Homology, Homotopy and Applications, vol.4(2), 2002, pp.357–396

NONCOMMUTATIVE DEFORMATIONS OF MODULES

O. A. LAUDAL

(communicated by Clas Löfwall)

Abstract

The classical deformation theory for modules on a k-algebra, where k is a field, is generalized. For any k-algebra, and for any finite family of r modules, we consider a deformation functor defined on the category of Artinian r-pointed (not necessarily commutative) k-algebras, and prove that it has a prorepresenting hull, which can be computed using Massey-type products in the Ext-groups. This is first used to construct k-algebras with a preassigned set of simple modules, and to study the moduli space of iterated extensions of modules. In forthcoming papers we shall show that this noncommutative deformation theory is a useful tool in the study of k-algebras, and in establishing a noncommutative algebraic geometry.

To Jan–Erik Roos on his sixty–fifth birthday

1. Introduction

In this paper I shall introduce a noncommutative deformation theory for modules on some k-algebra A, k a field. The basic idea of noncommutative deformation theory is very simple. Let \underline{a}_{r} denote the category of r-pointed not necessarily commutative k-algebras R. The objects are the diagrams of k-algebras,

k^{r} →^{ι}→ R →^{ρ}→ k^{r}

such that the composition of ι and ρ is the identity. Any such r-pointed k-algebra R is isomorphic to a k-algebra of r × r-matrices (R_{i,j}). The radical of R is the bilateral ideal Rad(R) := kerρ, such that R/Rad(R) ≃ k^{r}. The dual k-vectorspace of Rad(R)/Rad(R)^{2} is called the tangent space of R.

For r = 1, there is an obvious inclusion of categories

\underline{l} ⊆ \underline{a}_{1}

where \underline{l}, as usual, denotes the category of commutative local artinian k-algebras with residue field k.

Received March 15, 2001, revised August 13, 2001; published on July 12, 2002. 2000 Mathematics Subject Classification: 16.40, 16.90, 16.50, 14A22. Key words and phrases: Modules, deformations of modules, formal moduli, moduli of iterated extensions, Hochschild cohomology, Massey products, swarm of modules, algebra of observables, modular substratum, quivers. \begin{matrix}c \\ ◯ \end{matrix} 2002, O. A. Laudal. Permission to copy for private use granted.

Fix a not necessarily commutative k-algebra A and consider a right A-module M. The ordinary deformation functor

Def_{M} : \underline{l} → \underline{Sets}

is then defined. Assuming Ext_{A}^{i}(M, M) has finite k-dimension for i = 1, 2, it is well known, see [15], or [6], that Def_{M} has a noetherian prorepresenting hull H, the formal moduli of M. Moreover, the tangent space of H is isomorphic to Ext_{A}^{1}(M, M), and H can be computed in terms of Ext_{A}^{i}(M, M), i = 1, 2 and their matric Massey products, see [6], [7].

In the general case, consider a finite family V = {V_{i}}_{i=1}^{r} of A-modules. Assume that,

dim_{k}Ext_{A}^{1}(V_{i}, V_{j}) < ∞.

Any such family of A-modules will be called a swarm. Define a deformation functor,

Def_{V} : \underline{a}_{r} → \underline{Sets}

generalizing the functor Def_{M} above. Given an object ρ : R = (R_{i,j}) → k^{r} of \underline{a}_{r}, consider the k-vectorspace and R-left module (R_{i,j} ⊗_{k} V_{j}). ρ defines a k-linear and left R-linear map,

ρ(R) : (R_{i,j} ⊗_{k} V_{j}) → ⊕_{i=1}^{r}V_{i},

inducing a homomorphism of R-endomorphism rings,

ρ̃(R) : (R_{i,j} ⊗_{k} Hom_{k}(V_{i}, V_{j})) → ⊕_{i=1}^{r}End_{k}(V_{i}).

The right A-module structure on the V′_{i}s is defined by a homomorphism of kalgebras, η_{0} : A → ⊕_{i=1}^{r}End_{k}(V_{i}). Let

Def_{V}(R) ∈ \underline{Sets}

be the isoclasses of homomorphisms of k-algebras,

η′ : A → (R_{i,j} ⊗_{k} Hom_{k}(V_{i}, V_{j}))

such that,

ρ̃(R) ∘ η′ = η_{0},

where the equivalence relation is defined by inner automorphisms in the k-algebra (R_{i,j} ⊗_{k} Hom_{k}(V_{i}, V_{j})). One easily proves that Def_{V} has the same properties as the ordinary deformation functor:

Theorem 1.1. The functor Def_{V} has a prorepresentable hull, i.e. an object H of the category of pro-objects ˆ \underline{a}_{r} of \underline{a}_{r}, together with a versal family,

Ṽ = (H_{i,j} ⊗ V_{j}) ∈ \underleftarrow{lim}_{n⩾1} Def_{V}(H/𝔪^{n})

such that the corresponding morphism of functors on \underline{a}_{r},

ρ : Mor(H, −) → Def_{V}

is smooth (see §2), and an isomorphism on the tangent level. Moreover, H is uniquely determined by a set of matric Massey products of the form

Ext^{1}(V_{i}, V_{j_{1}}) ⊗ ⋯ ⊗ Ext^{1}(V_{j_{n−1}}, V_{j}) ⋯ → Ext^{2}(V_{i}, V_{j}),

see [7] and [16] for an exposition of the Massey product structure in the category of all O_{X}-modules for X a scheme defined on some field k.

The right action of A on Ṽ defines a homomorphism of k-algebras,

η : A ⟶ O(V) := End_{H}(Ṽ) = (H_{i,j} ⊗ Hom_{k}(V_{i}, V_{j})),

and the k-algebra O(V) acts on the family of A-modules V = {V_{i}}, extending the action of A. If dim_{k}V_{i} < ∞, for all i = 1, ..., r, the operation of associating (O(V), V) to (A, V) turns out to be a closure operation.

Moreover, if A is an object of \underline{a}_{r}, and if V = \{k_{i}\}_{i=1}^{r} is the correspondiong family of simple A-modules, then

η : A ⟶ H(V)

is an isomorphism.

In §3 we prove that there exists, in the noncommutative deformation theory, an obvious analogy to the notion of prorepresenting (modular) substratum H_{0} of the formal moduli H. The tangent space of H_{0} is determined by a family of subspaces

Ext_{0}^{1}(V_{i}, V_{j}) ⊆ Ext_{A}^{1}(V_{i}, V_{j}), i ≠ j

the elements of which should be called the almost split extensions (sequences) relative to the family V, and by a subspace,

T_{0}(Δ) ⊆ ∏_{i} Ext_{A}^{1}(V_{i}, V_{i})

which is the tangent space of the deformation functor of the full subcategory of the category of A-modules generated by the family V = {V_{i}}_{i=1}^{r}, see [6]. If V = {V_{i}}_{i=1}^{r} is the set of all indecomposables of some artinian k-algebra A, we show that the above notion of almost split sequence coincides with that of Auslander.

In Remark (3.7) we associate to the family V = {V_{i}}, and to any quotient R of H in \underline{a}_{r} a quiver with vertices contained in the set \{V_{i}\}. The Auslander-Reiten quiver then turns out to correspond to R = H_{0}/Rad(H_{0})^{2}.

Observe that, in general, the k-algebra H_{0} and its corresponding modular family Ṽ_{0} contains much more information than what may be deduced from the tangent level.

In §4, we first prove,

Theorem 1.2 (A generalized Burnside theorem). Let A be a finite dimensional k-algebra, k an algebraically closed field. Consider the family V = {V_{i}}_{i=1}^{r} of simple A-modules, then

A ≃ O(V) = (H_{i,j} ⊗ Hom_{k}(V_{i}, V_{j}))

Then we consider the general problem of classification of iterated extensions of a family of modules V = {V_{i}, }_{i=1}^{r}, and the corresponding classification of filtered modules with graded components in the family V. The remainder of the paragraph is concerned with properties of the O-construction, and some examples.

Based on this notion of noncommutative deformations, we have in [13] proposed a general definition of an affine noncommutative prescheme, and scheme, generalizing the classical notion of an affine algebraic scheme in the commutative case.

Earlier versions of this paper has appeared in the preprints [9]–[12].

