ON CUBIC FUNCTORS

YURI A. DROZD

Introduction

Polynomial functors appeared in algebraic topology ((cf. [8])) and proved to be useful for lots of questions in homotopy theory. That is why their study seems to be of interest. Mostly, one deals with “continuous” polynomial functors, which are defined by their values on free groups, and in what follows we consider only such ones. In [7] the author gave a description of all (finitely generated) quadratic functors. This description was done using the technique of the so called matrix problems, namely, representations of bunches of chains. The reduction to such a matrix problem depends on the presentation of quadratic functors as modules over a special ring, which happens to be an order in a semi-simple algebra. Then we used the procedure of [6]. Now we are going to pass the same way for cubic functors. We check that the corresponding ring is again an order in a semi-simple algebra. Unfortunately, in this case, the classification of modules is a wild problem, i.e., it includes, in some sense, the classification of all representations of all finitely generated algebras over the residue field ℤ/2 (cf. Section 2). Thus, there can be no “good” description of all cubic functors.

On the contrary, such a description becomes possible (and rather analogous to that of quadratic functors) if we “make 2 invertible,” i.e., consider cubic modules over the ring ℤ[ 1/2 ] . Such a description is given in Section 3. Just as for quadratic case, this classification problem is tame, i.e., indecomposable modules depend on some “discrete” combinatorial data and on at most one “continuous parametre,” which is an irreducible polynomial from ℤ/2[ t] (or, the same, a closed point of 𝔸_{ℤ/2}^{1} = Spec ℤ/2[ t] ). We also consider weakly alternative cubic functors F , i.e., those with F (ℤ) = 0 . Their classification given in Section 4 is also tame, though this time the corresponding ring has nilpotent ideals. At last, we give a classification of cubic functors with the image being vector spaces (Section 5) as well as torsion free ones (Section 6). These problems also happens to be tame. We end up with one conjecture concerning polynomial functors of higher degrees and some corollaries from this conjecture.

The author was partially supported by CRDF Award UM2-2094.

1. Generalities

We suppose all categories to be pre-additive, i.e., such that their morphism sets are abelian groups and the composition is bi-additive. On the contrary, we do not suppose the functors to be additive, though we always suppose that F (0) = 0 for a zero morphism. Remind that a fully additive category is an additive category such that every idempotent in it splits. If F : A → B is a functor from an additive category A to a fully additive category B and A = ⨁_{k=1}^{n} A_{k} is an object from A , consider the corresponding embeddings i_{k} : A_{k} → A and projections p_{k} : A → A_{k}. Put e_{k} = i_{k}p_{k} and f(k) = F (e_{k}) . Certainly, e_{k}, hence, f(k) , are pairwise orthogonal idempotents and Im f(k) ≃ F (A_{k}) . Moreover, put f(kl) = F (e_{k} + e_{l}) − f(k) − f(l) ( k < l). Then f(kl) are also idempotents, pairwise orthogonal and orthogonal to all f(k) . Hence, F (A) has a direct summand (⨁_{k} F (A_{k})) ⊕ (⨁_{k<l} F_{2}(A_{k}|A_{l})) , where one denotes F_{2}(A_{k}|A_{l}) = Im f(kl) . Define recursively for any 1 ≤ k_{1} < k_{2} < ⋯ < k_{r} ≤ n

f(k_{1}k_{2} … k_{m}) = F (e_{k_{1}} + … e_{k_{m}}) − ∑_{r<m} ∑_{1≤l_{1}<···<l_{r}≤m} f(k_{l_{1}}k_{l_{2}} … k_{l_{r}})

and F_{m}(A_{1}|A_{2}| … |A_{m}) = Im f(k_{1}k_{2} … k_{m}) . Then

F (A) = ⨁_{m≤n} ( ⨁_{k_{1}k_{2}...k_{m}} F_{m}(A_{k_{1}}|A_{k_{2}}| … |A_{k_{m}})) .

One easily sees that F_{m} is indeed a functor A^{m} → B . It is called the m-th cross-effect of F . Of course, F_{1} = F . If there is a positive integer n such that F_{m} = 0 for m > n, one calls F a polynomial functor. The smallest possible n with this property is called the degree of F . Certainly, the functors of degree 1 are just additive ones. The functors of degree 1 are called linear, of degree 2 quadratic, of degree 3 cubic, etc. For an arbitrary pre-additive category A , one can always consider its fully additive hull add A and we call polynomial functors A → B those from add A to B . Of course, for additive functors we get in this way nothing new as every additive functor A → B can be uniquely (up to isomorphism) prolonged to an additive functor add A → B .

In what follows, we consider the case when A = ℤ, the ring of integers, and B = R-Mod , the category of modules over a ring R , mainly even B = R-mod , the category of finitely generated R-modules. Note that add ℤ = fab , the category of finitely generated free abelian groups.

In this case polynomial functors are called polynomial R-modules and the category of finitely generated polynomial R-modules of degree at most n is denoted by R-mod^{n}. Of course, R-mod^{1} coincide with the category of finitely generated R-modules. If R = ℤ, we are just speaking of “polynomial modules” not precising the target category.

To define a polynomial functor F : fab → B , one only has to define the following data:

• objects F_{m} = F_{m}(ℤ|ℤ| … |ℤ) ;

• morphisms h_{k}^{m} : F_{m} → F_{m+1} that are compositions of the morphisms

F_{m}(A_{1}|A_{2}| … |A_{m}) → F (A_{1} ⊕ A_{2} ⊕ … ⊕ A_{m})
→ F (A_{1} ⊕ … ⊕ A_{k} ⊕ A_{k} ⊕ … ⊕ A_{m})
→ F_{m+1}(A_{1}| … |A_{k}|A_{k}| … |A_{m}) ,

where all A_{j} = ℤ, the first morphism is the embedding of the direct summand, the last one is the projection onto the direct summand and the middle one is F (γ) , γ|_{A_{j}} being identity if j ≠ k and γ|_{A_{k}} being the diagonal embedding A_{k} → A_{k} ⊕ A_{k};

• morphisms p_{k}^{m} : F_{m+1} → F_{m} that are compositions of the morphisms

F_{m+1}(A_{1}| … |A_{k}|A_{k}| … |A_{m}) → F (A_{1} ⊕ … ⊕ A_{k} ⊕ A_{k} ⊕ … ⊕ A_{m})
→ F (A_{1} ⊕ A_{2} ⊕ … ⊕ A_{m})
→ F_{m}(A_{1}|A_{2}| … |A_{m}) ,

where again all A_{j} = ℤ, the first morphism is the embedding of the direct summand, the last one is the projection onto the direct summand and the middle one is F (β) , β|_{A_{j}} being identity for j ≠ k, while β|_{A_{k}} being the codiagonal (summation) mapping A_{k} ⊕ A_{k} → A_{k}.

In particular, a cubic R-module F , where R is a ring, is defined by a diagram of R-modules:

F_{1}  ⟶^{h} ; ⟵^{p} F_{2}  ⟶^{h_{1}} ; ⟶^{h_{2}} ; ⟵^{p_{1}} ; ⟵^{p_{2}} F_{3} (1)

One can show that such a diagram corresponds to a cubic module if and only if the following relations hold:

\begin{matrix}h_{1}p_{2} = h_{2}p_{1} = 0 , h_{1}h = h_{2}h , pp_{1} = pp_{2} , \\ h_{i}p_{i}h_{i} = 2h_{i} , p_{i}h_{i}p_{i} = 2p_{i} (i = 1, 2) , \\ hph = 2(h + (p_{1} + p_{2})\overline{h}) , php = 2(p + \overline{p}(h_{1} + h_{2})) , \\ \overline{h}p + h_{1} + h_{2} = h_{1}p_{1}h_{2}p_{2}h_{1} + h_{2}p_{2}h_{1}p_{1}h_{2} , \\ h\overline{p} + p_{1} + p_{2} = p_{1}h_{2}p_{2}h_{1}p_{1} + p_{2}h_{1}p_{1}h_{2}p_{2} , \end{matrix} (2)

where \overline{h} = h_{1}h = h_{2}h and \overline{p} = pp_{1} = pp_{2} [3].

