André-Quillen homology via functor homology

Teimuraz Pirashvili^{1}

We obtain André-Quillen homology for commutative algebras using relative homological algebra in the category of functors on finite pointed sets.

Keywords: André-Quillen homology, relative homological algebra, functor homology, Gamma spaces AMS classification: 13D03, 18G25.

1 Introduction.

Let Γ be the small category of finite pointed sets. For any n ≥ 0, let [n] be the set {0, 1, ..., n} with basepoint 0. We assume that the objects of Γ are the sets [n]. A left Γ-module is a covariant functor Γ → Vect to the category of vector spaces over a field K. For a left Γ-module F we put

π_{0}(F) := Coker(d_{0} − d_{1} + d_{2} : F ([2]) → F ([1])),

where d_{1} is induced by the folding map [2] → [1], 1, 2 ↦ 1 while d_{0} and d_{2} are induced by the projection maps [2] → [1] given respectively by 1 ↦ 1, 2 ↦ 0 and 1 ↦ 0, 2 ↦ 1. The category Γ-mod of left Γ-modules is an abelian category with enough projective and injective objects. Therefore one can form the left derived functors of the functor π_{0} : Γ-mod → Vect, which we will denote by π_{∗}. Thanks to [5] and [6] we know that π_{∗}F is isomorphic to the homotopy of the spectrum corresponding to the Γ-space F according to Segal (see [9] and [1]).

Let A be a commutative algebra over a ground field K and let M be an Amodule. There exists a functor L(A, M) : Γ → Vect, which assigns M ⊗ A^{⊗n} to [n] (see [3] or section 3). Here all tensor products are taken over K. It was proved in [7] that π_{∗}(L(A, M)) is isomorphic to a brave new algebra version of André-Quillen homology H_{∗}^{Γ}(A, M) constructed by Alan Robinson and Sarah Whitehouse [10]. The main result of this paper shows that a similar isomorphism exists also for André-Quillen homology if one takes an appropriate relative derived functors of the same functor π_{0} : Γ-mod → Vect.

1The author was partially supported by the grant INTAS-99-00817 and by the TMR network K-theory and algebraic groups , ERB FMRX CT-97-0107.

2 A class of proper exact sequences

Thanks to the Yoneda lemma Γ^{n}, n ≥ 0, are projective generators of the category Γ-mod. Here

Γ^{n} : = K[Hom_{Γ}([n], −)].

and K[S] denotes the free vector space generated by a set S. For left Γ-modules F and T one defines the pointwise tensor product F ⊗ T to be the left Γ-module given by (F ⊗ T)([n]) = F ([n]) ⊗ T ([n]). Since Γ^{n} ⊗ Γ^{m} ≅ Γ^{n+m} one sees that the tensor product of two projective left Γ-modules is still projective. We also have Γ^{n} ≅ (Γ^{1})^{⊗n}.

A partition λ = (λ_{1}, ⋯, λ_{k}) is a sequence of natural numbers λ_{1} ≥ ⋯ ≥ λ_{k} ≥ 1. The sum of partition is given by s(λ) := λ_{1} + ⋯ + λ_{k}, while the group Σ(λ) is a product of the corresponding symmetric groups

Σ(λ) := Σ_{λ_{1}} × ⋯ × Σ_{λ_{k}}.

which is identified with the Young subgroup of Σ_{s(λ)}. Let us observe that Σ_{n} = Aut_{Γ}([n]) and therefore Σ_{n} acts on Γ^{n} ≅ (Γ^{1})^{⊗n}. For a partition λ with s(λ) = n we let Γ(λ) be the coinvariants of Γ^{n} under the action of Σ(λ) ⊂ Σ_{n}.

For a vector space V we let S^{∗}(V), Λ^{∗}(V) and D^{∗}(V) be respectively the symmetric, exterior and divided power algebra generated by V. Let us recall that S^{n}(V) = (V^{⊗n})/Σ_{n} is the space of coinvariants of V^{⊗n} under the action of the symmetric group Σ_{n}, while D^{n}(V) = (V^{⊗n})^{Σ_{n}} is the space of invariants. Moreover for a particiaon λ = (λ_{1}, ⋯, λ_{k}) we put

S^{λ} := S^{λ_{1}} ⊗ ⋯ ⊗ S^{λ_{k}}.