2. Homological preparations

Exts and Hochschild cohomology

Let k be a (usually algebraically closed) field, and let A be a k-algebra. Denote by A-mod the category of right A-modules and consider the exact forgetful functor

π : A − mod ⟶ k − mod

Given two A-modules M and N, we shall always use the identification

σ^{i} : Ext_{A}^{i}(M, N) ≃ HH^{i}(A, Hom_{k}(M, N)) for i = 0, 1, 2,

where Hom_{k}(M, N) is provided with the obvious left and right A-module structures. If L_{∗} and F_{∗} are A-free resolutions of M and N respectively, and if an element

ξ ∈ Ext_{A}^{1}(M, N)

is represented by the Yoneda cocycle,

ξ̂ = {ξ_{n}} ∈ ∏_{n} Hom_{A}(L_{n}, F_{n−1})

then σ^{1}(ξ) is gotten as follows. Let σ be a k-linear section of the augmentation morphism

ρ : L_{0} ⟶ M

and let for every a ∈ A and m ∈ M, σ(ma) − σ(m)a = d_{0}(x). Put,

σ^{1}(ξ̂)(a, m) = −μ(ξ_{1}(x))

where

μ : F_{0} ⟶ N

is the augmentation morphism of F_{∗}. Then,

σ^{1}(ξ̂) ∈ Der_{k}(A, Hom_{k}(M, N))

and its class in HH^{1}(A, Hom_{k}(M, N)) equals σ^{1}(ξ).

Recall the spectral sequence associated to a change of rings. If π : A ⟶ B is a surjective homomorphism of commutative k-algebras, M a B-module and N an A-module, then Ext_{A}^{∗}(M, N) is the abuttment of the spectral sequence given by,

E_{2}^{p,q} = Ext_{B}^{p}(M, Ext_{A}^{q}(B, N)).

There is an exact sequence,

0 ⟶ E_{2}^{1,0} ⟶ Ext_{A}^{1}(M, N) ⟶ E_{2}^{0,1} ⟶ E_{2}^{2,0},

which, for a B-module N, considered as an A-module, implies the exactness of

0 ⟶ Ext_{B}^{1}(M, N) ⟶ Ext_{A}^{1}(M, N)
⟶ Hom_{B}(M, Hom_{B}(I/I^{2}, N)) ⟶ Ext_{B}^{2}(M, N)

where I=ker π. The corresponding exact sequence,

0 → HH^{1}(B, Hom_{k}(M, N)) → HH^{1}(A, Hom_{k}(M, N))
→ Hom_{A⊗A^{op}}(I, Hom_{k}(M, N))

in the noncommutative case is induced by the sequence

0 → Der_{k}(B, Hom_{k}(M, N)) → Der_{k}(A, Hom_{k}(M, N))
→ Hom_{A⊗A^{op}}(I, Hom_{k}(M, N)).

Notice that in general we do not know that the last morphism is surjective. This, however, is true if B=A/rad(A), where rad(A) is the radical of A, and A is a finite dimensional, i.e. an artinian, k-algebra. In this case, B is semisimple and the surjectivity above follows from the Wedderburn-Malcev theorem. Notice also that in the commutative case,

Hom_{A⊗A^{op}}(I, Hom_{k}(M, N)) ≃ Hom_{B}(I/I^{2}, Hom_{B}(M, N))

as it must, since for ϕ ∈ Hom_{A⊗A^{op}}(I, Hom_{k}(M, N)), a ∈ A, and i ∈ I, ai = ia, and therefore

aϕ(i) = ϕ(ai) = ϕ(ia) = ϕ(i)a, i.e. ϕ(i) ∈ Hom_{B}(M, N).

This implies that for B = A/𝔭, M = A/𝔭, N = A/𝔮, where 𝔭 ⊆ 𝔮 are (prime) ideals of A,

Ext_{A}^{1}(A/𝔭, A/𝔮) ≃ Hom_{A}(𝔭/𝔭^{2}, A/𝔮)

and, in particular

Ext_{A}^{1}(A/𝔮, A/𝔮) ≃ Hom_{A}(𝔮/𝔮^{2}, A/𝔮) = N_{𝔮},

the normal bundle of V (𝔮) in Spec(A). If 𝔮 ⊂ 𝔭 and 𝔮 ≠ 𝔭 we find,

Ext_{A}^{1}(A/𝔭, A/𝔮) ≃ Ext_{A/𝔮}^{1}(A/𝔭, A/𝔮).

In [6], chapter 1., we considered the cohomology of a category \underline{c} with values in a bifunctor, i.e. in a functor defined on the category mor\underline{c} of morphismes of \underline{c}. Recall that a morphism between the objects ψ and ψ′ is a commutative diagram,

\xymatrix{c_{1} \ar[r]^{ψ} & c_{2} \ar[d] \\ c′_{1} \ar[u] \ar[r]^{ψ′} & c′_{2}.}

It is easy to see that this cohomology is an immediate generalization of the projective limit functor and its derivatives, or if one likes it better, the obvious generalization of the Hochschild cohomology of a ring. In fact, for every small category \underline{c} and for every bifunctor,

G : \underline{c} × \underline{c} ⟶ Ab

contravariant in the first variable, and covariant in the second, one obtains a covariant functor,

G : mor\underline{c} ⟶ Ab.

Consider now the complex,

D^{∗}(\underline{c}, G)

where,

D^{p}(\underline{c}, G) = ∏_{c_{0}→c_{1}···→c_{p}} G(c_{0}, c_{p})

where the indices are strings of morphisms ψ_{i} : c_{i} → c_{i+1} in \underline{c}, and the differential,

d^{p} : D^{p}(\underline{c}, G) ⟶ D^{p+1}(\underline{c}, G)

is defined as usual,

(d^{p}ξ)(ψ_{1}, …, ψ_{i}, ψ_{i+1}, …, ψ_{p+1}) = ψ_{1}ξ(ψ_{2}, …, ψ_{p+1})

+ ∑_{i=1}^{p}(−1)^{i}ξ(ψ_{1}, …, ψ_{i} ∘ ψ_{i+1}, …, ψ_{p+1}) + (−1)^{p+1}ξ(ψ_{1}, …, ψ_{p})ψ_{p+1}.

As shown in [6], the cohomology of this complex is the higher derivatives of the projective limit functor \underleftarrow{lim}_{mor\underline{c}}^{(∗)} applied to the covariant functor

G : mor\underline{c} ⟶ Ab.

This is the ”Hochschild” cohomology of the category \underline{c}, denoted

H^{∗}(\underline{c}, G) := H^{∗}(D^{∗}(\underline{c}, G)).

Example 2.1. Let \underline{c} be a multiplicative subset of a ring R, considered as a category with one object, and let R̃ : \underline{c} × \underline{c} ⟶ Ab be the functor, defined for ψ, ψ′ ∈ \underline{c}, by R̃(ψ, ψ′) = ψ^{∗}ψ′_{∗}, where ψ^{∗} is left multiplication on R by ψ, and where ψ′_{∗} is right multiplication on R by ψ′, then

H^{0}(\underline{c}, R̃) = \{ϕ ∈ R| ϕψ = ψϕ for all ψ ∈ \underline{c}\},

i.e. the commutant of \underline{c} in R.

Given a k-algebra A, and consider a subcategory \underline{c} of the category of right Amodules. Let, as above π : \underline{c} → k − mod be the forgetful-functor, and consider the bifunctor,

Hom_{π} : \underline{c} × \underline{c} ⟶ k − mod

defined by

Hom_{π}(V_{i}, V_{j}) = Hom_{k}(V_{i}, V_{j}).

Put,

O_{0}(\underline{c}, π) := H^{0}(\underline{c}, Hom_{π}).

It is clear that O_{0}(\underline{c}, π) is a k-algebra, and that there is a canonical homomorphism of k-algebras,

η_{0}(\underline{c}, π) : A ⟶ O_{0}(\underline{c}, π),

see §5.

Example 2.2. Example 1.2 Let A be a commutative k-algebra of finite type, k algebraically closed, and let Spec(A) be the subcategory of A-mod consisting of the modules A/𝔭, where 𝔭 runs through Spec(A), the morphisms being only the obvious ones. It is easy to see that the homomorphism

η_{0}(Spec(A), π) : A ⟶ O_{0}(Spec(A), π)

identifies A/rad(A) with O_{0}(Spec(A), π). If rad(A) = 0 we even find an isomorphism,

η_{0}(Simp^{∗}(A), π) : A ≃ O_{0}(Simp^{∗}(A), π).

Here Simp^{∗}(A) is the subcategory of A-mod where the objects are A and the simple A-modules, A/𝔪, i.e. the closed points of Spec(A), and where the morphisms are the obvious quotient morphisms A → A/𝔪. η_{0}(Simp^{∗}(A), π) is, however not, in general, an isomorphism. This is easily seen when A is a local k-algebra. To remedy this situation we shall in [13] introduce and study a generalization O(\underline{c}, π) of O_{0}(\underline{c}, π) defined in terms of the noncommutative deformation theory, see the next §.