Consider the pre-additive category A with three objects 1, 2, 3 and generating morphisms

h : 1 → 2 , p : 2 → 1 , h_{i} : 2 → 3 , p_{i} : 3 → 2 (i = 1, 2)

subject to the relations (2). Then cubic R-modules are just linear functors A → R-mod . Sometimes we identify A with the endomorphism ring End(1 ⊕ 2 ⊕ 3) . Obviously, the categories of modules over A and over this ring coincide.

2. Wildness

We are going to show that the description of all cubic functors is a wild problem in the sense of the representation theory (cf. [5]). Moreover, we show that it is even true for those cubic functors “freely generated by their 1-part,” or, the same, for A(1, 1)-modules.

Proposition 2.1. The ring A(1, 1) is isomorphic to the subring of ℤ^{3} with the basis { (1, 1, 1), (2, 0, 0), (0, 6, 0) }.

Proof. Put a = ph − \overline{ph}, b = ph. One easily verifies that these two elements generate A(1, 1) , ab = ba = 0 , a^{2} = 2a and b^{2} = 6b. Hence, one gets an isomorphism by mapping a ↦ (2, 0, 0) , b ↦ (0, 6, 0) .

The category of A(1, 1)-modules can be embedded into that of Amodules. Namely, every A(1, 1)-module M gives rise to a cubic module F , where

F_{1} = M , F_{2} = A(1, 2) ⊗_{A(1,1)} M and F_{3} = A(1, 3) ⊗_{A(1,1)} M

with the obvious action of morphisms. It is known (and easy to check) that this procedure really defines a functor which is a full embedding.

Let Σ_{n} be the free (non-commutative) algebra ℤ/4⟨ x_{1}, x_{2}, …, x_{n} ⟩ with n generators over the residue ring ℤ/4 .

Proposition 2.2. Denote by φ : A(1, 1) → Σ_{2} the homomorphism mapping a to 2x_{1} and b to 2x_{2} . For any Σ_{2}-module L, let ^{φ}L be the A(1, 1)-module obtained from L via the “change of rings” φ. Then, for any Σ_{2}-modules L, L′ which are free as ℤ/4-modules,

• ^{φ}L ≃ ^{φ}L′ as A(1, 1)-modules if and only if L/2L ≃ L′/2L′ as Σ_{2}-modules;

• A(1, 1)-module ^{φ}L is indecomposable if and only if so is Σ_{2}- module L/2L.

The proof is evident.

Corollary 2.3. For every n, there is a cubic module M ∈ C(Σ_{n}) such that, for any Σ_{n}-modules L, L′ which are free over ℤ/4 ,

• M ⊗_{Σ_{n}} L ≃ M ⊗_{Σ_{n}} L′ if and only if L/2 ≃ L′/2 ;

• M⊗_{Σ_{n}} L is indecomposable if and only if L/2 is indecomposable.

Proof. One only has to consider the Σ_{2}- Σ_{n}-bimodule N which is free of rank n+1 over Σ_{n}, while the action of Σ_{2} is given by the following rules:

• x_{1} is acting as the (upper) Jordan cell;

• x_{2} is acting as the matrix

(\begin{matrix}0 & & & 0 & & & … & & 0 & & 0 \\ x_{1} & & & 0 & & & … & & 0 & & 0 \\ 0 & & & x_{2} & & & … & & 0 & & 0 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 0 & & & 0 & & & … & & x_{n} & & 0 \end{matrix})

Afterwards, one puts M = ^{φ}N , with φ defined as in Proposition 2.2. All properties are verified immediately (cf., e.g., [5]).

This Corollary shows that the classification of all cubic modules (even of A(1, 1)-modules) is at least so complicated as the classification of n-tuples of matrices over the filed ℤ/2 up to conjugations, for all n. It is just the situation when one calls such a problem “wild.”

3. 2-divisible case

In this section we consider the case of cubic 2-divisible modules, i.e., cubic functors F : fab → ℤ′-mod , where ℤ′ = ℤ[1/2] . It happens that here we are able to describe all (finitely generated) cubic modules. Moreover, this description is quite alike that of quadratic ℤ-modules given in [7]. Certainly, 2-divisible cubic modules correspond to the modules over the category A′ = A[1/2] .

Proposition 3.1. The category A′ is Morita-equivalent to the direct product ℤ′ × ℤ′ × B , where B is the subring in ℤ′ × Mat(2, ℤ′) × Mat(2, ℤ′) × ℤ′ consisting of all quadruples of the form:

(a, (\begin{matrix}b_{1} & 3b_{2} \\ b_{3} & b_{4} \end{matrix}) , (\begin{matrix}c_{1} & 3c_{2} \\ c_{3} & c_{4} \end{matrix}) , d)

with

a ≡ b_{1}, b_{4} ≡ c_{1}, c_{4} ≡ d (mod 3)

Remark. One can see that a module over the ring B is given by a diagram of abelian (2-divisible) groups of the following shape:

M_{1}  ⟶^{α_{1}} ; ⟵^{β_{1}} M_{2}  ⟶^{α_{2}} ; ⟵^{β_{2}} M_{3}, (3)

satisfying the relations:

α_{1}β_{1}α_{1} = 3α_{1} , β_{1}α_{1}β_{1} = 3β_{1} ,

α_{2}β_{2}α_{2} = 3α_{2} , β_{2}α_{2}β_{2} = 3β_{2} ,
β_{1}β_{2} = 0 , α_{2}α_{1} = 0 ,
α_{1}β_{1} + β_{2}α_{2} = 3 id_{2} .

We always identify B-modules with such diagrams.

Proof. Put A_{i} = A′(i, i) ( i = 1, 2, 3 ) and consider the following elements:

in A_{1} :

f_{1} = \frac{ph − \overline{p}\overline{h}}{2}, e_{1} = id_{1} −f_{1}, a_{1} = \frac{\overline{p}\overline{h}}{2};

in A_{2} :

e_{2} = \frac{p_{1}h_{1}}{2} , f_{2} = \frac{p_{2}h_{2}}{2} , g = id_{2} −e_{2} − f_{2},

u = hp, g_{1} = \frac{gu}{2}, g_{2} = g − g_{1},

v_{i} = p_{i}h_{j} (i, j = 1, 2, i ≠ j), v′_{1} = \frac{5v_{1} − v_{1}v_{2}v_{1}}{2} ,

a_{2} = v_{1}v_{2}, b_{2} = v_{2}v_{1};

in A_{3} :

u_{i} = h_{i}p_{i} (i = 1, 2), e_{3} = \frac{u_{1}}{2} , f_{3} = id_{3} −e_{3},

\overline{u} = \overline{h}\overline{p} + e_{3}, a_{3} = 2u_{2}e_{3} − \overline{u}, b_{3} = 2e_{3}u_{2} − \overline{u}; in A′(2, 1) :

p′ = p − \frac{\overline{p}(h_{1} + h_{2})}{2}.