A similar meaning has also Λ^{λ} and D^{λ}. It follows from the definition that

Γ(λ) ≅ S^{λ} ∘ Γ^{1}.

In particular Γ(1, ⋯, 1) ≅ Γ^{n} and Γ(n) ≅ S^{n} ∘ Γ^{1}.

Let

0 → T_{1} → T → T_{2} → 0

be an exact sequence of left Γ-modules. It is called a Y-exact sequence if for any partition λ with s(λ) = n the induced map

T ([n])^{Σ(λ)} → T_{2}([n])^{Σ(λ)}

is surjective. Here and elsewhere, M^{G} denotes the subspace of G-fixed elements of a G-module M. For a Y-exact sequence 0 → T_{1} → T → T_{2} → 0 the sequence

0 → T_{1}([n])^{Σ(λ)} → T ([n])^{Σ(λ)} → T_{2}([n])^{Σ(λ)} → 0

is also exact. Following to Section XII.4 of [4] we introduce the related notions. An epimorphism f : F → T is called Y-epimorphism if

0 → Ker(f) → F → T → 0

is a Y-exact sequence. Similarly, a monomorphism f : F → T is called Ymonomorphism if

0 → F → T → Coker(f) → 0

is a Y-exact sequence. A morphism f : F → T is called Y-morphism if F → Im(f) is a Y-epimorphism and Im(f) → T is a Y-monomorphism. A left Γ-module Z is called Y-projective if for any Y-epimorphism f : F → T and any morphism g : Z → T there exist a morphism h : Z → F such that g = fh.

Lemma 2.1 i) If a short exact sequence is isomorphic to a Y-exact sequence, then it is also a Y-exact sequence.

ii) A split short exact sequence is Y-exact.

iii) A composition of two Y-epimorphisms is still a Y-epimorphism.

iv) If f and g are two composable epimorphism and fg is a Y-epimorphism, then f is also a Y-epimorphism.

v) A composition of two Y-monomorphisms is still a Y-monomorphism.

vi) If f and g are two composable monomorphism and fg is a Y-monomorphism, then g is also a Y-monomorphism.

Proof. The properties i)- iv) are clear. Let f : B → C and g : A → B be monomorphisms. One can form the following diagram

\xymatrix{ &  & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & A \ar[r]^{g} \ar[d]^{1_{A}} & B \ar[r] \ar[d]^{f} & X \ar[r] \ar[d] & 0 \\ 0 \ar[r] & A \ar[r]^{fg} & C \ar[r] \ar[d] & Z \ar[r] \ar[d] & 0 \\  &  & Y \ar[r]^{1_{Y}} \ar[d] & Y \ar[d] \\  &  & 0 & 0}

Assume f and g are are Y-monomorphisms, then for any partition λ with s(λ) = n one has a commutative diagram

\xymatrix{ &  & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & A([n])^{Σ(λ)} \ar[r] \ar[d]^{1_{A}} & B([n])^{Σ(λ)} \ar[r] \ar[d] & X([n])^{Σ(λ)} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & A([n])^{Σ(λ)} \ar[r] & C([n])^{Σ(λ)} \ar[r]^{h} \ar[d] & Z([n])^{Σ(λ)} \ar[d] \\  &  & Y ([n])^{Σ(λ)} \ar[r]^{1_{Y}} \ar[d] & Y ([n])^{Σ(λ)} \\  &  & 0}

The diagram chasing shows that h is an epimorphism and therefore fg is a Ymonomorphism and v) is proved. Assume now fg is a Y-monomorphism. Them we have the following commutative diagram

\xymatrix{ &  & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & A([n])^{Σ(λ)} \ar[r] \ar[d]^{1_{A}} & B([n])^{Σ(λ)} \ar[r]^{l} \ar[d] & X([n])^{Σ(λ)} \ar[d] \\ 0 \ar[r] & A([n])^{Σ(λ)} \ar[r] & C([n])^{Σ(λ)} \ar[r] \ar[d] & Z([n])^{Σ(λ)} \ar[d] \ar[r] & 0 \\  &  & Y ([n])^{Σ(λ)} \ar[r]^{1_{Y}} & Y ([n])^{Σ(λ)}}

The diagram chasing shows that l is an epimorphism and therefore f is a Ymonomorphism and therefore we get vi). □

As an immediate corollary we obtain that the class of all Y-exact sequences is proper in the sense of Mac Lane [4]. We now show that there are enough Y-projective objects.