3. §2. Noncommutative deformations

The category \underline{a}_{r}, test algebras and liftings of modules

Let \underline{a}_{r} be the category of “r-pointed” artinian k-algebras. Recall that an object R of \underline{a}_{r} is a diagram of morphism of artinian k-algebras,

\xymatrix{k^{r} \ar[dr]_{=} \ar[r]^{ι} & R \ar[d]^{ρ} \\  & k^{r}}

Put, Rad(R) := kerρ, such that,

R/Rad(R) ≃ ∏_{j=1}^{r} k_{j}, k_{j} ≃ k.

A morphism ϕ : R → S of \underline{a}_{r} is a morphism of such diagrams inducing the identity on k^{r}, implying that the induced map,

k^{r} ≃ R/Rad(R) → S/Rad(S) ≃ k^{r}

is the identity. Pick idempotents e_{i} ∈ k^{r} ⊆ R such that

∑_{i=1}^{r} e_{i} = 1, e_{i}e_{j} = 0 if i ≠ j.

For every (i, j), we shall consider the subspace R_{ij} := e_{i}Re_{j} ⊆ R, and the pairing

R_{ij} ⊗_{k} R_{jk} → R_{ik}

given in terms of the multiplication in R.

Let

R′ = (R_{ij})

be the matrix algebra, the elements of which are matrices of the form

(α_{ij})

with α_{ij} ∈ R_{ij}, i, j = 1, ⋯, r. There is an obvious isomorphism of k-algebras

ϕ : R → R′

defined by

ϕ(α) = (e_{i}αe_{j}).

identifying the sub k-algebra k^{r} of R with the algebra of diagonal r × r-matrices. Now, for any pair (i, j), i, j = 1, …, r, consider the symbol ϵ_{ij}, and let’s agree to put all products of such symbols equal to zero. Then we define the (i, j)-test algebra R(i, j) as the matrix algebra

R(i, j) = k^{r} ⊕ ( \begin{matrix}& & j \\ & 0 & ⋮ & 0 \\ i & ⋯ & k · ϵ_{ij} & ⋯ \\ & 0 & ⋮ & 0 \end{matrix} ) for i ≠ j

R(i, i) = ( \begin{matrix}& & i \\ & k & ⋮ & 0 \\ i & ⋯ & k[ϵ_{ii}] & ⋯ \\ & 0 & ⋮ & k \end{matrix} ) for i = j

Denote by HH^{∗}(A, −) the Hochschild cohomology of the k-algebra A. If W is an A-bimodule denote by Der_{k}(A, W) the k-vectorspace of derivations of A in W. Thus ψ ∈ Der_{k}(A, W) is a linear map from A to W such that ψ(a_{1} · a_{2}) = a_{1}ψ(a_{2}) + ψ(a_{1})a_{2}.

In particular, any element w ∈ W determines a derivation i(w) ∈ Der_{k}(A, W) defined by i(w)(a) = aw − wa. There is an exact sequence

0 → HH^{0}(A, W) → W → Der_{k}(A, W) → HH^{1}(A, W) → 0

If V_{i}, V_{j} are right A-modules, then

Hom_{k}(V_{i}, V_{j})

is an A-bimodule. In fact if ϕ ∈ Hom_{k}(V_{i}, V_{j}), then aϕ is defined by (aϕ)(v) = ϕ(va), and ϕa is defined by (ϕa)(v) = ϕ(v)a.

Moreover, we know that

HH^{0}(A, Hom_{k}(V_{i}, V_{j})) = Hom_{A}(V_{i}, V_{j})
HH^{1}(A, Hom_{k}(V_{i}, V_{j})) = Ext_{A}^{1}(V_{i}, V_{j}).

Fix a finite family V = {V_{i}}_{i=1}^{r} of right A-modules, and consider for every

ψ ∈ Der_{k}(A, Hom_{k}(V_{i}, V_{j}))

the left R(i, j)-module and right A-module,

V_{ij}(ψ) = ( \begin{matrix}& & & & j \\ & V_{1} & & & ⋮ \\ i & ⋯ & V_{i} & ⋯ & ϵ_{ij}V_{j} & ⋯ \\ & & & & V_{j} \\ & & & & ⋮ & V_{r} \end{matrix} )

defined by

(\begin{matrix}v_{1} \\ & v_{i} & ϵ_{ij}v′_{j} \\ & & v_{j} \\ & & & v_{r} \end{matrix}) · a = (\begin{matrix}v_{1}a \\ & v_{i}a & ϵ_{ij}(ψ(a, v_{i}) + v′_{j}a) \\ & & v_{j}a \\ & & & v_{r}a \end{matrix})

and the obvious left R(i, j)-action. The R(i, j)- and the A-action commute, therefore we have got a R(i, j) ⊗ A-module, such that

k^{r} ⊗_{R(i,j)} V_{ij}(ψ) ≃ ⊕_{i=1}^{r}V_{i} .

V_{ij}(ψ) is called a lifting of V to R(i, j). It is easy to see that if ψ maps to zero in HH^{1}(A, Hom_{k}(V_{i}, V_{j})) = Ext_{A}^{1}(V_{i}, V_{j}) then the lifting V_{ij}(ψ) is trivial, i.e. isomorphic to the trivial one. Conversely, if V_{ij}(ψ) is trivial, then ψ maps to zero in Ext_{A}^{1}(V_{i}, V_{j}).

The noncommutative deformation functor

We are now ready to start the study of noncommutative deformations of the family V = {V_{i}}_{i=1}^{r}. We shall assume that V is a swarm, i.e. that for all i, j = 1, 2, ..., r,

dim_{k}Ext_{A}^{1}(V_{i}, V_{j}) < ∞.

Given an object ρ : R = (R_{i,j}) → k^{r} of \underline{a}_{r}, consider the left R-module (R_{i,j}⊗_{k}V_{j}). ρ defines a k-linear and left R-linear map,

ρ(R) : (R_{i,j} ⊗_{k} V_{j}) → ⊕_{i=1}^{r}V_{i},

inducing a homomorphism of R-endomorphism rings,

ρ̃(R) : (R_{i,j} ⊗_{k} Hom_{k}(V_{i}, V_{j})) → ⊕_{i=1}^{r}End_{k}(V_{i}).

The right A-module structure on the V′_{i}s is defined by a homomorphism of kalgebras,

η_{0} : A → ⊕_{i=1}^{r}End_{k}(V_{i}).

Definition 3.1. The deformation functor

Def_{V} : \underline{a}_{r} → Sets

is defined for every R ∈ \underline{a}_{r}, as the set,

Def_{V}(R) ∈ \underline{Sets}

of isoclasses of homomorphisms of k-algebras,

{η′ : A → (R_{i,j} ⊗_{k} Hom_{k}(V_{i}, V_{j}))}/ ∼

such that,

ρ̃(R) ∘ η′ = η_{0},

where the equivalence relation ∼ is defined by inner automorphisms in the k-algebra

End_{R}((R_{i,j} ⊗_{k} V_{j})) = (R_{i,j} ⊗_{k} Hom_{k}(V_{i}, V_{j})).

Any such isoclass η̃′ will be called a deformation or a lifting of V to R, and usually denoted V_{R}.

One easily proves that Def_{V} has the same properties as the ordinary deformation functor.

Let π : R → S be a morphism of \underline{a}_{r}, such that Rad(R)·ker π = 0. Morphisms like this will be called small. If V_{R} ∈ Def_{V} (R) it is easy to see that V_{S} := S ⊗_{R} V_{R} ∈ Def_{V} (S) and that \overline{V} = ker\{V_{R} → S ⊗_{R} V_{R}\} is, as a left R-module, an R/Rad(R) = k^{r}-module. Put ker π = (K_{ij}), then \overline{V} = (\overline{V}_{ij}) where \overline{V}_{ij} = K_{ij} ⊗_{k} V_{j}.

Consider now the k-vector spaces

E_{ij}^{d} = Ext_{A}^{d}(V_{i}, V_{j})^{∗}

i.e. the dual k-vectorspaces of Ext_{A}^{d}(V_{i}, V_{j}), and consider the k-algebra of matrices,

T_{2}^{d} = (\begin{matrix}k & & & & 0 \\ & . & . & . \\ 0 & & & & k \end{matrix}) + (ϵ_{ij}E_{ij}^{d})

where as above, we assume all products of the ϵ_{ij}’s are equal to zero. Now let for every i, j = 1, …, r, and d = 1, 2,

{t_{ij}^{d}(ℓ)}_{ℓ=1}^{e_{ij}^{d}}

be a basis of E_{ij}^{d}, and let {ψ_{ij}^{d}(ℓ)}_{ℓ=1}^{e_{ij}^{d}} be the dual basis. Thus e_{ij}^{k} = dim_{k}E_{ij}^{k}. Consider the k-algebra

T^{d} = (\begin{matrix}k & & & & 0 \\ & . & . & . \\ 0 & & & & k \end{matrix}) + (Ẽ_{ij}^{d})

freely generated as matrix algebra by the generators {t_{ij}^{d}(ℓ)}_{ℓ=1}^{e_{ij}^{d}}. An element of Ẽ_{ij}^{d} is then a matrix where the elements are linear combinations of elements of the form:

τ_{ij} = t_{ij_{1}}^{d}(l_{1}) ⊗ t_{j_{1}j_{2}}^{d}(l_{2}) ⊗ ⋯ ⊗ t_{j_{m−1}j_{m}}^{d}(l_{m})
j = j_{m}, 1 ⩽ l_{s} ⩽ e_{j_{s−1}j_{s}}^{d}, 1 ⩽ j_{s} ⩽ r, m ⩾ 1
of E_{ij_{1}}^{d} ⊗ E_{j_{1}j_{2}}^{d} ⊗ ⋯ ⊗ E_{j_{m−1}j}^{d}.