On can verify that:

• elements e_{i}, f_{i}, g_{i} are orthogonal idempotents;

• elements e_{1}, f_{1}, a_{1} form a ℤ′-basis of A_{1} ;

• a_{1} = e_{1}a_{1}e_{1} and a_{1}^{2} = 3a_{1} ;

• elements e_{2}, f_{2}, g_{1}, g_{2}, a_{2}, b_{2}, v_{1}, v_{2}, v_{1}b_{2}, a_{2}v_{1} form a ℤ′-basis of A_{2} ;

• gA_{2} = A_{2}g = ⟨ g_{1}, g_{2} ⟩;

• a_{2} = e_{2}a_{2}e_{2} , b_{2} = f_{2}b_{2}f_{2} , a_{2}^{2} = 3a_{2} , b_{2}^{2} = 3b_{2} ;

• v′_{1}(v_{2}/2) = e_{2} , (v_{2}/2)v′_{1} = f_{2} ;

• p′(h/2) = f_{1} , (h/2)p′ = g_{1} ;

• elements e_{3}, f_{3}, a_{3}, b_{3}, a_{3}b_{3}, b_{3}a_{3} form a ℤ′-basis of A_{3} ;

• a_{3} = f_{3}a_{3}e_{3} , b_{3} = e_{3}b_{3}f_{3} , a_{3}b_{3}a_{3} = 3a_{3} , b_{3}a_{3}b_{3} = 3b_{3} ;

• p′(h/2) = f_{1} and (h/2)p′ = g_{1} ;

• (p_{1}/2)h_{1} = e_{2} and h_{1}(p_{1}/2) = e_{3} ;

• gp_{i} = h_{i}g = 0 ( i = 1, 2 ) and g_{2}h = pg_{2} = 0 ;

• f_{2}h_{i} = p_{i}f_{2} = 0 ( i = 1, 2 ). Consider the projective A′-modules:

E_{1} = A′(1, )e_{1}, F_{1} = A′(1, )f_{1},
E_{2} = A′(2, )e_{2}, F_{2} = A′(2, )f_{2}, G_{i} = A′(2, )g_{i},
E_{3} = A′(3, )e_{3}, F_{3} = A′(3, )f_{3},

They are projective generators of the category of A′-modules. The above equalities imply that:

F_{1} ≃ G_{1}, E_{2} ≃ F_{2}, E_{2} ≃ E_{3}; Hom(G_{i}, G_{j}) = 0 if i ≠ j; Hom(G_{i}, E_{j}) = Hom(E_{j}, G_{i}) = Hom(G_{i}, F_{3}) = Hom(F_{3}, G_{i}) = 0; Hom(E_{1}, F_{3}) = Hom(F_{3}, E_{1}) = 0; End E_{1} ≃ e_{1}A_{1}e_{1} = ⟨ e_{1}, a_{1} ⟩ ≃ D; End E_{2} ≃ e_{2}A_{2}e_{2} = ⟨ e_{2}, a_{2} ⟩ ≃ D; End F_{3} ≃ f_{3}A_{3}f_{3} = ⟨ f_{3}, a_{3}b_{3} ⟩ ≃ D; End G_{i} = ℤ′; Hom(E_{3}, F_{3}) = ⟨ b_{3} ⟩ ≃ ℤ′;

Hom(F_{3}, E_{3}) = ⟨ a_{3} ⟩ ≃ ℤ′;
Hom(E_{1}, E_{3}) = ⟨ \overline{p} ⟩ ≃ ℤ′;
Hom(E_{3}, E_{1}) = ⟨ \overline{h} ⟩ ≃ ℤ′.

Here D denotes the ring ℤ′[t]/(t^{2} −3) , which is isomorphic to the subring in ℤ′ ×ℤ′ consisting of the pairs { (a, b) | a ≡ b (mod 3) }. Therefore, the category of A′-modules is equivalent to that of E-modules, where E = End(G_{1} ⊕ G_{2} ⊕ E_{1} ⊕ F_{3} ⊕ E_{3}) ≃ ℤ′ × ℤ′ × B .

One can verify that the A′-modules G_{i} correspond to the functors S^{2} and Λ^{2} , where S and Λ^{r} denote, as usually, the r-th symmetric and exterior powers. Actually, these functor are quadratic, and it is the main reason that they “stand apart” really cubic ones.

Now the theory of 2-divisible cubic modules becomes quite parallel to that of quadratic modules [7] and one gets the main results just following the same way. As the proofs are also almost the same, we replace them by the exact references to the corresponding items from [7]. We denote by e_{i} ( i = 1, 2, 3 ) the natural idempotents in B :

e_{1} = (1, (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) , (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}) , 0) ,
e_{2} = (0, (\begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}) , (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) , 0) ,
e_{1} = (0, (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}) , (\begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}) , 1) .

Put B_{i} = Be_{i}; they are just indecomposable projective B-modules. Denote by L_{1} the projection of B_{1} onto the first component, by L_{2} its projection onto the second component; by L_{3} the projection of B_{2} onto the second component, by L_{4} its projection onto the third component; by L_{5} the projection of B_{3} onto the third and by L_{6} its projection onto the forth component. Then L_{i} are just irreducible torsion free B-modules. Put ℤ_{(p)} = { a/b | a, b ∈ ℤ, p ∤ b }, M_{(p)} = ℤ_{(p)} ⊗ M .

Theorem 3.2. 1. 2-divisible cubic modules M, N are isomorphic if

and only if M_{(p)} ≃ N_{(p)} for every odd prime p.

2. Given a set { N_{p} } ( p > 2 ) where N_{p} is a cubic ℤ_{(p)}-module, there is a 2-divisible cubic module M such that M_{(p)} ≃ N_{p} for all p if and only if almost all (i.e., all but a finite set) of them are torsion free (maybe, zero) and ℚ ⊗ N_{p} ≃ ℚ ⊗ N_{q} for all p, q. In this case, ℚ ⊗ M = ℚ ⊗ N_{p} and tM = ⨁_{p} tN_{p}.

Cf. [7, Theorem 2.1].

Now we define our main personages: string and band modules. We rearrange a bit the description given in [7, Section 4] to make it more comprehensible. We start with some notations and definitions.

Definition 3.3. 1. First define on the set { 1, 2, 3, 4, 5, 6 } an equivalence relation − with the only non-trivial equivalences 2 − 3 and 4−5 and a symmetric relation ∼ (not equivalence!) putting 1 ∼ 2, 3 ∼ 4, 5 ∼ 6 .

2. Define also the following mappings acting on every diagram of the form (3):

\begin{matrix}θ(11) = 3 id_{1} −β_{1}α_{1} , & θ(22) = β_{1}α_{1} , \\ θ(23) = α_{1} , & θ(32) = β_{1} , \\ θ(33) = β_{2}α_{2} , & θ(44) = α_{2}β_{2} \\ θ(45) = α_{2} , & θ(54) = β_{2} , \\ θ(55) = α_{2}β_{2} & θ(66) = 3 id_{3} −α_{2}β_{2} . \end{matrix} (4)

3. Define the function ν : { { 1, 2 } , { 3, 4 } , { 5, 6 } } → ℕ putting ν { 1, 2 } = 1, ν { 3, 4 } = 2, ν { 5, 6 } = 3 .

4. A primary polynomial over a field k is, by definition, a power of an irreducible polynomial with the leading coefficient 1 .

Now the string and band modules can be defined as follows.

Definition 3.4. 1. A string diagram D is a diagram of one of the following types:

\begin{aligned}j_{1} & & & & j_{2}j_{3} & & & & j_{2n−2}j_{2n−1} \\ & k_{1} & & k_{2} & & k_{3} & & ⋯ & & k_{2n−1} \\ & & i_{1}i_{2} & & & & i_{3}i_{4} & & & & i_{2n−1} \end{aligned} , (i)

or

\begin{aligned}& & j_{2}j_{3} & & & & j_{2n−2}j_{2n−1} \\ & k_{2} & & k_{3} & & ⋯ & & k_{2n−1} \\ i_{2} & & & & i_{3}i_{4} & & & & i_{2n−1} \end{aligned} , (ii)

or

\begin{aligned}j_{1} & & & & j_{2}j_{3} & & & & j_{2n} \\ & k_{1} & & k_{2} & & ⋯ & & k_{2n} \\ & & i_{1}i_{2} & & & & i_{2n−1}i_{2n} \end{aligned} , (iii)

where k ∈ ℕ, i_{m}, j_{m} ∈ { 1, 2, 3, 4, 5, 6 }, satisfying the following conditions:

• i_{2m−1} ∼ i_{2m} for every m = 1, …, n. (This condition is empty for diagrams (i) and (ii) if m = n and for diagrams (ii) if m = 1 ; nevertheless, in these cases we define i_{1} or i_{2n} so that this condition holds.)