Lemma 2.2 i) For any partition λ the left Γ-module Γ(λ) is a Y-projective object.

ii) A morphism f : F → T of left Γ-modules is Y-epimorphism iff for any partition λ the induced morphism

Hom_{Γ −mod}(Γ(λ), F) → Hom_{Γ −mod}(Γ(λ), T)

is an epimorphism.

iii) For any left Γ-module F there is a Y-projective object Z and Y-epimorphism f : Z → F.

iv) Any projective Y-module is a direct summand of the sum of objects of the form Γ(λ).

v) The tensor product of two Y-projective left Γ-modules is still Y-projective.

Proof. Let λ be a partition with s(λ) = n. By definition Γ(λ) = H_{0}(Σ(λ), Γ^{n}). Hence for any left Γ-module F one has

Hom_{Γ −mod}(Γ(λ), F) ≅ H^{0}(Σ(λ), Hom_{Γ −mod}(Γ^{n}, F) ≅ F (n)^{Σ(λ)}.

The assertions i) and ii) are immediate consequence of this isomorphisms. To proof iii) we set

X(λ) := Hom_{Γ −mod}(Γ(λ), F).

Moreover, for each x ∈ X(λ) we let f_{x} : Γ(λ) → F be the corresponding morphism. Take Z = ⨁_{λ} ⨁_{x∈X(λ)} Γ(λ). Then the collection f_{x}, x ∈ X(λ) yields the morphism f : Z → F. We have to show that it is a Y-epimorphism. Let g : Γ(λ) → F be a morphism of left Γ-modules. By ii) we need to lift g to Z. By our construction g ∈ X(λ) and therefore the inclusion Γ(λ) → Z corresponding to the summand g ∈ X(λ) is an expected lifting and iii) is proved. The proof of iii) shows that one can assume P to be a sum of Γ^{λ} and iv) follows. To proof the last statement one observes that, for any partitions λ and μ one has

Γ(λ) ⊗ Γ(μ) ≅ (Γ^{s(λ)} ⊗ Γ^{s(μ)})^{Σ(λ)×Σ(μ)} = (Γ^{s(λ)+s(μ)})^{Σ(λ)×Σ(μ)}

and therefore Γ(λ) ⊗ Γ(μ) is Y-projective. □

3 Definition of André-Quillen homology and the functor L

The definition André-Quillen homology is based on the framework of homotopical algebra [8] and it is given as follows. We let C_{∗}(V_{∗}) be the chain complex associated to a simplicial vector space V_{∗}. Let A be a commutative algebra over a ground field K and let M be an A-module. A simplicial resolution of A is an augmented simplicial object P_{∗} → A in the category of commutative algebras, which is a weak equivalence (in other words C_{∗}(P_{∗}) → A is a weak equivalence). A simplicial resolution is called free if P_{n} is a polynomial algebra over K for all n ≥ 0. Any commutative algebra posses a free simplicial resolution which is unique up to homotopy. Then the André-Quillen homology is defined by

D_{∗}(A, M) := H_{∗}(C_{∗}(Ω_{P_{∗}}^{1} ⊗_{P_{∗}} M)),

where Ω^{1} is the Kähler 1-differentials and P_{∗} → A is a free simplicial resolution. In the dimension 0 we have D_{0}(A, M) ≅ Ω_{A}^{1} ⊗_{A} M.

As we mentioned the functor L(A, M) : Γ → Vect is given on objects by [n] ↦ M ⊗ A^{⊗n}. For a pointed map f : [n] → [m], the action of f on L(A, M) is given by

f_{∗}(a_{0} ⊗ ⋯ ⊗ a_{n}) : = b_{0} ⊗ ⋯ ⊗ b_{m},

where b_{j} = ∏_{f(i)=j} a_{i}, j = 0, ⋯, n.

Example 3.1 Let M = A = K[t]. In this case one has an isomorphism

L(K[t], K[t]) ≅ S^{∗} ∘ Γ^{1}.