Obviously

T_{2}^{1} = T^{1}/Rad(T^{1})^{2}.

where Rad(T^{1}) is the two-sided ideal of T^{1} generated by (Ẽ_{ij}^{1}).

Lemma 3.1. Let R be an object of \underline{a}_{r} and suppose that there exists a surjective homomorphism

ϕ_{2} : T_{2}^{1} → R/Rad(R)^{2},

then there exists a surjective homomorphism

ϕ : T^{1} ⟶ R

which lifts ϕ_{2}.

Definition 3.2. For every object R of ˆ \underline{a}_{r}, put

T_{R} = (Rad(R)/Rad(R)^{2})^{∗}

and call it the tangent space of R.

Lemma 3.2. Let ϕ : R → S be a morphism of \underline{a}_{r}. Assume ϕ induces a surjective homomorphism

ϕ^{1} : T_{R}^{∗} → T_{S}^{∗}

(or an injective homomorphism on the tangent space level). Then ϕ is surjective.

Notice that if we pick any finite dimensional k-vectorspaces F_{ij}, then there is a unique maximal pro-algebra F = F (F_{ij}) in \underline{a}_{r} with tangent space

T_{F} ≃ (F_{ij}^{∗})

F is defined in the same way as T^{d}, above, with E^{d} replaced by F.

To prove the existence of a hull for the deformation functor Def_{V} the basic tool is the obstruction calculus, which in this case is easily established:

Proposition 3.1. Suppose R →^{ϕ}→ S is a surjective small morphism of \underline{a}_{r}, i.e. suppose kerϕ · Rad(R) = 0. Put kerϕ = (I_{ij}). Consider any V_{S} ∈ Def_{V} (S). Then there exists an obstruction

o(ϕ, V_{S}) ∈ (I_{ij} ⊗_{k} Ext_{A}^{2}(V_{i}, V_{j}))

which is zero if and only if there exists a lifting V_{R} ∈ Def_{V} (R) of V_{S}. The set of isomorphism classes of such liftings is a psevdotorsor under

(I_{ij} ⊗_{k} Ext_{A}^{1}(V_{i}, V_{j})) .

Proof. As a k-vectorspace V_{R} = (R_{ij} ⊗ V_{j}) maps onto V_{S} = (S_{ij} ⊗ V_{j}). Since the right action of A commutes with the left S-action the action of an element a ∈ A on V_{S} is uniquely given in terms of a family of k-linear maps,

a_{ij} : V_{i} → S_{ij} ⊗ V_{j}.

We may of course lift these to k-linear maps

σ(a)_{ij} : V_{i} → R_{ij} ⊗ V_{j}

inducing a lift of the action of each element of A on

⨁_{j=1}^{r}S_{ij} ⊗ V_{j}

to a k-linear action on

⨁_{j=1}^{r} R_{ij} ⊗ V_{,}.

The obstruction for this to be an A-module structure is, as usual, the Hochschild 2-cocycle

ψ^{2}(a, b) = σ(ab) − σ(a) · σ(b) ∈ (I_{ij} ⊗_{k} Hom_{k}(V_{i}, V_{j})) .

The fact that this is a 2-cocycle follows from

1. σ(c) · ψ^{2}(a, b) = c · ψ^{2}(a, b)

2. ψ^{2}(a, b) · σ(c) = ψ^{2}(a, b) · c

and the obvious relation

dψ^{2}(a, b, c) = aψ^{2}(b, c) − ψ^{2}(ab, c) + ψ^{2}(a, bc) − ψ^{2}(a, b) · c
= σ(a)(σ(bc) − σ(a)σ(c)) − (σ(abc) − σ(ab)σ(c)) + (σ(abc) − σ(a)σ(bc))
− (σ(ab) − σ(a)σ(b))σ(c) ≡ 0

Suppose the class of ψ^{2} in (I_{ij} ⊗_{k} Ext_{A}^{2}(V_{i}, V_{j})) is zero. This means that ψ^{2} = dϕ, where ϕ ∈ Hom_{k}(A, (I_{ij} ⊗_{k} Hom_{k}(V_{i}, V_{j})), ψ^{2}(a, b) = dϕ(a, b) = aϕ(b) − ϕ(ab) + ϕ(a)b. Let σ′ = σ + ϕ and consider,

σ′(ab) − σ′(a)σ′(b) = σ(ab) − σ(a)σ(b) + ϕ(ab) − σ(a)ϕ(b) − ϕ(a)σ(b) − ϕ(a)ϕ(b) .

Since the matrix ϕ(a)ϕ(b) = 0 as I_{ij} ·I_{jk} = 0, ∀i, j, k and since σ(a)ϕ(b) = aϕ(b), ϕ(a)σ(b)=ϕ(a)b for the same reason, we find that σ′(ab) − σ′(a)σ′(b)=0, i.e. there is a lifting of the A-module action to V_{R} = (R_{ij} ⊗ V_{j}).

If we have given one A-module action σ on V_{R} lifting the action on V_{S}, then for any other σ′ we may consider the difference

σ′ − σ : A → (I_{ij} ⊗_{k} Hom_{k}(V_{i}, V_{j}))

Consider

d(σ′ − σ)(a, b) = a(σ′(b) − σ(b)) − (σ′(ab) − σ(ab)) + (σ′(a) − σ(a))b

As above we may substitute σ′(a) for a and σ(b) for b, and the expression becomes zero. Thus σ′ − σ = ξ̄ defines a class

ξ ∈ (I_{ij} ⊗_{k} Ext_{A}^{1}(V_{i}, V_{j})) .

If ξ = 0, then ξ̄ = dϕ, ϕ ∈ (I_{ij} ⊗_{k} Hom_{k}(V_{i}, V_{j})) such that σ′(a) − σ(a) = aϕ − ϕa. Let ϕ = (ϕ_{ij}), then ϕ_{ij} defines an isomorphism

ϕ̄ = id + ϕ : ⨁_{j} R_{ij} ⊗ V_{j} → ⨁_{j} R_{ij} ⊗ V_{j}

lifting the identity of ⨁_{j} S_{ij} ⊗ V_{j}. Moreover

σ(a)(id + ϕ)(v_{i}) = σ(a)v_{i} + aϕ(v_{i})
= σ′(a)(v_{i}) + ϕ(av_{i}) = (id + ϕ)σ′(a)(v_{i})

since ϕ(σ′(a)v_{i}) = ϕ(av_{i}).

Therefore the A-module structures on

V_{R} = (R_{ij} ⊗ V_{j})

defined by σ and σ′ are isomorphic. The rest is clear.

Theorem 3.1. The functor Def_{V} has a prorepresentable hull, or a formal moduli of V, H ∈ ˆ \underline{a}_{r}, together with a versal family

Ṽ = (H_{i,j} ⊗ V_{j}) ∈ \underleftarrow{lim}_{n⩾1} Def_{V}(H/Rad(H)^{n})

such that the corresponding morphism of functors on \underline{a}_{r},

ρ : Mor(H, −) → Def_{V}

is smooth and an isomorphism on the tangent level. Moreover, H is uniquely determined by a set of matric Massey products defined on subspaces,

D_{n} ⊂ ⨁_{p=2}^{n} Ext^{1}(V_{i}, V_{j_{1}}) ⊗ ⋯ ⊗ Ext^{1}(V_{j_{p−1}}, V_{j}),

with values in Ext^{2}(V_{i}, V_{j}).

Proof. Notice first that ρ being an isomorphism at the tangent level means that ρ is an isomorphism for all objects R of \underline{a}_{r} for which Rad(R)^{2} = 0.

Word for word we may copy the proof (4.2) of [6], and the proof of [7]. In particular H/Rad(H)^{2} ≃ T_{2}^{1} and

Mor(H, R(i, j)) ≃ Hom_{k}(E_{ij}^{1}, k) ≃ Ext_{A}^{1}(V_{i}, V_{j}) ≃ Def_{V}(R(i, j)).

