{n−1}]} & ^{^{h_{n!}}  \overline{→}} \end{aligned} (4.1.2)

A. Technical results

A.1. An equivalence of equivariant derived categories

Let P be a linear complex algebraic group, let U, L be its subgroups such that U is normal in P and the manifold U is an affine space. Assume that P is a semidirect product of U and L, that is, the multiplication map · : U × L → P is an isomorphism of manifolds. Denote by ι : L^{⊂} → P and κ : P ▷ L the natural inclusion and projection. Let Y be a P -variety, let E be an L-variety, let i : E → Y be a ι-map and let p : Y → E be a κ-map identifying E with the quotient U\Y such that p ∘ i = id_{E}. Consider

W = P ×_{L} E = (P × E)/L, (p, e).l =^{def} (pl, l^{−1}e).

The variety W is a P -variety via left translations on P. As a P -variety W is isomorphic to U × E, where P acts by

(u, l).(v, e) = (u · (lvl^{−1}), l.e).

There is a surjective map of P -varieties

π : W ≃ U × E → Y, (u, e) ↦ u.i(e).

Assume furthermore that π is smooth (a locally trivial fibration), and its fibres are affine spaces. Then π : W → Y is ∞-acyclic in the sense of [BL94].

A.1.1. Proposition ([Lyu]). The functors

\frac{π}{1}^{∗} :\frac{Y}{P} → \frac{W}{P} ,
\frac{p}{κ}^{∗} :\frac{E}{L} → \frac{Y}{P} ,
\frac{i}{ι}^{∗} :\frac{Y}{P} → \frac{E}{L}

are equivalences, and the last two are quasi-inverse to each other.

Using the fact that \frac{i}{ι}^{∗} is an equivalence, we define unambiguously an isomorphism

\begin{matrix}\frac{Y}{P} & \overline{^{\frac{i}{ι}^{∗}} →} & \frac{E}{L} \\ \frac{Y}{P} & ↓^{\frac{p}{κ}^{∗}} & ^{s^{Id}^{=⇀ ⇁}} \end{matrix}

by the equation

\begin{matrix}\frac{Y}{P} & \overline{^{\frac{i}{ι}^{∗}} →} & \frac{E}{L} \\ \frac{E}{L} & \begin{aligned}\frac{i}{ι}^{∗} & & = \\ & ↓ \end{aligned} & ← _{\frac{i}{ι}^{∗}} & \frac{Y}{P} & ↓^{\frac{p}{κ}^{∗}} & ^{s^{=⇀ ⇁}} \end{matrix} = \begin{matrix}\frac{Y}{P} & \overline{^{\frac{i}{ι}^{∗}} →} & \frac{E}{L} \\ \frac{E}{L} & \begin{aligned}\frac{i}{ι}^{∗} & & = \\ & ↓ \end{aligned} & ← _{\frac{i}{ι}^{∗}} & a ; ↼ ↽= & \frac{Y}{P} & ↓^{\frac{p}{κ}^{∗}} \end{matrix}  (A.1.1)

B. Affine linear bundles

B.1. Non-equivariant case

An affine linear bundle of rank d over a complex manifold B is a morphism of complex manifolds h : E → B, together with additional data consisting of an open covering {U_{i}} of B, and isomorphisms ψ_{i} : h^{−1}(U_{i}) → ℂ^{d} × U_{i} over U_{i}, such that for any i, j, and for U = U_{i} ∩ U_{j}, the automorphism ψ = ψ_{j} ∘ψ_{i}^{−1} of ℂ^{d} ×U is fiberwise affine-linear, i.e., ψ(x, u) = (A(u)x+b(u), u), u ∈ U, for some ℂ-linear maps A(u) : ℂ^{d} → ℂ^{d} and some vectors b(u) ∈ ℂ^{d}.

We are going to discuss a functor in derived category, which is a kind of braiding in a monoidal 2-category.