To see this isomorphism, one observes that Γ^{1} assigns the free vector space on a set [n] to [n] and therefore both functors in the question assign the ring K[t_{0}, ⋯, t_{n}] to [n]. An important consequence of this isomorphism is the fact that the functor L(K[t], K[t]) is Y-projective.

Lemma 3.2 For any commutative algebra A and any A-module M, one has a natural isomorphism π_{0}(L(A, M)) ≅ Ω_{A}^{1} ⊗_{A} M.

Proof. This is a consequence of Proposition 1.15 and Proposition 2.2 of [5]. □

Lemma 3.3 i) Let A be a commutative algebras and let

0 → M_{1} → M → M_{2} → 0

be a short exact sequence of A-modules. Then

0 → L(A, M_{1}) → L(A, M) → L(A, M_{2}) → 0

is a Y-exact sequence.

ii) Let f : B → A be a surjective homomorphism of commutative algebras, then for any A-module M the induced morphism of left Γ-modules

L(B, M) → L(A, M)

is a Y-epimorphism.

Proof. One observes that for any partition λ with s(λ) = n one has

(L(A, M)([n]))^{Σ(λ)} = (M ⊗ A^{⊗n})^{Σ(λ)} ≅ M ⊗ D^{λ}(A).

Since we are over field the tensor product is exact and we obtain i). By the same reason f has a linear section, which yields also a linear section of D^{λ}(B) → D^{λ}(A), because D^{λ} is a functor defined on the category of vector spaces . □

4 Relative derived functors

By Lemma 2.2 the class of Y-exact sequences has enough projective objects. Thanks to [4] this allows us to construct the relative derived functors. Let us recall that an augmented chain complex X_{∗} → F is called a Y-resolution of F if it is exact (that is H_{i}(X_{∗}) = 0, for i > 0 and H_{0}(X_{∗}) ≅ F) and all boundary maps X_{n+1} → X_{n} are Y-morphisms, n ≥ 0. It follows from Lemma 2.2 that X_{∗} → F is a Y-resolution iff for any partition λ the augmented complex

Hom_{Γ −mod}(Γ(λ), X_{∗}) → Hom_{Γ −mod}(Γ(λ), F)

is exact. A Y-resolution Z_{∗} → F is called Y-projective resolution if for all n ≥ 0 the left Γ-nodule Z_{n} is a Y-projective object. We define π_{∗}^{Y}(F) using relative derived functors of the functor π_{0} : Γ-mod → Vect. In other words we put

π_{n}^{Y}(F) := H_{n}(π_{0}(Z_{∗})), n ≥ 0,

where Z_{∗} → F is a Y-projective resolution. By [4] this gives the well-defined functors π_{n}^{Y} : Γ-mod → Vect, n ≥ 0.

Lemma 4.1 If K is a field of chracteristic zero, then π_{∗}(F) ≅ π_{∗}^{Y}(F).

Proof. In this case all exact sequences are Y-exact, because for any finite group G, the functor M ↦ M^{G} is exact. □

Lemma 4.2 For left Γ-modules F, T one has an isomorphism

π_{∗}^{Y}(F ⊗ T) ≅ π_{∗}^{Y}(F) ⊗ T ([0]) ⊕ F ([0]) ⊗ π_{∗}^{Y} (T).

Proof. The result in the dimension 0 is known (see Lemma 4.2 of [5]). Let Z_{∗} → F and R_{∗} → T be Y-projective resolutions. By Lemma 2.2 Z_{∗} ⊗ R_{∗} → F ⊗ T is aslo Y-projective resolution. Thus

π_{∗}^{Y}(F ⊗ T) = H_{∗}(π_{0}(Z_{∗} ⊗ R_{∗})) ≅
H_{∗}(π_{0}^{Y}(Z_{∗}) ⊗ R_{∗}([0]) ⊕ Z_{∗}([0]) ⊗ π_{0}^{Y} (R_{∗})) ≅
π_{∗}^{Y}(F) ⊗ T ([0]) ⊕ F ([0]) ⊗ π_{∗}^{Y} (T),

where the last isomorphism follows from the Eilenberg-Zilber theorem and Künneth theorem. □

Lemma 4.3 Let ϵ : X_{∗} → A be a simplicial resolution in the category of commutative algebras and let M be an A-module. Then the associated chain complex of the simplicial Γ-module C_{∗}(L(X_{∗}, M)) → L(A, M) is a Y −resolution.