Notice that the universal lifting of V to T_{2}^{1} is the T_{2}^{1} ⊗_{k} A-module Ṽ_{2}

(\begin{matrix}V_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & V_{2} \end{matrix}) + (E_{ij}^{1} ⊗_{k} V_{j})

with the obvious left T_{2}^{1}-action and the right A-action defined as,

((1 ⊗ v_{i}) · a)_{ii} = 1 ⊗ v_{i} · a + ∑_{ℓ} t_{i,j}^{1}(ℓ) ⊗ (ψ_{i,j}^{1}(ℓ)(a, v_{i}))

where v_{i} ∈ V_{i}, and where {t_{ij}^{1}(ℓ)}_{ℓ=1}^{e_{ij}} is the chosen basis of E_{ij}^{1}. Recall that {ψ_{ij}^{1}(ℓ)}_{ℓ=1}^{e_{ij}}, the dual base, consists of elements ψ_{ij}^{1}(ℓ) ∈ Ext_{A}^{1}(V_{i}, V_{j}), which may be represented as elements of Der_{k}(A, Hom_{k}(V_{i}, V_{j})).

To obtain H we kill obstructions for lifting Ṽ_{2} successively, to T_{3}^{1} := T^{1}/Rad(T^{1})^{3}, T_{4}^{1} etc. just like in the commutative case. The proof of the existence of a prorepresentable hull for Def_{V} can, of course, also be modeled on the classical proof of M.Schlessinger [15]. This has been carried out by Runar Ile, see [2].

A general structure theorem for artinian k-algebras

For every deformation V_{R} ∈ Def_{V}(R) there exists, by definition an, up to inner automorphisms, unique homomorphism of k-algebras,

η_{V_{R}} : A → End_{R}(V_{R}) = (R_{ij} ⊗ Hom_{k}(V_{i}, V_{j})).

Definition 3.3. Let V = {V_{i}}_{i=1}^{r} be any finite swarm of A-modules, and let H := H(V) be the formal moduli for V, and Ṽ the versal family. The k-algebra of observables of the family V is the k-algebra,

O(V) := End_{H}(Ṽ) = (H_{ij} ⊗ Hom_{k}(V_{i}, V_{j}))

We would like to describe the kernel and the image of the map,

η : A → O(V)

To do this we need to consider the matric Massey products of the form,

D_{n} → Ext_{A}^{2}(V_{i_{1}}, V_{i_{n}}),

the obvious generalizations of the matrix Massey products introduced in [7].

Here we shall describe these products using Hochschild cohomology. This is a more convenient way of describing the map η and maybe also an easier way of understanding the nature of the Massey products.

To simplify the notations, put

Ext_{A}^{1}(V) = (Ext_{A}^{1}(V_{i}, V_{j}))

For l = 2, the Massey product above is simply the cup product

Ext_{A}^{1}(V) ⊗ Ext_{A}^{1}(V) → Ext_{A}^{2}(V)

defined by: Let (ψ_{ij}^{1}), (ψ_{ij}^{2}) ∈ Ext_{A}^{1}(V), and express ψ_{ij}^{k} as 1-Hochschild cocycles, i.e. ψ̄_{ij}^{1} ∈ Der_{k}(A, Hom_{k}(V_{i}, V_{j})), ψ̄_{ij}^{2} ∈ Der_{k}(A, Hom_{k}(V_{i}, V_{j})). The cup product (ψ_{ij}^{1}) ∪ (ψ_{ij}^{2}) ∈ Ext_{A}^{2}(V), now denoted

⟨(ψ_{ij}^{1}), (ψ_{ij}^{2})⟩ ∈ Ext_{A}^{2}(V)

is defined by the 2-cocycle in the Hochschild complex

⟨(ψ_{ij}^{1}), (ψ_{ij}^{2})⟩_{ik}(a, b) = ∑_{j} ψ̄_{ij}^{1}(a) ∘ ψ̄_{jk}^{2}(b) ∈ Hom_{k}(V_{i}, V_{k})

Suppose ⟨(ψ_{ij}^{1}), (ψ_{ij}^{2})⟩ = 0, this means that there exists, for each pair (i, k) a 1cochain ϕ_{ik}^{12} in the Hochschild complex, i.e. a map

ϕ_{ik}^{12} ∈ Hom_{k}(A, Hom_{k}(V_{i}, V_{k}))

such that dϕ_{ik}^{12} = ⟨(ψ_{ij}^{1}), (ψ_{ij}^{2})⟩_{ik}, i.e. such that for all a, b ∈ A,

aϕ_{ik}^{12}(b) − ϕ_{ik}^{12}(ab) + ϕ_{ik}^{12}(a)b = ∑_{j} ψ̄_{ij}^{1}(a) ∘ ψ̄_{jk}^{2}(b)

Given classes ψ^{1} = (ψ_{ij}^{1}), ψ^{2} = (ψ_{ij}^{2}), ψ^{3} = (ψ_{ij}^{3}) ∈ Ext_{A}^{1}(V) such that ⟨ψ^{1}, ψ^{2}⟩ = ⟨ψ^{2}, ψ^{3}⟩ = 0 there exists ϕ^{12} = (ϕ_{ik}^{12}), ϕ^{23} = (ϕ_{ik}^{23}) ∈ Hom_{k}(A, Hom_{k}(V_{i}, V_{k})) such that

dϕ^{12} = ⟨ψ^{1}, ψ^{2}⟩, dϕ^{23} = ⟨ψ^{2}, ψ^{3}⟩ .

Then there exists a matrix Massey product

⟨ψ^{1}, ψ^{2}, ψ^{3}⟩ ∈ Ext_{A}^{2}(V)

defined by the 2-cocycle

⟨ψ^{1}, ψ^{2}, ψ^{3}⟩_{ik}(a, b) = ∑_{j} ϕ_{ij}^{12}(a)ψ_{jk}^{3}(b) − ∑_{j} ψ_{ij}^{1}(a)ϕ_{jk}^{23}(b)

in Hom_{k}(A ⊗_{k} A, Hom_{k}(V_{i}, V_{j})).

As in [7] we may go on and obtain a sequence of defining systems {D_{n}}_{n=2}^{∞} and Massey products, computing the relations of H(V).

Now if a ∈ A, denote by ã_{i} ∈ Hom_{k}(V_{i}, V_{i}) its action on V_{i}, i = 1, …, d. Let End_{0}(V) be the diagonal matrix (End_{k}(V_{i}, V_{i})), contained in the matrix End_{k}(V) := (End_{k}(V_{i}, V_{j})). Put,

End(V)a = (ã_{1}, …, ã_{d}) ∈ End_{0}(V) ⊆ End(V)

If a ∈ A is such that End(V)a = 0, this means that a acts trivially on each V_{i}. Let ψ ∈ Ext_{A}^{1}(V) be represented by 1-cocycles ψ_{ij} ∈ Der_{k}(A, End_{k}(V_{i}, V_{j})). If End(V)a = End(V)b = 0, we find,

ψ_{ij}(ab) = aψ_{ij}(b) + ψ_{ij}(a)b = 0

This shows that ψ ∈ Ext_{A}^{1}(V) defines a unique k-linear map,

ψ : {a ∈ A | End(V)a = 0} → End_{k}(V)

vanishing on all squares.

Let a ∈ A, End(V)a = 0, and put

Ext_{A}^{1}(V)a = 0

when ψ(a) = 0, ∀ψ ∈ Ext_{A}^{1}(V). Consider the sub k-vector space of A

K_{1} = {a ∈ A | End(V)a = Ext_{A}^{1}(V)a = 0}

Let ∑ α_{ij}ψ^{i} ⊗ ψ^{j} ∈ Ext_{A}^{1}(V) ⊗ Ext_{A}^{1}(V) such that its Massey (cup-)product is zero, i.e. such that:

∑ α_{ij}⟨ψ^{i}, ψ^{j}⟩ = 0

Then there exists a 1-cochain ϕ ∈ Hom_{k}(A, (Hom_{k}(V_{i}, V_{j}))) such that

dϕ = ∑_{ij} α_{ij}⟨ψ^{i}, ψ^{j}⟩

Since dϕ = 0 implies that ϕ represents an element of Ext_{A}^{1}(V) it is clear that ϕ defines a unique k-linear map

ϕ : K_{1} → End_{k}(V) .

Let us denote by

ker⟨Ext_{A}^{1}(V), Ext_{A}^{1}(V)⟩

the subset of Ext_{A}^{1}(V) ⊗ Ext_{A}^{1}(V) for which the Massey product (i.e. the cup product) is zero. Then we may put

ker⟨Ext_{A}^{1}(V), Ext_{A}^{1}(V)⟩a = 0

if for every dϕ ∈ ker⟨Ext_{A}^{1}(V), Ext_{A}^{1}(V)⟩, ϕ(a) = 0.