• j_{2m+1} ∼ j_{2m} for every m = 1, …, n − 1 .

• i_{k} − j_{k} for every m = 1, …, 2n.

(This condition is empty for diagrams (i) and (ii) if m = 2n and for diagrams (ii) if m = 1 .)

2. The string B-module corresponding to a string diagram D is the B-module M = M^{D} with the generators

g_{1}, g_{2}, …, g_{n} , g_{m} ∈ M_{ν{ i_{2m−1},i_{2m} }} ,

and the defining relations

3^{k_{2m}}θ(i_{2m}j_{2m})g_{m} = 3^{k_{2m+1}}θ(i_{2m+1}j_{2m+1})g_{m+1}

for i = 0, …, n; we put here g_{0} = g_{n+1} = 0 and omit the relation with m = n for diagrams (i), (ii) and with m = 1 for diagrams (ii).

3. A string cubic module is one corresponding to a string B-module via the Morita-equivalence of Proposition 3.1.

Definition 3.5. 1. A string diagram D of type (iii) is said to be non-periodic if it cannot be obtained by a repetition of a smaller diagram D_{1} of the same type: D ≠ D_{1}D_{1} … D_{1} .

2. A band data is a pair B = (D, f(t)) consisting of a non-periodic string diagram of type (iii) with the additional condition:

• j_{2n} ∼ j_{1} , and of a primary polynomial f(t) = λ_{1} + λ_{2}t + ⋯ + λ_{d}t^{d−1} + t^{d} over the residue field ℤ/3 with λ_{1} ≠ 0 .

3. The band B-module corresponding to a band data B is the Bmodule M = M^{B} with the generators

g_{ml} (m = 1, …, n, l = 1, …, d) , g_{ml} ∈ M_{ν{ i_{2m−1},i_{2m} }} ,

and the defining relations:

3^{k_{2m}}θ(i_{2m}j_{2m})g_{ml} = 3^{k_{2m+1}}θ(i_{2m+1}j_{2m+1})g_{m+1,l} if 1 ≤ m < n;
3^{k_{2n}}θ(i_{2n}j_{2n})g_{nl} = 3^{k_{1}}θ(i_{1}j_{1})g_{1,l+1} if 1 ≤ l < d;
3^{k_{2n}}θ(i_{2n}j_{2n})g_{nd} = −3^{k_{1}}θ(i_{1}j_{1}) ∑_{ν=1}^{d} λ_{ν}g_{1ν} .

4. A band cubic module is one corresponding to a band B-module via the Morita-equivalence of Proposition 3.1.

Put also L(i, p, k) = L_{i}/p^{k}L_{i}, where k is a positive integer, p > 3 is a prime number, i ∈ { 1, 2, 4, 6 }; Λ^{2}(p, k) = Λ^{2}/p^{k}Λ^{2} and S^{2}(p, k) = S^{2}/p^{k}S^{2} , where p ≥ 3 is again a prime number.

Theorem 3.6. 1. The string and band cubic modules defined above, the modules L(i, p, k) and the modules Λ^{2}(p, k), S^{2}(p, k) are just all finitely generated indecomposable 2-divisible cubic modules.

2. The only isomorphisms between these modules are the following:

• M^{D} ≃ M^{D^{∗}}, where D^{∗} denotes the diagram symmetric to D , the latter being of type (ii) or (iii);

• M^{B} ≃ M^{B^{(s)}} , where B^{(s)} = (D^{(s)}, f(t)) and D^{(s)} is the s-th shift of the diagram D , i.e., the diagram

j_{2s+1}  j_{2s+2}j_{2s+3}  j_{2s}
k_{2s+1}  k_{2s+2}  ⋯  k_{2s}
i_{2s+1}i_{2s+2}  i_{2s−1}i_{2s}

• M^{B} ≃ M^{B^{∗}^{(s)}}, where (D, f(t))^{∗} = (D^{∗}, f^{∗}(t)) and f^{∗}(t) = λ_{1}^{−1}t^{d}f(1/t) ;

Cf. [7, Theorem 5.1]

Corollary 3.7. 1. A 2-divisible cubic module M is of finite projective dimension if and only if it contains no string summands of types (i) and (iii). In this case either M is projective (hence, a direct sum of modules isomorphic to B_{i}) or pr. dim M = 1 . Otherwise, pr. dim M = ∞.

2. Every 2-divisible cubic module M has a periodic projective resolution

⋯ → P_{n+1} ⟶^{γ_{n}} P_{n} ⟶^{γ_{n−1}} … ⟶^{γ_{1}} P_{1} ⟶^{γ_{0}} P_{0} → M → 0

of period 6, namely, such that γ_{n+6} = γ_{n} for every n ≥ 2 .

3. If M is an indecomposable non-projective 2-divisible cubic module, tM is its torsion part, then

M/tM ≃ \begin{cases}L_{i_{2n}} & if M is a string module of type (i) \\ L_{i_{1}} ⊕ L_{i_{2n}} & if M is a string module of type (ii) \\ 0 & otherwise \end{cases}

(Remind that here i_{2n} denotes the unique index such that i_{2n−1} ∼ i_{2n} and i_{1} denotes the unique index such that i_{2} ∼ i_{1} .)

Cf. [7, Corollaries 5.3, 5.5].

4. Weakly alternative functors

Consider now the case of cubic ℤ-modules F such that F (ℤ) = 0 . We call them weakly alternative. They are actually modules over the category A^{a} = A/(id_{1}) . The functors Λ^{3} and Λ^{2} ⊗Id are of this sort. Here again we are able to obtain a complete description.

Denote by C_{0} the subring in ℤ × Mat(2, ℤ) consisting of the pairs (a, B) with b_{12} ≡ 0, a ≡ b_{11} (mod 3) .

Proposition 4.1. The category A^{a} is Morita equivalent to the semidirect product C = (ℤ × C_{0}) ⋉ T , where T is the elementary 2-group with three generators ξ, η, θ such that the following relations hold:

θ = ξη, ηξ = 0,
ηe = η, eξ = ξ, where e = (1, 0, 0),
ξ (0, a, B) = aξ and (0, a, B) η = aη

for any pair (a, B) ∈ C_{0} .

Proof. The category A^{a} has two objects: 2, 3 , and is generated by the morphisms h_{i} : 2 → 3, p_{i} : 3 → 2 (i=1,2) subject to the following relations, cf. diagram (1):

h_{i}p_{i}h_{i} = 2h_{i} , p_{i}h_{i}p_{i} = 2p_{i} , h_{i}p_{j} = 0 if i ≠ j ,
h_{1} + h_{2} = h_{1}p_{1}h_{2}p_{2}h_{1} + h_{2}p_{2}h_{1}p_{1}h_{2} ,
p_{1} + p_{2} = p_{1}h_{2}p_{2}h_{1}p_{1} + p_{2}h_{1}p_{1}h_{2}p_{2} ,

that imply:

h_{i}p_{i} = h_{i}p_{i}h_{j}p_{j}h_{i}p_{i} ,
2p_{i} = 2p_{i}h_{j}p_{j}h_{i}p_{i} ,
2h_{i} = 2h_{i}p_{i}h_{j}p_{j}h_{i}

(always i, j ∈ { 1, 2 } , i ≠ j). Then e_{i} = p_{i}h_{j}p_{j}h_{i} are orthogonal idempotents in A^{a}(2, 2) which are conjugate as e_{1}p_{1}h_{2} = p_{1}h_{2}e_{2} and e_{2}p_{2}h_{1} = p_{2}h_{1}e_{1} . Moreover, one can check that e_{3}p_{1}h_{2} = p_{1}h_{2}e_{3} = e_{3}p_{2}h_{1} = p_{2}h_{1}e_{3} where e_{3} = 1 − e_{1} − e_{2} . Denote this product by θ. Then 2θ = 0 and p_{i}h_{i} = θ + 2e_{i}, so A^{a}(2, 2) is Morita equivalent to ℤ × ℤ[ θ] .