Proof. Since ϵ is weak equivalence of simplicial algebras it is a homotopy equivalence in the category of simplicial vector spaces. Thus M⊗D^{λ}(X_{∗}) → M⊗D^{λ}(A_{∗}) is also a homotopy equivalence, for any particion λ. It follows that

L(X_{∗}, M)([n])^{Σ(λ)} → L(A, M)([n])^{Σ(λ)}

is also a homotopy equivalence of simplicial vector spaces. □

The following is our main result.

Theorem 4.4 For any commutative ring A and any A-module M, there is a canonical isomorphism

D_{i}(A, M) ≅ π_{i}^{Y}(L(A, M)), i ≥ 0

between the André-Quillen homology and relative derived functors of π_{0} applied on the functor L(A, M).

Proof. Thanks to Lemma 3.2 the result is true for i = 0. First consider the case, when M = A = K[t]. In this case André-Quillen homology vanishes in positive dimensions by definition. On the other hand L(K[t], K[t]) is Y-projective thanks to Example 3.1 and therefore π_{i}^{Y}(L(A, M)) vanishes for all i > 0. One can use Lemma 4.2 to conclude that π_{i}^{Y}(L(A, A)) vanishes for all i > 0 provided A is a polynomial algebra. For the next step, we proof that the result is true if A is a polynomial algebra and M is any A-module. We have to prove that π_{i}^{Y}(L(A, M)) also vanishes for i > 0. We already proved this fact if M = A. By additivity the functor π_{i}^{Y}(L(A, −)) vanishes on free A-modules. By Lemma 3.3 the functor π_{∗}^{Y}(L(A, −)) assignes the long exact sequence to a short exact sequence of Amodules. Therefore we can consider such an exact sequence associated to a short exact sequence of A-modules

0 → N → F → M → 0

with free F. Since the result is true if i = 0 one obtains by induction on i, that π_{i}^{Y}(L(A, M)) = 0 provided i > 0. Now consider the general case. Let P_{∗} → A be a free simplicial resolution in the category of commutative algebras. Then we have

Ω_{P_{∗}}^{1} ⊗_{P_{∗}} M ≅ π_{0}^{Y}(L(P_{∗}, M))

Thanks to Lemma 4.3 C_{∗}(L(P_{∗}, M)) → L(A, M) is a Y-resolution consisting with π_{∗}^{Y}-acyclic objects and the result follows. □

The main theorem together with the main result of [7] yields:

Corollary 4.5 If Char(K) = 0, then for any commutative algebra A and any A-module M one has a natural isomorphism

D_{∗}(A, M) ≅ H_{∗}^{Γ}(A, M).

This fact was also proved in [10] based on the combinatorical and homotopical analysis of the space of fully grown trees.

Remarks. i) We let t : Γ^{op} → Vect be the functor which assignes the vector space of all maps f : [n] → K, f(0) = 0 to [n]. Then t⊗_{Γ}F ≅ π_{0}(F) (see Proposition 2.2 [5]). Hence π_{∗}^{Y} van be also defined as the relative derived functors of the functor t⊗_{Γ} (−) : Γ-nod → Vect. More generally one can take any functor T : Γ^{op} → Vect and define Tor_{∗}^{Y}(T, F) as the value of the relative derived functors (with respect of Y-exact sequences) of the functor T ⊗_{Γ} (−) : Γ-nod → Vect. Then our result claims that

D_{∗}(A, M) ≅ Tor_{∗}^{Y}(t, L(A, M)).

Based on the Proposition 1.15 of [5] the argument given above shows that

D_{∗}^{{n}}(A, M) ≅ Tor_{∗}^{Y}(Λ^{n} ∘ t, L(A, M)),

where D_{∗}^{{n}}(A, M) are defined using Kähler n-differentials:

D_{∗}^{{n}}(A, M) := H_{∗}(C_{∗}(Ω_{P_{∗}}^{n} ⊗_{P_{∗}} M))

and for n = 1 one recovers the main theorem.

ii) All results remains true if K is any commutative ring and A and M are projective as K-mudule.

Acknowledgments

This work was written during my visit at MPI at Bonn.

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