Let

K_{2} = {a ∈ A | End(V)a = Ext_{A}^{1}(V)a = ker⟨Ext_{A}^{1}(V), Ext_{A}^{1}(V)⟩a = 0}

Continuing in this way we find a sequence of ideals {K_{n}}_{n⩾0}, where K_{0} = ker{A → End(V)} and, in general, K_{n} = {a ∈ A | D_{n}a = 0}.

Theorem 3.2. Let A be any k-algebra and let V = {V_{i}}_{i=1}^{r} be a swarm of Amodules. Then the kernel of the canonical map

η : A → O(V)

is determined by the matrix Massey product structure of Ext_{A}^{i}(V), i = 1, 2. In fact

ker η = ⋂_{n⩾0} K_{n}

Proof. By definition, the homomorphism of k-algebras

η : A → O(V)

lifts the k-algebra homomorphism,

η_{0} : A → ∏_{i=1}^{d} End_{A}(V_{i}).

Modulo Rad(H)^{2} η induces the homomorphism,

η_{1} : A → ∏_{i=1}^{d} End_{k}(V_{i}) ⊕ (E_{ij}^{1} ⊗ Hom_{k}(V_{i}, V_{j}))

with,

η_{1}(a)_{ij} = δ_{ij} ⊗ η_{0}(a)_{i} + ∑_{l} t_{ij}(l) ⊗ ψ_{ij}^{1}(l)(a_{i}, −) , δ_{ij} = \begin{cases}0 & i ≠ j \\ 1 & i = j \end{cases}

Now, by construction H is the quotient of the formally free k-algebra T^{1} generated by the independent variables {t_{ij}(l), l = 1, …, l_{ij}} as explained above. The relations of T^{1} are generated by linear combinations of monomials in these variables of the form,

y_{ik} = ∑_{r=1}^{∞} ∑_{\underline{j},\underline{l}} α_{i,j_{1},...,j_{r−1},k}^{l_{1},...,l_{r}}t_{ij_{1}}(l_{1})t_{j_{1}j_{2}}(l_{2}) ⋯ t_{j_{r−1},k}(l_{r}),

corresponding to elements,

y_{ik} ∈ Ext_{A}^{2}(V_{i}, V_{k})^{∗}.

The coefficients α are expressed in terms of partially, but inductively well defined, matric Massey products,

⟨ ⟩\underline{r} : D\underline{r} ⟶ Ext_{A}^{2}(V)

such that, if the Massey product ⟨ψ_{ij_{1}}^{1}(l_{1}), …, ψ_{j_{r−1},k}^{1}(l_{r})⟩ is defined, then

y_{ik}(⟨ψ_{ij_{1}}^{1}(l_{1}), …, ψ_{j_{r−1},k}^{1}(l_{r})⟩) = α_{ij_{1},...,j_{r−1},k}^{l_{1},...,l_{r}}.

We therefore obtain a basis for H, as k-vector space, by picking, in a coherent way, a k-basis for

coker\{Ext_{A}^{2}(V)^{∗} → D∗ ; \underline{r}\} = (ker⟨ ⟩\underline{r})^{∗}

Since K_{r} = ker⟨ ⟩\underline{r}, the conclusion of the Theorem follows.

Remark 3.1. Let E_{ij} be an extension of V_{i} by V_{j}, then as a k-vector space E_{ij} = V_{j} ⊕ V_{i} and the right action by A is defined for (v_{j}, v_{i}) ∈ E_{ij}, a ∈ A by,

(v_{j}, v_{i})a = (v_{j}a + ψ_{ij}^{1}(a, v_{i}), v_{i}a),

where,

ψ_{ij}^{1} ∈ Der_{k}(A, Hom_{k}(V_{i}, V_{j}))

defines an element,

ψ̄_{ij}^{1} ∈ Ext_{A}^{1}(V_{i}, V_{j})

corresponding to E_{ij}. Suppose we consider an extension E_{ijk} of E_{ij} by V_{k}. Then as a k-vector space E_{ijk} ≃ V_{k} ⊕ E_{ij} = V_{k} ⊕ V_{j} ⊕ V_{i} and the action by A is defined by

(v_{k}, v_{j}, v_{i})a = (v_{k}a + ϕ(a, (v_{j}, v_{i})), v_{j}a + ψ_{ij}^{1}(a, v_{i}), v_{i}a).

By additivity

ϕ(a, (v_{j}, v_{i})) = ϕ(a, (v_{j}, 0)) + ϕ(a, (0, v_{i})) .

Put

ψ_{ij}^{1,0}(a, v_{i}) = ψ_{ij}^{1}(a, v_{i}), ψ_{jk}^{0,1}(a, v_{j}) = ϕ(a, (v_{j}, 0)), ψ_{ik}^{1,1}(a, v_{i}) = ϕ(a, (0, v_{i})), then the conditions on the action imply

ψ_{jk}^{0,1} ∈ Der_{k}(A, Hom_{k}(V_{j}, V_{k}))
ψ_{jk}^{1,1} ∈ Hom_{k}(A, Hom_{k}(V_{i}, V_{k}))

and

dψ_{ik}^{1,1} = ψ_{jk}^{0,1} ∘ ψ_{ij}^{1,0}.

This means that ψ̄_{jk}^{0,1} ∈ Ext_{A}^{1}(V_{j}, V_{k}) and that the cup product,

ψ̄^{0,1} ∪ ψ̄^{1,0} ∈ Ext_{A}^{2}(V_{i}, V_{k})

is zero.

Now, consider an extension E_{ijkl} of E_{ijk} by V_{l}. As before the action of A on E_{ijkl} is given by

(v_{l}, v_{k}, v_{j}, v_{i}) · a

= (v_{l} · a + ϕ(a, v_{k}, v_{j}, v_{i}), v_{k} · a + ψ_{ik}^{2}(a, v_{i}) + ψ_{jk}^{0,1}(a, v_{j}), v_{j} · a + ψ_{ij}^{1}(a, v_{i}), v_{i} · a).

Put, as above,

ψ^{1,0,0} = ψ^{1,0}
ψ^{0,1,0} = ψ^{0,1}
ψ_{kl}^{0,0,1}(a, v_{k}) = ϕ(a, v_{k}, 0, 0)
ψ_{jl}^{0,1,1}(a, v_{j}) = ϕ(a, 0, v_{j}, 0)
ψ_{ik}^{1,1,1}(a, v_{i}) = ϕ(a, 0, 0, v_{i})

The conditions on ϕ are expressed by:

dψ_{kl}^{0,0,1} = 0
dψ_{jl}^{0,1,1} = ψ_{kl}^{0,0,1} ∘ ψ_{jk}^{0,1,0}
dψ_{il}^{1,1,1} = ψ_{jl}^{0,1,1} ∘ ψ_{ij}^{1,0,0} + ψ_{kl}^{0,0,1} ∘ ψ_{ik}^{1,1,0}

This means that ψ̄_{kl}^{0,0,1} ∈ Ext_{A}^{1}(V_{k}, V_{l}), that the cup product ψ̄_{kl}^{0,0,1} ∪ ψ̄_{jk}^{0,1,0} ∈ Ext_{A}^{2}(V_{j}, V_{l}) is zero, and that the Massey product

⟨ψ̄_{kl}^{0,0,1}, ψ̄_{jk}^{0,1,0}, ψ̄_{ij}^{1,0,0}⟩ ∈ Ext_{A}^{2}(V_{i}, V_{l})

is zero.

It is clear how to continue.

Corollary 3.1. Suppose the k-algebra A is of finite dimension, and let the finite swarm V = {V_{i}}_{i=1}^{r} contain all simple representations, then

η : A → O(V)

is injective.