Just in the same way one can verify that A^{a}(3, 3) = ⟨ f_{1}, f_{2}, α_{1}, α_{2}, β ⟩, where f_{1} = h_{1}p_{1}h_{2}p_{2}, f_{2} = 1 − f_{1}, α_{i} = h_{i}p_{i}, β = α_{2}α_{1} . Moreover, f_{i}α_{i} = α_{i}f_{j} = α_{i}, α_{1}α_{2} = 3f_{1} and β^{2} = 3β. Therefore, A^{a}(3, 3) is isomorphic to the subring C_{0} .

Let u = p_{1}h_{2}p_{2}, v = h_{1}p_{1}h_{2}p_{2}h_{1} . Then e_{1}u = uf_{1}, f_{1}v = ve_{1}, uv = e_{1}, vu = f_{1} , so e_{1} and f_{1} are conjugate in A^{a}. Put ξ = p_{1} − p_{1}h_{2}p_{2}h_{1}p_{1}, η = h_{1} − h_{1}p_{1}h_{2}p_{2}h_{1} . One can also check that

2ξ = 0, 2η = 0, ηξ = 0, ξη = θ, ξf_{3} = ξ, f_{3}η = η,
e_{3}A^{a}(3, 2) = ⟨ ξ ⟩, A^{a}(2, 3)e_{3} = ⟨ η ⟩ .

Hence, putting E_{3}^{a} = A^{a}(2, )e_{3} , we get that A^{a} is Morita equivalent to the endomorphism ring End(E_{3}^{a} ⊕ A^{a}(3, )) which is isomorphic to C .

Describe now (finitely generated) C-modules or, the same, weakly alternative cubic functors. If M is such a module, put \overline{M} = M/T M . It is a module over ℤ × C , so \overline{M} = M_{1} ⊕ M_{2} , where M_{1} is an abelian group with C_{0}M_{1} = 0 and M_{2} is a C_{0}-module. Again, the same observations as in [7, Theorem 2.1] yield the following result.

Proposition 4.2. 1. Weakly alternative cubic modules M, N are

isomorphic if and only if M_{(p)} ≃ N_{(p)} for every prime p.

2. Given a set { N_{p} } where N_{p} is a weakly alternative cubic ℤ_{(p)}- module, there is a weakly alternative cubic module M such that M_{(p)} ≃ N_{p} for all p if and only if almost all of them are torsion free (maybe, zero) and ℚ⊗N_{p} ≃ ℚ⊗N_{q} for all p, q. In this case ℚ ⊗ M = ℚ ⊗ N_{p} and tM = ⨁_{p} tN_{p}.

Let \overline{C} = C/T = ℤ × C_{0} , C_{1} = L_{1} = \overline{C}e, C_{2} = \overline{C}(0, 1, e_{11}) and C_{3} = L_{4} = \overline{C}(0, 0, e_{22}) . They are all indecomposable projective \overline{C}- modules. Put also L_{2} = \overline{C}(0, 1, 0) and L_{3} = \overline{C}(0, 0, e_{11}) . They are all non-projective indecomposable torsion free \overline{C}-modules. If p > 2 , the localization C_{(p)} is torsion free. If, moreover, p > 3 , this localization is hereditary, hence, any C_{(p)}-module splits into a direct sum of a torsion free and a torsion one, the latter being a direct sum of modules isomorphic to L(i, k, p) = L_{i}/p^{k} for i ∈ { 1, 2, 3 } , k ∈ ℕ.

The description of C_{(3)}-modules is similar to that of B-modules in the preceding section. One only has to consider the set { 1, 2, 3, 4 } instead of { 1, 2, 3, 4, 5, 6 } with the relations − and ∼ defined as follows: 3 − 4 and 2 ∼ 3 .

At last, C_{(2)} = C′ × C′′ where C′ = (ℤ_{(2)} × ℤ_{(2)}) ⋉ T and C′′ ≃ Mat(2, ℤ_{(2)}) . A C′′-module is a direct sum of several copies of (L_{3})_{(2)} and of modules isomorphic to L(3, k, 2) = L_{3}/2^{k}. A C′-module W is given by a diagram of ℤ_{(2)}-modules of the form:

W_{1}  ⟶^{ξ} ; ⟵_{η} W_{2}

such that 2ξ = 0, 2η = 0, ηξ = 0 . Split both W_{1} and W_{2} into a direct sum of cyclic modules C_{i} = ℤ/2^{i} and C_{∞} = ℤ_{(2)} . Then one can check that such a diagram is a direct sum of diagrams W (ω) and W (ω, π) . Here ω is a (finite) word of the form

… ξ^{i_{m}}η_{j_{m}}ξ^{i_{m+1}}η_{m+1}ξ … (i_{l}, j_{l} ∈ ℕ ∪ { ∞ })

containing no subwords of the form _{∞}ξ, ^{∞}η, η_{1}ξ. In W (ω, π) , ω must be of the form

jξ^{i_{1}}η_{j_{1}}ξ^{i_{2}}η_{j_{2}}ξ …^{i_{m}} η_{j}

with the same restrictions as above and π(t) ≠ t^{n} is a primary polynomial over ℤ/2 . Namely, if W = W (ω) , then W_{1} = ⨁_{m} C_{i_{m}}, W_{2} = ⨁_{m} C_{j_{m}}, while ξ(C_{i_{m}}) ⊆ C_{j_{m−1}} , η(C_{j_{m}}) ⊆ C_{i_{m}} and the induced mappings are the unique non-zero ones of period 2 (we denote them by γ). If W = W (ω, π) and deg π = n, then W_{1} = ⨁_{m} nC_{i_{m}}, W_{2} = nC_{j} ⊕ (⨁_{m} nC_{j_{m}}) ; ξ(nC_{i_{m}}) ⊆ nC_{j_{m−1}} , η(nC_{j_{m}}) ⊆ nC_{i_{m}}, ξ(nC_{i_{1}}) ⊆ nC_{j}, η(nC_{j}) ⊆ nC_{i_{m}} and the induced mappings are given by the matrices γI, except the last one, given by γΦ , where Φ is the Frobenius cell corresponding to the polynomial π.

Note that the torsion free part of W (ω, π) is zero, that of W (ω) consists of at most one cyclic summand, and that of an indecomposable C_{(3)}-module either is trivial, or consists of one or of two cyclic summand (for string modules M^{D} of type, respectively, (i) or (ii), cf. page 9). So, in accordance with Proposition 4.2, the indecomposable C-modules M that are not torsion have the following local components M_{(p)} (we only describe M_{(2)} and M_{(3)} as all other ones are torsion free, hence, uniquely defined):

1. M_{(2)} = W (ω) where ω contains ξ^{∞}, M_{(3)} = M^{D} where D is a string of type (i) or (ii) with i_{2n−1} = 2 or i_{2} = 2 (if both i_{2} = i_{2n−1} = 2 in a string of type (ii), there are two possibilities for such M ).

2. M_{(2)} = W (ω)⊕W (ω′) where both ω and ω′ contain ξ^{∞}, M_{(3)} = M^{D} where D is a string of type (ii) with i_{2} = i_{2n−1} = 2 .