B.1.1. Theorem. Let h : E → B be an affine linear bundle of dimension d over a complex manifold B. Then there is an isomorphism of functors

λ_{B} : [D^{b,c}(B) ^{h^{∗}}→ D^{b,c}(E) ^{h_{!}}→ D^{b,c}(B)] ≃ [D^{b,c}(B) ^{[−2d]} → D^{b,c}(B)].

B.1.2. Proposition. Let f : F → B and h : E → B be affine linear bundles, and let y : F → E be a surjective map of affine linear bundles over B with its natural structure of an affine linear bundle. Then the following isomorphisms are equal

\begin{matrix}D^{b,c}(E) & ←^{h^{∗}} & D^{b,c}(B) \\ D^{b,c}(F) & =^{λ_{y}} ⇒ & \underline{←}^{y^{∗}} & D^{b,c}(B) & ↓^{⟹^{λ_{h}} [−2 rk h]} & ^{^{h_{!}}  \overline{→}} \\ = \\ D^{b,c}(E) & ↓^{[−2 rk y]} & \overline{_{_{h_{!}} →}} & ^{^{y_{!}}  \overline{→}} & D^{b,c}(B) & ↓^{[−2 rk y]} \end{matrix} = \begin{aligned}D^{b,c}(E) & ←^{h^{∗}} & D^{b,c}(B) \\ D^{b,c}(F) & =^{λ_{f}} ⇒ & \underline{←}^{f^{∗}} & \underline{←}^{y^{∗}} \\ D^{b,c}(E) & \overline{_{_{h_{!}}→}} & ^{^{y_{!}}  \overline{→}} & D^{b,c}(B) & ↓^{[−2 rk f]} & ^{^{f_{!}}  \overline{→}} \end{aligned}

B.2. Equivariant affine linear bundles

B.2.1. Definition. An affine linear bundle h : E → B, which is a G-equivariant map, is called an affine linear G-bundle if elements of G map fibers to fibers, preserving affine linear structure.

B.2.2. Lemma. Let h : E → B be an affine linear G-bundle, and let K be a normal subgroup of G. Assume that B is a free K-space. Then the quotient K\h : K\E → K\B is an affine linear G/K-bundle.

One of the sources of affine linear G-bundles is the following. Let M be a G-module over ℂ, and let N be its G-submodule. Then the projection π : M → M/N equipped with the linear combination map

ψ : ℂ × M ×_{M/N} M → M, (λ, m, m′) ↦ λm + (1 − λ)m′,

is an affine linear G-bundle. Indeed, denote L = M/N and fix λ ∈ ℂ. To prove that

\begin{matrix}M ×_{L} M & \overline{^{ψ(λ)} →} & M \\ M ×_{L} M & ↓^{g×g} & \overline{^{ψ(λ)} →} & M & ↓^{g} \end{matrix}

commutes choose a splitting i : L → M of π, so that M = i(L) ⊕ N as a vector space, and use g.(i(l), n) = (i(g.l), ν(g, l) + n). (If N has a direct complement G-submodule in M, then π : M → M/N has an obvious structure of a vector G-bundle.)

The above consideration apply, in particular, to P_{𝒲} ∩P_{𝒱}-modules N = f_{𝒲} ∩f_{𝒱} ⊂ F_{𝒲} ∩F_{𝒱} = M, giving an affine P_{𝒲} ∩ P_{𝒱}-bundle

F_{𝒲} ∩ F_{𝒱} → (F_{𝒲} ∩ F_{𝒱})/(f_{𝒲} ∩ f_{𝒱}),

or, among many others, an affine P_{Y_{−}} ∩ P_{Y^{−}}-bundle

F_{Y_{−}} ∩ F_{Y^{−}} → (F_{Y_{−}} ∩ F_{Y^{−}})/(f_{Y_{−}} ∩ f_{Y^{−}}).