Proof. Let a ∈ A, and suppose η(a) = 0. Since A as a right A-module is an extension of the V′_{i}s we may assume there are exact sequences of right A-modules

\xymatrix{0 \ar[r] & Q_{1} \ar[r] & A \ar[r] & _{⨁i∈I_{1}}V_{i} \ar[r] & 0 \\ 0 \ar[r] & Q_{2} \ar[r] & Q_{1} \ar[r] & _{⨁i∈I_{2}}V_{i} \ar[r] & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 \ar[r] & Q_{N} \ar[r] & Q_{N−1} \ar[r] & _{⨁i∈I_{N}} V_{i} \ar[r] & 0}

with Q_{N} = ⨁_{i∈I_{N+1}} V_{i}, Q_{N+1} = 0. Since End(V)a = 0 it follows from the first exact sequence above that 1 · a = a ∈ Q_{1}. Consider the exact sequence

0 ⟶ ⨁_{i∈I_{2}}V_{i} ⟶ A/Q_{2} ⟶ ⨁_{i∈I_{1}}V_{i} ⟶ 0

Since Ext_{A}^{1}(V)a = 0 it follows that 1 · a = a ∈ Q_{2}. In fact, multiplication by a is zero on V_{i}, i = 1, …, r and on A/Q_{2} it is therefore given by the elements in Ext_{A}^{1}(V). Continuing in this way, we consider the extensions of extensions,

\xymatrix{0 \ar[r] & _{⨁i∈I_{3}}V_{i} \ar[r] & A/Q_{3} \ar[r] & A/Q_{2} \ar[r] & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 \ar[r] & _{⨁i∈I_{N+1}}V_{i} \ar[r] & A \ar[r] & A/Q_{N} \ar[r] & 0.}

Refering to (2.9), we know that the multiplication by a ∈ A on the right in the middle term is given inductively, by a family of cochains ψ\underline{ϵ} ; ij ∈ Hom_{k}(A, Hom_{k}(V_{i}, V_{j})), with \underline{ϵ} ∈ \{0, 1\}^{n}, for 2 ⩽ n, such that

dψ\underline{ϵ} ; ik = ∑_{{\underline{ϵ}_{1}+\underline{ϵ}_{2}=\underline{ϵ} \atop j}} ψ_{ij}^{\underline{ϵ}_{1}} ∘ ψ_{jk}^{\underline{ϵ}_{2}} .

Now, this means that all these extensions are defined in terms of a series of well defined Massey products each one containing 0. By the proof of Theorem (2.8), we find that for all i, j and all \underline{ϵ}, ψ\underline{ϵ} ; ij(a, −) = 0.

This means that the action of a ∈ A must be 0, so a = 0.

The same proof works for the following, Corollary 3.2 (2.10 bis). Suppose the k-algebra A is an iterated extension of the objects in the finite swarm V = {V_{i}}_{i=1}^{r}. Then

η : A → O(V)

is injective.

Corollary 3.3. Suppose A is an object of \underline{a}_{r}, and let V = \{V_{i}\}_{i=1}^{r} be the family of simple representations, with V_{i} ≃ k_{i}. Then

A ≃ H

Proof. Obviously A is a left A-and a right A-module, flat over A, therefore A ∈ Def_{V} (A). Let R ∈ \underline{a}_{r} and pick an element V_{R} ∈ Def_{V}(R). Since

End(V) = (\begin{matrix}k & ⋯ & k \\ ⋮ & & ⋮ \\ k & ⋯ & k \end{matrix}) ,

this amounts to a homomorphism of k-algebras A → End_{R}(V_{R}) = R, implying that A is versal. But then the unicity of the hull of Def_{V} gives us an isomorphism:

ϕ : H → A

Remark 3.2. (Reconstructing an ordered set Λ and k[Λ], from the swarm of simple modules)

Let Λ be an ordered set, see §1, and let A = k[Λ], V = {k_{λ}}_{λ∈Λ}. Then the Corollary above implies that H ≃ k[Λ].

1. By the general theory we know that A = k[Λ] is the matrix algebra generated freely by the immediate relations λ_{1} ≫ λ_{2}, i.e. those for which {λ‘ ∈ Λ|λ_{1} > λ‘ > λ_{2}} = ∅, modulo relations of the form

(λ‘ > λ_{2}^{1})(λ_{2}^{1} > λ_{3}^{1}) ⋯ (λ_{n_{1}}^{1} > λ)
= (λ‘ > λ_{2}^{2})(λ_{2}^{2} > λ_{3}^{2}) ⋯ (λ_{n_{2}}^{2} > λ)

They correspond to the first obstructions, given by the n_{i} term well defined Massey products

Ext_{A}^{1}(k_{λ′}, k_{λ_{2}^{1}}) ⊗ ⋯ ⊗ Ext_{A}^{1}(k_{λ_{n_{1}}^{1}} , k_{λ}) → Ext_{A}^{2}(k_{λ′}, k_{λ})
Ext_{A}^{1}(k_{λ′}, k_{λ_{2}^{2}}) ⊗ ⋯ ⊗ Ext_{A}^{1}(k_{λ_{n_{2}}^{2}} , k_{λ}) → Ext_{A}^{2}(k_{λ′}, k_{λ})

There are as many relations as there are base elements of Ext_{A}^{2}(k_{λ′}, k_{λ}). 2. Let us check this for the diamond, i.e. for Λ:

[figure: →^{1} ∘; →^{2} ∘; →^{3} ∘; →_{4} ∘]

One easily computes the Ext’s,

Ext_{A}^{1}(k_{λ_{i}}, k_{λ_{j}}) = \begin{cases}0 & i = j \\ k & for i = 1, j = 2, 3 \\ k & for i = 2, 3, j = 4 \end{cases}

Ext_{A}^{2}(k_{λ_{i}}, k_{λ_{j}}) = \begin{cases}0 & for (i, j) ≠ (1, 4) \\ k & for i = 1, j = 4 \end{cases}

The two cup-products

Ext_{A}^{1}(k_{λ_{1}}, k_{λ_{j}}) ⊗ Ext_{A}^{1}(k_{λ_{j}}, k_{λ_{4}}) → Ext_{A}^{2}(k_{λ_{1}}, k_{λ_{4}}) for j = 2, 3,

are non-trivial. At the tangent level we have:

H_{2} = (\begin{matrix}k & k & k & 0 \\ 0 & k & 0 & k \\ 0 & 0 & k & k \\ 0 & 0 & 0 & k \end{matrix})

Therefore H must be a quotient of the matrix ring,

T^{1} = (\begin{matrix}k & t_{12} · k & t_{13} · k & (t_{12}t_{24} · k + t_{13}t_{34} · k) \\ 0 & k & 0 & t_{24} · k \\ 0 & 0 & k & t_{34} · k \\ 0 & 0 & 0 & k \end{matrix})

The kernel of T^{1} → H is given in terms of the cup products above. In fact, since we have t_{13}^{∗} ∪ t_{34}^{∗} = t_{12}^{∗} ∪ t_{24}^{∗} = y^{∗} where y^{∗} is the generator of Ext_{A}^{2}(k_{λ_{1}}, k_{λ_{4}}), the kernel of T^{1} → H is simply t_{13} ⊗ t_{34} + t_{12} ⊗ t_{24} such that

H = (\begin{matrix}k & k & k & k \\ 0 & k & 0 & k \\ 0 & 0 & k & k \\ 0 & 0 & 0 & k \end{matrix}) ≃ k[Λ]

as it should.

In general, we may reconstruct Λ from the tangent space T_{H} and the Masseyproducts above.

The corresponding problem for finite groups, i.e. reconstructing G from k[G] is called the isomorphism problem. Due to some nice examples of Dade, we know that this is hopeless. In fact there are two non isomorphic finite groups such that their group algebras are isomorphic for all fields.

§3. Noncommutative modular deformations

Let V be any right A-module such that dim_{k}Ext_{A}^{1}(V, V) < ∞. Consider the formal moduli H^{A} =: H, the formal versal family Ṽ = H⊗V, and the corresponding morphism of functors,

ρ : Mor_{\underline{a}_{r}}(H, −) → Def_{V} .

We know that ρ is not, in general, injective. However, V is also a right A⊗End_{A}(V)- module. As such it has a formal moduli H^{A,End}, and there is a natural k-algebra homomorphism, H^{A} → H^{A,End}. Let H_{0}^{A} be the unique maximal common quotient of H^{A} and H^{A,End}. Using the same construction as in [8], §2, we prove that the composition,

ρ_{0} : Mor_{\underline{a}_{r}}(H_{0}, −) → Mor_{\underline{a}_{r}}(H, −) → Def_{V}

is injective.

At the tangent level, the homomorphisms,

H^{A} → H^{A,End} ← H^{End},

looks like the canonical homomorphisms,

Ext_{A}^{1}(V, V) ← Ext_{A⊗_{k}End}^{1}(V, V) → Ext_{End}^{1}(V, V).

Representing elements of the Ext-groups as derivations, it is easy to see that the two images are contained in the subspace Ext_{A}^{1}(V, V)^{End}, respectively Ext_{End}^{1}(V, V)^{A}. Therefore the tangent space of H_{0} must be contained in the subspace of invariants under End_{A}(V) of the tangent space of H, Ext_{A}^{1}(V, V)^{End}.

The tangent space of the modular (prorepresenting) substratum, and almost split sequences

Consider as above a swarm V = {V_{i}}_{i=1}^{r} of A-modules, and consider the k^{r}- algebra

End_{A}(V) = (Hom_{A}(V_{i}, V_{j})).