3. M_{(3)} is a string module of type (i) or (ii), M_{(2)} is torsion free (thus, uniquely defined).

(Note that the case when M_{(3)} is torsion free is a part of case (1) above.)

5. Cubic vector spaces

Now we consider the cubic vector spaces, i.e., cubic functors F : fab → k-mod where k is a field. If char k ≠ 2 they are a partial case of the functors considered in Section 3; hence, we always suppose that char k = 2 . In this case a cubic functor F is given by a diagram of k-vector spaces of the same shape (1) with the relations:

\begin{matrix}h_{1}p_{2} = h_{2}p_{1} = 0 , h_{1}h = h_{2}h , pp_{1} = pp_{2} , \\ h_{i}p_{i}h_{i} = 0 , p_{i}h_{i}p_{i} = 0 (i = 1, 2) , \\ hph = 0 , php = 0 , \\ \overline{h}p + h_{1} + h_{2} = h_{1}p_{1}h_{2}p_{2}h_{1} + h_{2}p_{2}h_{1}p_{1}h_{2} , \\ h\overline{p} + p_{1} + p_{2} = p_{1}h_{2}p_{2}h_{1}p_{1} + p_{2}h_{1}p_{1}h_{2}p_{2} \end{matrix} (5)

(just as before, \overline{h} = h_{1}h = h_{2}h and \overline{p} = pp_{1} = pp_{2} ). Denote by C the k-linear category with objects 1, 2, 3 and generating morphisms h : 1 → 2, p : 2 → 1, h_{i} : 2 → 3, p_{i} : 3 → 1 ( i = 1, 2 ) satisfying the relations (5). So, a cubic vector space is the same as a C-module (i.e., a linear functor C → k-mod ).

The last two equations imply that

\overline{h}\overline{p} + h_{i}p_{i} = h_{i}p_{i}h_{j}p_{j}h_{i}p_{j} (i, j = 1, 2; i ≠ j),

whence p_{i}\overline{h}\overline{p} = \overline{h}\overline{p}h_{i} = 0 for i = 1, 2 . Hence, the elements e_{i} = h_{i}p_{i}h_{j}p_{j} as well as the elements f_{i} = p_{i}h_{j}p_{j}h_{i} ( i, j = 1, 2; i ≠ j) are orthogonal idempotents, respectively, in C(3, 3) and in C(2, 2) . Thus, in add C , 3 ≃ 3_{0} ⊕ 3_{1} ⊕ 3_{2} and 2 ≃ 2_{0} ⊕ 2_{1} ⊕ 2_{2} , so that the identity morphisms of 3_{i} are identified with e_{i} (with e_{0} = 1 − e_{1} − e_{2} ) and those of 2_{i} are identified with f_{i} (with f_{0} = 1 − f_{1} − f_{2} ). In what follows, we write C(x, y) for the set of morphisms x → y in add C . An easy calculation shows that the four objects 2_{i}, 3_{i} ( i = 1, 2 ) are isomorphic in add C . For instance, as p_{1}h_{2}p_{2} = p_{1}h_{2}p_{2}h_{1}p_{1}h_{2}p_{2} , this element lies in C(3_{1}, 2_{1}) ; the element h_{1}p_{1}h_{2}p_{2}h_{1} lies in C(2_{1}, 3_{1}) and their products are just f_{1} and e_{1} , whence 2_{1} ≃ 3_{1} , etc. So we only have to take into account one of these objects, say 3_{1} . One can also easily check that C(3_{1}, 3_{1}) = ke_{1} , while C(x, 3_{1}) = C(3_{1}, x) = 0 if x ∈ { 1, 2_{0}, 3_{0} }. Thus, the category C is Morita equivalent to the direct product of the trivial category with one object 3_{1} and the full subcategory C^{∗} of add C with the objects 1, 2_{0}, 3_{0} . One easily check that the cubic module T corresponding to the (unique) indecomposable representation of the trivial part is the following one:

\begin{matrix}T_{1} = 0 , T_{2} = ⟨ u_{1}, u_{2} ⟩ , T_{3} = ⟨ v_{1}, v_{2} ⟩; \\ h_{1}(u_{1}) = v_{1} , h_{1}(u_{2}) = 0 ; h_{2}(u_{1}) = 0 , h_{2}(u_{2}) = v_{2} ; \\ p_{1}(v_{1}) = 0 , p_{1}(v_{2}) = u_{1} ; p_{2}(v_{1}) = u_{2} , p_{2}(v_{2}) = 0 . \end{matrix} (6)

(This cubic module corresponds to the functor k ⊗ Λ^{2} ⊗ Id .) So, from now on, we only consider the representations of C^{∗} and for every morphism α from C we denote by the same letter α its restriction onto C^{∗}. As such restrictions of e_{i} and f_{i} are zero for i = 1, 2 , one obtains the relations:

h\overline{p} + p_{1} + p_{2} = 0 , \overline{h}p + h_{1} + h_{2} = 0 ,

so we may exclude p_{2}, h_{2} from the generating set. Therefore, C^{∗}- modules are just diagrams of vector spaces

F_{1}  ⟶^{h} ; ⟵^{p} F_{2}  ⟶^{h_{1}} ; ⟵^{p_{1}} F_{3} (7)

with the relations

hph = php = h_{1}p_{1}h_{1} = p_{1}h_{1}p_{1} = 0 , h_{1}p_{1} = h_{1}hpp_{1} . (8)

(One easily checks that they imply all relations (5) if we put h_{2} = h_{1} + h_{1}hp and p_{2} = p_{1} + hpp_{1} .)

Consider the subdiagram

F_{1}  ⟶^{h} ; ⟵^{p} F_{2} .

As hph = php = 0 , it decomposes into a direct sum of the following shape:

\begin{aligned}& & U_{1} & U_{2} & U_{3} & & U_{4} & U_{5} & U_{6} \\ & ↗ & ↓ & ↓ & ↑ & ↙ & & ↑ \\ V_{6} & & V_{1} & V_{2} & V_{3} & & V_{4} & V_{5} \end{aligned}

( U_{i} are the direct summands of F_{1} , V_{i} those of F_{2} , the arrows show the action of h and p when it is non-zero, the corresponding maps being isomorphisms.) It means that h and p are given by the following matrices:

h = (\begin{matrix}I & 0 & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ) , p = (\begin{matrix}0 & 0 & 0 & 0 & 0 & I \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & I & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & I & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}) . (9)

( I denotes the identity matrices.)

With respect to the decomposition of F_{2} , h_{1} and p_{1} are given by the matrices H = (H_{1}, H_{2}, H_{3}, H_{4}, H_{5}, H_{6}) and P = (P_{1}, P_{2}, P_{3}, P_{4}, P_{5}, P_{6})^{⊤}, where H_{i} : V_{i} → F_{3}, P_{i} : F_{3} → V_{i}. For these matrices the following conditions hold:

• the number of rows of H equals the number of columns of P ;

• the number of columns of H_{i} is the same as the number of rows of P_{i} for every i; • the number of columns of H_{1} is the same as the number of rows

of H_{6} . When one applies the isomorphisms of the spaces F_{i} which do not

destroy the form (9) of h and p, they result in elementary transformations of the columns of the matrices H and P such that:

• the transformations of P are contragredient to those of H (e.g., when we add the k-th column of H to the l-th one, we have to subtract the l-th column of P from the k-th one, etc.);

• the transformations inside H_{1} are the same as those inside H_{6} ;

• one can only add columns of H_{i} to those of H_{j} if i ≤ j and

(i, j) ≠ (3, 4) . Using such transformations, one can reduce the matrix H to the following form:

( \begin{matrix}0 & 0 & I & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & I & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & I & 0 & 0 & I & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & I & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & I & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & I & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & I \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} )

(the vertical lines denote the borders of the matrices H_{i}). Certainly, we have to subdivide in the same manner the rows and columns of the matrix P . Denote the corresponding blocks by P_{ij} ( 1 ≤ i ≤ 17, 1 ≤ j ≤ 10 ); the horizontal (vertical) stripes of P will be denoted by R_{i} (respectively, S_{j}). Then the relations (8) for h_{1} and p_{1} are equivalent to the following conditions for these blocks:

P_{ij} = 0 if i ∈ { 4, 6, 9, 12, 14, 16, 18 } ,

P_{ij} = 0 if i ∈ { 3, 17 } and j ≠ 10 ,
P_{8j} = P_{11,j} for all j ,
P_{3,10} = P_{17,10} and P_{i1}P_{3,10} = 0 for all i .