B.2.3. A tower of affine line bundles. Identifying U_{Y_{−}} with 1 + 𝔫_{Y_{−}}, where 𝔫_{Y_{−}} is a Lie algebra of U_{Y_{−}} (strictly uppertriangular matrices), we could make U_{Y_{−}} into a P_{Y_{−}}-module via conjugation. We need, however, to deal with the P_{Y_{−}} ∩ P_{Y^{−}}-variety U_{Y_{−}}/(U_{Y_{−}} ∩ P_{Y^{−}}). The polynomial action of P_{Y_{−}} ∩ P_{Y^{−}} can be presented as action in a tower of affine linear bundles as follows.

Consider the partition (Young diagram) λ = (m − 1, m − 2, …, 2, 1), identified with the shape of strictly uppertriangular m × m matrices. Consider a (non-standard, row strict) Young tableau for this diagram, which we define specialising the usual term [Ful97, Mac95] to our needs as an m × m matrix T with integer coefficients, such that (i) for some n ∈ ℤ the set of values {T_{kl}} is [0, n], (ii) T_{kl} = 0 iff k ⩾ l, (iii) l < l′ implies T_{kl} < T_{kl′}, and (iv) k < k′ implies T_{kl} ⩾ T_{k′l}. A Young tableau is standard if each value from [1, n] occurs exactly once in (T_{kl}). To a Young tableau T corresponds a series of normal subgroups of U = U_{Y_{−}}, namely,

U = U^{(1)} ⊃ U^{(2)} ⊃ ⋯ ⊃ U^{(n)} ⊃ U^{(n+1)} = 1,  (B.2.1)

where

U^{(p)} = 1 + {A = (A_{kl})_{k,l=1}^{m}, A_{kl} ∈ Hom_{ℂ}(Y_{l}, Y_{k}) | A_{kl} ≠ 0 ⟹ T_{kl} ⩾ p}.

For instance, the lower central series U = U_{1} ⊃ U_{2} ⊃ ⋯ ⊃ U_{m} = 1, where U_{k+1} = {U, U_{k}}, corresponds to n = m − 1 and

T = (\begin{matrix}0 & 1 & ... & m − 2 & m − 1 \\ 0 & 0 & ... & m − 3 & m − 2 \\ ... & ... & ... & ... & ... \\ 0 & 0 & ... & 0 & 1 \\ 0 & 0 & ... & 0 & 0 \end{matrix})  (B.2.2)

A Young tableau T defines a sequence of Young diagrams

λ = T_{1} ⊃ ⋯ ⊃ T_{k} ⊃ T_{k+1} ⊃ ⋯ ⊃ T_{m} = ∅,

where T_{k} is the union of places (i, j) for which T_{ij} ⩾ k. The corresponding ideals of the algebra of triangular matrices are

𝔫_{V_{−}}^{T_{k}} = {A = (A_{ij})_{i,j=1}^{m}, A_{ij} ∈ Hom_{ℂ}(V_{j}, V_{i}) | A_{ij} ≠ 0 ⟹ (i, j) ∈ T_{k}},

and the corresponding normal subgroups of U_{V_{−}} are U_{V_{−}}^{T_{k}} = 1+𝔫_{V_{−}}^{T_{k}}. It follows from our definition of Young tableau that

T_{k} − T_{k+1} has no more than 1 element in each row.  (B.2.3)

B.2.4. Lemma. A pair of embedded Young diagrams T_{k} ⊃ T_{k+1} satisfying condition (B.2.3) gives rise to an affine linear P_{V_{−}} ∩ P_{V^{−}}-bundle

h_{k} : E_{k+1} = U_{V_{−}}/[U_{V_{−}}^{T_{k+1}}(U_{V_{−}} ∩ P_{V^{−}})] → U_{V_{−}}/[U_{V_{−}}^{T_{k}}(U_{V_{−}} ∩ P_{V^{−}})] = E_{k}.