Thus, we only have to consider the matrix \overline{P} obtained from P by crossing out all zero horizontal stripes as well as the stripes R_{3} and R_{8} . A straightforward calculation shows that the automorphisms of the spaces F_{3} and V_{i} which do not destroy the shape of H give rise to the elementary transformations of the matrix \overline{P} satisfying the following conditions:

• the transformations inside R_{1} are the same as those inside R_{15} ;

• the transformations inside S_{2} are the same as those inside S_{9} ;

• the transformations inside R_{2} are contragredient to those inside S_{8} ;

• the transformations inside R_{11} are contragredient to those inside S_{4} ;

• the transformations inside R_{17} are contragredient to those inside S_{1} ;

• one can add the columns of S_{i} to those of S_{j} if and only if i ≤ j and (i, j) ≠ (5, 6) ;

• one can add the rows of R_{i} to those of R_{j} if and only if i ≥ j

and (i, j) ≠ (10, 7) . Obviously, one may suppose that the matrix P_{17,10} is of the form

(\begin{matrix}0 & I \\ 0 & 0 \end{matrix}). Then the whole stripe S_{1} has the form (0 S′_{1}), the num-

ber of columns in S′_{1} being the same as that of zero rows in P_{17,10} . Moreover, using the elementary transformations described above, one may suppose that the remaining part of the stripe S_{10} is of the form (S′_{10} 0), the number of columns in S′_{10} being the same as in the zero part of P_{17,10} . In what follows, we omit the dash and write S_{1} , S_{10} , P_{i1} and P_{i,10} for the remaining (non-zero) parts of the corresponding matrices. Of course, we should no more consider the stripe R_{17} .

Now one immediately sees that we are again in the situation considered in [1]. Namely, we have got the following semi-chains:

\begin{aligned}𝔈 & = \{ R_{i} | R_{1} > R_{2} > R_{5} > R_{7} > R_{11} > R_{13} > R_{15}, R_{5} > R_{10} > R_{11} \} , \\ 𝔉 & = \{ S_{j} | S_{1} < S_{2} < S_{3} < S_{4} < S_{5} < S_{7} < S_{8} < S_{9} < S_{10}, S_{4} < S_{6} < S_{7} \} \end{aligned}

with the involution σ such that

σ(R_{1}) = R_{15} , σ(R_{2}) = S_{8} , σ(R_{11}) = S_{4} , σ(S_{2}) = S_{9}

and σ(x) = x if x ∉ { R_{1}, R_{2}, R_{11}, R_{15}, S_{2}, S_{4}, S_{8}, S_{9} }. Hence, one can deduce from [1] a list of canonical forms of the matrix P and, therefore, of cubic functors fab → k-mod . It can again be arranged in the form of “strings and bands,” though a trifle more sophisticated than in the preceding sections.

Definition 5.1. Put 𝔛 = (𝔈 ∪ 𝔉) \ { R_{10}, S_{6} }. Write x − y if x ∈ 𝔈, y ∈ 𝔉 or vice versa; x ∼ y if σ(x) = y ≠ x or x = y ∈ { R_{7}, S_{5} }. Call R_{7}, S_{5} special elements. An 𝔛-word (or simply word) is a sequence w = x_{1}r_{2}x_{2}r_{3} … r_{n}x_{n}, where x_{k} ∈ 𝔛, r_{k} ∈ { ∼, − }, satisfying the following conditions:

• x_{k−1}r_{k}x_{k} (in the above defined sense) for all k = 2, …, n;

• r_{k} ≠ r_{k−1} for all k = 3, …, n; Such a word is called full if the following conditions hold:

• either r_{2} = ∼ or x_{1} ≁ y for every y ≠ x_{1} ;

• either r_{n} = ∼ or x_{n} ≁ y for every y ≠ x_{n}. A full word is called special if x_{1} or x_{n}, but not both of them, is a special element, bispecial if both x_{1} and x_{n} are special, and ordinary if neither x_{1} nor x_{n} is special. A word w is called non-symmetric if w ≠ w^{∗}, where w^{∗} is the inverse word: w^{∗} = x_{n}r_{n}x_{n−1} … x_{2}r_{2}x_{1} .

An 𝔛-word is called cyclic if r_{2} = r_{n} = − and x_{n} ∼ x_{1} . A cyclic word is called aperiodic if it is not of the form v ∼ v ∼ ⋯ ∼ v for a shorter word v. The s-th shift of a cyclic word w is the word w^{(s)} = x_{2s+1}r_{2s+2}x_{2s+2} … x_{n} ∼ x_{1}r_{2} … r_{2s}x_{2s}. A cyclic word w is said to be shift-symmetric if w^{(s)} is symmetric for some s. Note that the length n of a cyclic word is always even.

Definition 5.2. A string datum D is:

• either an ordinary non-symmetric 𝔛-word w (ordinary string datum);

• or a pair (w, δ) consisting of a special word w and δ ∈ { 0, 1 } (special string datum);

• or a quadruple (w, δ_{1}, δ_{2}, m) consisting of a bispecial non-symmetric

word w, δ_{1}, δ_{2} ∈ { 0, 1 } and m ∈ ℕ (bispecial string datum). Put D^{∗} = w^{∗} in the first case, D^{∗} = (w^{∗}, δ) in the second and D^{∗} = (w^{∗}, δ_{2}, δ_{1}, m) in the third one.

A band datum B is a pair (w, π(t)) consisting of an aperiodic cyclic word w and of a primary polynomial π(t) ∈ k[t] (i.e., a power of an irreducible one) such that π(t) ≠ t^{d} if w is not shift-symmetric and π(t) ≠ (t − 1)^{d} if w is shift-symmetric. Put B^{(s)} = (w^{(s)}, f(t)) and B^{∗} = (w^{∗}^{(s)}, λ^{−1}t^{d}π(1/t)) , where w is not shift-symmetric, d = deg π and λ = π(0) .

Now the results of [1] immediately imply the following

Theorem 5.3. 1. Every string data D defines an indecomposable

cubic vector space V^{D} called string cubic space.

2. Every band data B defines an indecomposable cubic vector space V^{B} called band cubic space.

3. Every indecomposable cubic vector space, except k ⊗ Λ^{2} ⊗ Id , is isomorphic either to a string or to a band one.

4. The only isomorphisms between string and band spaces are the following:

• V^{D} ≃ V^{D^{∗}} where D is a string data; • V^{B} ≃ V^{B^{(s)}} and V^{B} ≃ V^{B^{∗}^{(s)}} where B is a band data and

s ∈ ℕ.

Moreover, one can deduce from [1] and the preceding considerations an explicit construction of string and band spaces, though it is rather cumbersome and we will not include it here.