Proof. Let e, f ∈ E_{k+1} lie over the same point a = h_{k}(e) = h_{k}(f) ∈ E_{k}. Their linear combination λe + (1 − λ)f ∈ E_{k+1}, λ ∈ ℂ, is defined as follows. Consider ẽ, f̃ ∈ U_{V_{−}} such that e = ẽU_{V_{−}}^{T_{k+1}}(U_{V_{−}} ∩ P_{V^{−}}), f = f̃U_{V_{−}}^{T_{k+1}}(U_{V_{−}} ∩ P_{V^{−}}). Then we define λe + (1 − λ)f ∈ E_{k+1} as

(λẽ + (1 − λ)f̃)U_{V_{−}}^{T_{k+1}}(U_{V_{−}} ∩ P_{V^{−}}).  (B.2.4)

Let us check consistency of this definition.

We have a = h_{k}(e) = ẽU_{V_{−}}^{T_{k}}(U_{V_{−}} ∩ P_{V^{−}}), a = h_{k}(f) = f̃U_{V_{−}}^{T_{k}}(U_{V_{−}} ∩ P_{V^{−}}). Let g ∈ U_{V_{−}}^{T_{k}}(U_{V_{−}} ∩ P_{V^{−}}) be such that ẽ = f̃g. Then

λẽ + (1 − λ)f̃ = f̃(λg + 1 − λ).

Hence, point (B.2.4) lies over

(λẽ + (1 − λ)f̃)U_{V_{−}}^{T_{k}}(U_{V_{−}} ∩ P_{V^{−}}) = f̃(λg + 1 − λ)U_{V_{−}}^{T_{k}}(U_{V_{−}} ∩ P_{V^{−}}) = a,

since b =^{def} λg + 1 − λ ∈ U_{V_{−}}^{T_{k}}(U_{V_{−}} ∩ P_{V^{−}}).

Let e′ = ẽu, f′ = f̃v be another representatives of e, f for u, v ∈ U_{V_{−}}^{T_{k+1}}(U_{V_{−}} ∩ P_{V^{−}}). Then

λe′ + (1 − λ)f′ = λẽu + (1 − λ)f̃v = f̃[λgu + (1 − λ)v] = f̃[λg + (1 − λ)vu^{−1}]u

= f̃(λg + 1 − λ)(λg + 1 − λ)^{−1}[λg + (1 − λ) + (1 − λ)m]u = [λẽ+ (1 − λ)f̃][1 + (1 − λ)b^{−1}m]u,

where we denote vu^{−1} = 1 + m, m ∈ 𝔫_{V_{−}}^{T_{k+1}} + (𝔫_{V_{−}} ∩ P_{V^{−}}). Let us decompose some terms in this formula as follows: (1 − λ)m = m′ + m′′, m′ ∈ 𝔫_{V_{−}}^{T_{k+1}}, m′′ ∈ 𝔫_{V_{−}} ∩ P_{V^{−}}, and b^{−1} = 1 + b′ + b′′, b′ ∈ 𝔫_{V_{−}}^{T_{k}}, b′′ ∈ 𝔫_{V_{−}} ∩ P_{V^{−}}. Then

b^{−1}(1 − λ)m = (1 + b′ + b′′)(m′ + m′′) = (1 + b′ + b′′)m′ + (1 + b′′)m′′ + b′m′′

∈ 𝔫_{V_{−}}^{T_{k+1}} + 𝔫_{V_{−}} ∩ P_{V^{−}} + 𝔫_{V_{−}}^{T_{k}} · 𝔫_{V_{−}}.

Condition (B.2.3) implies that 𝔫_{V_{−}}^{T_{k}} · 𝔫_{V_{−}} ⊂ 𝔫_{V_{−}}^{T_{k+1}}, hence,

λe′ + (1 − λ)f′ ∈ [λẽ + (1 − λ)f̃]U_{V_{−}}^{T_{k+1}}(U_{V_{−}} ∩ P_{V^{−}}).

Therefore, linear combinations of elements are well-defined.

The obtained affine linear bundle is a P_{V_{−}} ∩ P_{V^{−}}-bundle, on which the group acts by conjugation. Indeed, let p ∈ P_{V_{−}} ∩ P_{V^{−}}. If h_{k}(e) = h_{k}(f) as above, then h_{k}(p.e) = h_{k}(p.f). Furthermore,

p.[λe + (1 − λ)f] = p[λẽ + (1 − λ)f̃]p^{−1}U_{V_{−}}^{T_{k+1}}(U_{V_{−}} ∩ P_{V^{−}})

= [λpẽp^{−1} + (1 − λ)pf̃p^{−1}]U_{V_{−}}^{T_{k+1}}(U_{V_{−}} ∩ P_{V^{−}}) = λ(p.e) + (1 − λ)(p.f).

Thus, linear combinations are equivariant.

B.2.5. Theorem. Let h : E → B be an affine linear bundle G-bundle of dimension d over a complex manifold B. Then there is an isomorphism of functors

λ : [D_{G}^{b,c}(B) ^{h^{∗}}→ D_{G}^{b,c}(E) ^{h_{!}}→ D_{G}^{b,c}(B)] ^{∼}→ [D_{G}^{b,c}(B) ^{[−2d]} → D_{G}^{b,c}(B)].

B.2.6. Proposition. Let f : F → B and h : E → B be affine linear G-bundles, and let y : F → E be a G-equivariant surjective map of affine linear bundles over B with its natural structure of an affine linear G-bundle. Then the following isomorphisms are equal

\begin{aligned}\frac{E}{G} & ← ^{h^{∗}} & \frac{B}{G} \\ \frac{F}{G} & =^{λ_{y}} ⇒ & \underline{←}^{y^{∗}} & \frac{B}{G} & ↓^{⟹^{λ_{h}} [−2 rk h]} & ^{^{h_{!}}  \overline{→}} \\ & = \\ \frac{E}{G} & ↓^{[−2 rk y]} & \overline{_{_{h_{!}} →}} & ^{^{y_{!}}  \overline{→}} & \frac{B}{G} & ↓^{[−2 rk y]} \end{aligned} = \begin{aligned}\frac{E}{G} & ← ^{h^{∗}} & \frac{B}{G} \\ \frac{F}{G} & =^{λ_{f}} ⇒ & \underline{←}^{f^{∗}} & \underline{←}^{y^{∗}} \\ \frac{E}{G} & \overline{_{_{h_{!}} →}} & ^{^{y_{!}}  \overline{→}} & \frac{B}{G} & ↓^{[−2 rk f]} & ^{^{f_{!}}  \overline{→}} \end{aligned}

Proof. To define inverse image functors it suffices to work with pull-back resolutions. These will be used also for direct image with proper support functors. Thus, for an arbitrary G-resolution p : P → B we pull back the affine linear G-bundle structures and take their quotients by G:

\overline{P ×_{B} F} \underline{←}  P ×_{B} F  \overline{→} F

\overline{P ×_{B} E} \underline{←}^{^{\overline{1×y}} \overline{→}}  P ×_{B} E^{^{1×y} \overline{→}}  \overline{→} E^{^{y}  \overline{→}}

\overline{P} ↓^{\overline{pr_{1}}} \underline{←} \underline{←}^{\overline{pr_{1}}}  P ↓^{pr_{1}}  \overline{→} \underline{←}^{pr^{1}}  B ↓^{f} \underline{←}^{h}

Therefore, the proposition reduces to its non-equivariant version Proposition B.1.2.

B.2.7. Example. Let K ⊂ N ⊂ M be embedded G-modules. There are quotient maps

M  \overline{^{y}  →} M/K

=

M/N\underline{←}^{h}^{^{f}  \overline{→}}

According to Section B.2.2 each of these three maps has a natural structure of an affine linear G-bundle. Furthermore, y : M → M/K is a surjective map between affine linear G-bundles f and h over M/N. The arising structure of y as an affine linear G-bundle coincides with that of Section B.2.2, since both come, in the final account, from the vector space structure of M. Therefore, Proposition B.2.6 applies in this situation.