6. Torsion free cubic modules

In this section we consider torsion free cubic modules, i.e., cubic functors fab → fab . Again, we first study them locally, i.e., describe cubic functors fab → fab_{(p)} , the latter being the category of torsion free finitely generated ℤ_{(p)}-modules. Denote also by ℤ_{p} the ring of p-adic integers. The latter has the advantage of being complete, that guarantees lifting idempotent endomorphisms modulo p. Note that the calculations of Section 3 easily imply that A ⊗ ℚ ≃ ℚ^{3} × Mat(2, ℚ) × Mat(4, ℚ)^{2} . In particular, it is a semi-simple split ℚ-algebra, i.e., End V ≃ ℚ for every simple module V . Thus, it follows from the standard results of the theory of lattices over orders (cf. [9, Chapter 4, § 1]) that every torsion free A_{p}-module is actually a completion of a torsion free A_{(p)}-module. Hence, lifting idempotents is possible for A_{(p)}-modules too. Together with the calculations of Section 5 it implies the following result.

Proposition 6.1. The ring A_{(2)} is isomorphic to the subring of ℤ_{(2)}^{3}× Mat(2, ℤ_{(2)})×Mat(4, ℤ_{(2)})^{2} consisting of all sextuples (a_{1}, a_{2}, a_{3}, B, C, D) satisfying the following congruences modulo 2 :

a_{1} ≡ b_{11} ≡ c_{11} , a_{2} ≡ b_{22} ≡ c_{22} ≡ c_{33} , a_{3} ≡ c_{44} ,

b_{12} ≡ 0 and c_{ij} ≡ 0 if i < j . (*)

One can easily check that this ring is an example of the so called Backström order [10], i.e., its Jacobson radical coincides with that of an hereditary order H . Thus, torsion free A_{(2)}-modules are in a natural one-to-one correspondence with the representations of a quiver Q over the field ℤ/2 . Namely, the vertices of Q are just simple A_{(2)}- modules A_{1}, …, A_{r} and simple H-modules H_{1}, …, H_{s}, all arrows are from some A_{i} to some H_{j}, and the number of such arrows equals the multiplicity of A_{i} in H_{j}. In our example H consists of the sextuples satisfying the congruences (*) only, r = 4, s = 10 and the quiver Q consists of 4 connected components:

\begin{matrix}H_{1} ⟵ A_{1} ⟶ H_{4} ; ↓ ; H_{6} & H_{5} ; ↑ ; H_{2} ⟵ A_{2} ⟶ H_{7} ; ↓ ; H_{8} \\ H_{3} ⟵ A_{3} ⟶ H_{9} & A_{4} ⟶ H_{10} \end{matrix}

(The numeration is the natural one with respect to the description of 𝔸_{(2)} and H above.) This quiver is tame and the list of its representations is well known (cf., e.g., [2]). Hence, we can derive the description of torsion free A_{(2)}-modules.

The description of indecomposable torsion free A_{(3)}-modules is given in Section 3. For any other prime p, the localization A_{(p)} is just ℤ_{(p)}^{3}× Mat(2, ℤ_{(p)})×Mat(4, ℤ_{(p)})^{2} , hence, there are exactly 6 indecomposable (and irreducible) torsion free modules. Therefore, the standard “local– global” procedure [9, 4] implies the following result on torsion free cubic modules.

Proposition 6.2. Torsion free cubic modules are in one-to-one correspondence with the pairs (M_{2}, M_{3}) , where M_{p} is a torsion free A_{(p)} - module ( p = 2, 3 ) and ℚ ⊗ M_{2} ≃ ℚ ⊗ M_{3} .

Proof. It follows from [9, Chapter 4] that such a pair (M_{2}, M_{3}) always defines a cubic module M up to genus. (Remind that two modules M, N belong to the same genus if M_{p} ≃ N_{p} for all prime p.) Note that Γ = ℤ^{3} × Mat(2, ℤ) × Mat(4, ℤ)^{2} is a maximal order containing A and two torsion free Γ-modules belonging to the same genus are isomorphic. Applying the results of [4], we see that the isomorphism classes of modules belonging to the same genus as M are in one-to-one correspondence with the double cosets

Aut(ΓM)\ Aut(ΓM_{2}) × Aut(ΓM_{3})/ Aut(M_{2}) × Aut(M_{3}) ,

where ΓM denotes the Γ-submodule in ℚ ⊗ M generated by M . As A ⊃ 6Γ , we can replace these cosets by

\overline{Aut}(ΓM)\ Aut(ΓM/6ΓM)/ Aut(M/6ΓM) ,

where \overline{Aut}(ΓM) denotes the image of Aut(ΓM) in Aut(ΓM/6ΓM) . But End(ΓM) is just a direct product of matrix algebras Mat(n_{i}, ℤ) , and any matrix invertible modulo 6 is an image of an invertible integer matrix. Hence, \overline{Aut}(ΓM)) = Aut(ΓM/6ΓM) , so every genus only contains one module up to isomorphism.

Using the arguments analogous to those of [7, Theorem 2.1], one gets the following corollary for all cubic functors extending Theorem 3.2.

Corollary 6.3. 1. Cubic modules M, N are isomorphic if and only

if M_{(p)} ≃ N_{(p)} for every prime p.

2. Given a set { N_{p} } where N_{p} is a cubic ℤ_{(p)}-module, there is a cubic module M such that M_{(p)} ≃ N_{p} for all p if and only if almost all of them are torsion free (maybe, zero) and ℚ ⊗ N_{p} ≃ ℚ ⊗ N_{q} for all p, q. In this case ℚ ⊗ M = ℚ ⊗ N_{p} and tM = ⨁_{p} tN_{p}.

It seems plausible that the analogous result is no more true for the polynomial functors of degree 4 , but at the moment we do not have a corresponding counterexample.

7. One conjecture

Description of quadratic modules and of cubic 2-divisible modules as well as some other evidence give rise to the following conjecture concerning polynomial modules of higher degrees.

Put ℤ_{<p} = ℤ[ 1/(p − 1)! ] and denote by A^{(p)} the subring of the direct product ℤ_{<p} × ℤ_{<p} × Mat(2, ℤ_{<p})^{p−1} consisting of all (p + 1)- tuples

(a, b, C^{1}, …, C^{p−1})

such that a ≡ c_{11}^{1} (mod p), b ≡ c_{22}^{p−1} (mod p) and c_{22}^{i} ≡ c_{11}^{i+1} (mod p) for every i = 1, …, p − 2 .

Conjecture 7.1. The category M(p) = ℤ_{<p}-mod^{p} is equivalent to the category of modules over A^{(p)} × ℤ_{<p}^{r} for appropriate r (depending on p).

As the ring A^{(p)} fits the conditions of [6], it would give a complete description of such polynomial functors in strings and bands terms quite similar to that of Section 3. Moreover, easy calculations show that this conjecture would imply the following properties of such polynomial functors, analogous to those of quadratic and 2-divisible cubic functors:

• Two modules from M(p) are isomorphic if and only if all their localizations are isomorphic.

• Given a set { N_{q} } of polynomial functors (N_{q} ∈ ℤ_{(q)}-mod^{p}, q runs through all primes ≥ p), there is a functor M ∈ M(p) such that M_{(q)} ≃ N_{q} for all q if and only if almost all N_{q} are torsion free and ℚ ⊗ N_{q} ≃ ℚ ⊗ N_{q′} for all q, q′. In this case tM = ⨁_{q} tN_{q} and ℚ ⊗ M ≃ ℚ ⊗ N_{q}.

• Any functor from M(p) is either projective, or of projective dimension 1, or of infinite projective dimension.

• Any functor from M(p) has a periodic projective resolution of period 2p.

• If a functor F ∈ M(p) is indecomposable and non-projective, its torsion free part is a direct sum of at most two irreducible torsion free modules.

Acknowledgments

This research has mainly been performed when the author was at the Max-Plank-Institut für Mathematik. I am extremely grateful to the Institute for excellent opportunity and working conditions. I am also grateful to H.-J. Baues who enthusiastically supported this research and to W. Dreckman for useful discussions.

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Max-Plank-Institut für Mathematik and Kiev Taras Shevchenko University

E-mail address: yuriy@drozd.org