B.2.8. Relations between towers of affine line bundles. Let

h = (E_{1} ^{h_{1}}→ E_{2} ^{h_{2}}→ … ^{h_{n−1}} → E_{n} ^{h_{n}}→ B)

be a sequence of affine linear G-bundles. Suppose that each map h_{k} : E_{k} → E_{k+1} equals to a composition of several maps

h_{k} = (E_{k} = E_{k}^{0} ^{h_{k}^{1}} ▷ E_{k}^{1} ^{h_{k}^{2}} ▷ … E_{k}^{m−1} ^{h_{k}^{m}}▷ E_{k}^{m} = E_{k+1})

(m can be 1 as well as bigger integer), where each composite map E_{k}^{j} → E_{k+1} is an affine linear G-bundle, and h_{k}^{j} : E_{k}^{j−1} → E_{k}^{j} is a surjection of affine linear bundles. Then h_{k}^{j} is given its natural structure of affine linear G-bundle, and we can form a sequence (h_{k}^{j}) : E_{1} → ⋯ → B of affine linear G-bundles. We may associate type V^{−1}) isomorphisms Λ : h_{!}h^{∗} → [−2d] given by (4.1.2) to both sequences (h_{k}) and (h_{k}^{j}). Repeated application of Proposition B.2.6 shows that both isomorphisms Λ coincide.

Examples of such situations are provided by Section B.2.3. To normal series (B.2.1) corresponds a sequence of affine linear P_{Y_{−}}-bundles

U = U/U^{(n+1)} ▷ … ▷ U/U^{(k+1)} ▷ U/U^{(k)} ▷ … ▷ U/U^{(1)} = pt

and a series of affine linear P_{Y_{−}} ∩ P_{Y^{−}}-bundles

h = (U/(U ∩ P_{Y^{−}}) ▷ … ▷ U/[U^{(k+1)}(U ∩ P_{Y^{−}})] ▷ U/[U^{(k)}(U ∩ P_{Y^{−}})] ▷ … pt).

Let Γ be the following graph, whose vertices are Young tableaux of shape λ. An edge connects tableaux T and T′ if for some 1 ⩽ p < n we have {T_{kl}} = [1, n], {T′_{kl}} = [1, n+1], T_{kl} < p implies T′_{kl} = T_{kl}, T_{kl} > p implies T′_{kl} = T_{kl}+1, and T_{kl} = p implies that T′_{kl} ∈ {p, p+1}. Notice that if two tableaux T and T′ are connected by an edge, then the corresponding isomorphisms Λ : h_{!}h^{∗} → [−2d] are equal by Proposition B.2.6.

The graph Γ is connected. Indeed, any tableau is connected by a path with a standard tableau. Any standard tableau is connected to the tableau

\overline{T} = (\begin{matrix}0 & 1 & m & ... & n − 2 & n \\ 0 & 0 & 2 & ... & n − 4 & n − 1 \\ 0 & 0 & 0 & ... & ... & n − 3 \\ ... & ... & ... & ... & ... & ... \\ 0 & 0 & 0 & ... & 0 & m − 1 \\ 0 & 0 & 0 & ... & 0 & 0 \end{matrix}) ,  (B.2.5)

where n = m(m − 1)/2, by a sequence of double moves – pairs of edges, which replace values p, p + 1 of a pair of entries (not contained in the same row or column) with values p, p, and then with p + 1, p. Let us say that the rating of entry (k, l) is \overline{T}_{kl}. Start with the highest rated entry for which T_{kl} ≠ \overline{T}_{kl}, hence, T_{kl} < \overline{T}_{kl}. Replace T_{kl} with T_{kl} + 1 using a double move, then with T_{kl} + 2 and so on until T_{kl} becomes equal to \overline{T}_{kl}. Iterating this procedure we transform T in \overline{T}.

Since Γ is connected, all isomorphisms Λ : h_{!}h^{∗} → [−2d] obtained above from Young tableaux are equal.

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