Suppose from now on that the modules V_{i} are non-isomorphic, indecomposables, and that for each i = 1, .., r. End_{A}(V_{i}) is a commutative local ring with maximal ideal \underline{m}_{i}.

Lemma 3.3. Under the above assumptions, the radical of End_{A}(V) has the form

rad(V) = (\begin{matrix}m_{1}End_{A}(V_{1}) & & & ⋮ \\ ⋯ & ⋯ & \underline{m}_{i}End_{A}(V_{i}) & Hom_{A}(V_{i}, V_{j}) & ⋯ \\ & & & \underline{m}_{j}End_{A}(V_{j}) \\ & & & ⋮ & \underline{m}_{r}End_{A}(V_{r}) \end{matrix} )

Proof. We need only check that rad(V) is an ideal, and this amounts to proving that if ϕ_{ij} ∈ Hom(V_{i}, V_{j}) i ≠ j and ϕ_{ji} ∈ Hom(V_{j}, V_{i}) then

ϕ_{ji}ϕ_{ij} ∈ \underline{m}_{i} ⊆ End_{A}(V_{i}).

Suppose ϕ_{ji}ϕ_{ij} is not in \underline{m}_{i}, then ϕ_{ji}ϕ_{ij} is an isomorphism, and we may as well assume that ϕ_{ji}ϕ_{ij} = id_{V_{i}}. But then V_{j} ≃ V_{i} ⊕ ker ϕ_{ji} which contradicts the indecomposability of V_{j}.

In particular this lemma proves that if A is artinian and all V_{i} are of finite type, then for some N,

rad(V)^{N} = 0

Obviously there is a left and a right action of End_{A}(V) on

T_{H} = (Ext_{A}^{1}(V_{i}, V_{j})).

The difference between these actions defines the action of the Lie algebra End_{A}(V) on T_{H}. The invariants of T_{H} under the Lie algebra rad(V), is equal to the invariants under End_{A}(V), therefore equal to,

T_{H_{0}} := {ξ ∈ T_{H}|∀ϕ ∈ End_{A}(V), ϕξ − ξϕ = 0},

containing the tangent space of the modular, or the prorepresentable substratum H_{0} of H.

Lemma 3.4. Let ξ ∈ T_{H_{0}}, with ξ = (ξ_{i,j}), then for all ϕ = (ϕ_{k,l}) ∈ rad(V) we have for i≠ j, and all l,

ϕ_{l,i}ξ_{i,j} = 0
ξ_{i,j}ϕ_{j,l} = 0.

Moreover, for all i, j

ϕ_{i,j}ξ_{j,j} = ξ_{i,i}ϕ_{i,j}

Proof. Just computation.

Definition 3.4. In the above situation, an extension ξ ∈ Ext_{A}^{1}(V_{i}, V_{j}) is called a left almost split extension (resp. a right almost split extension), lase (resp. rase) for short, if for all ϕ_{ki} ∈ r(V)_{ki} (resp. ϕ_{jk} ∈ r(V)_{jk})

ϕ_{ki}ξ = 0 (resp. ξϕ_{jk} = 0).

An extension ξ which is both a lase and a rase is called an ase, an almost split extension.

This, of course, is nothing but a trivial generalization of the notion of almost split sequence, due to Auslander, see [14].

Denote by Ext_{l}^{1}(V_{i}, V_{j}) (resp. Ext_{r}^{1}(V_{i}, V_{j})) the subspace of Ext_{A}^{1}(V_{i}, V_{j}) formed by the lase’s (resp. rase’s), and put

T_{H}^{l} = (Ext_{l}^{1}(V_{i}, V_{j})) ⊆ T_{H}
T_{H}^{r} = (Ext_{r}^{1}(V_{i}, V_{j})) ⊆ T_{H}
T_{H}^{a} = T_{H}^{l} ∩ T_{H}^{r} =: (Ext_{a}^{1}(V_{i}, V_{j})) ⊆ T_{H}.

Observe that since the left and the right action of End(V) on T_{H} commute, End(V) acts at right on T_{H}^{l} and at left on T_{H}^{r}. Moreover, by the lemma above

T_{H}^{a} = T_{H}^{l} ∩ T_{H}^{r} ⊆ T_{H_{0}}.

Observe also that if End_{A}(V_{i}) = k ⊕ \underline{m}_{i} the diagonal part of T_{H_{0}} is exactly the tangent space of the deformation functor of the full subcategory of mod_{A} generated by V, see [6].

The structure of the modular substratum, and the existence of almost split sequences for artinian k-algebras

Assume that A is artinian, and that the V′_{i}s are of finite type. Then T_{H} is a kvectorspace of finite dimension, and the radical rad(V) of End(V) acts nilpotently on T_{H}.

Corollary 3.4. Given i ∈ {1, …, r}, assume there exists one j ∈ {1, …, r} such that Ext_{A}^{1}(V_{i}, V_{j}) ≠ 0. Then there exists a τ(i) ∈ {1, …, r} such that,

Ext_{r}^{1}(V_{i}, V_{τ(i)}) ≠ 0.

Proof. This is simply Engels theorem for the right action of rad(V) on T_{H}.

Theorem 3.3. Suppose V is such that every extension ξ ∈ Ext_{A}^{1}(V_{i}, V_{j}) is of the form 0 → V_{j} → E → V_{i} → 0 with E a direct sum of V_{k}’s. Then, for every i = 1, …, r, such that there exists a j = 1, …, r for which Ext_{A}^{1}(V_{i}, V_{j}) ≠ 0, there is a unique ase of the form

0 → V_{τ(i)} → E_{i} → V_{i} → 0

Moreover, if we agree to put τ(i) = i for those i’s for which Ext_{A}^{1}(V_{i}, V_{k}) = 0 for all k, then

τ : {1, …, r} → {1, …, r}

is a permutation.

Proof. We already know that there exists a rase of the form

ξ_{i} : 0 → V_{τ(i)} → E_{i} → V_{i} = 0.

We shall prove that ξ_{i} is also a lase. Let ϕ_{ki} ∈ Hom_{A}(V_{k}, V_{i}) for k ≠ i, or pick an element ϕ_{ii} ∈ \underline{m}_{i} ⊆ End(V_{i}), and consider the commutative diagram,

\xymatrix{ & 0 \ar[d] \\ 0 \ar[r] & V_{τ(i)} \ar[d] \ar[r] & E_{i} \ar[r] & V_{i} \ar[r] & 0,  ξ_{i} \\  & E_{k} = V_{k} →_{V_{i}} ×E_{i} \ar@{-->}[ur]^{ψ_{ki}} \ar[d] \\  & V_{k} \ar[uurr]_{ϕ_{ki}} \ar[d] \\  & 0}

Suppose V_{τ(i)} → E_{k} is not split, then

0 → E_{k} → E_{k} →^{V_{τ(i)}} ⊕E_{i} → V_{i} → 0

is split, since ξ_{i} is a rase. Let pr.: E_{k} →^{V_{τ(i)}} ⊕E_{i} → E_{k} be the splitting. But then the two following diagrams commute:

[figure: 0; 0; V_{τ(i)}; E_{i}; ψ_{ik}; V_{i}; ϕ_{ik}; 0; E_{k}; ψ_{ki}; V_{k}; ϕ_{ki}; 0]

Here ψ_{ik} is the composition of E_{i} → E_{k} →^{V_{τ(i)}} ⊕E_{i} and the projection

E_{k} →^{V_{τ(i)}} ⊕E_{i} → E_{k} and ϕ_{ik} the induced map.

This means that (ϕ_{ki}ϕ_{ik})ξ_{i} = ξ_{i} which is impossible since (ϕ_{ki}ϕ_{ik}) acts nilpotently on Ext_{A}^{1}(V_{i}, V_{τ(i)}), and ξ is nonzero. Therefore V_{τ(i)} → E_{k} splits and ξ_{i} is also a lase, therefore an ase.

The unicity and the permutation property follows immediately from the following: Assume there exist two ase’s ξ_{i} and ξ′_{i} of the form:

[figure: 0; 0; V_{τ(i)}; ϕ_{i}; E_{i}; ρ_{i}; ψ_{i}; V_{i}; 0; ξ_{i} :; E′_{i}; ψ′_{i}; ρ′_{i}; V_{τ(i′)}; ϕ′_{i}; 0]

Then, since ρ′_{i} is not split, there exist liftings ψ_{i}, ψ′_{i} inducing morphisms ϕ_{i}, ϕ′_{i}. But then (ϕ_{i}ϕ′_{i})ξ_{i} = ξ_{i} which means that ξ_{i} is zero. Therefore an ase is unique and in particular, τ(i) = τ(i′). Dually we prove that τ(i) = τ(i′) implies i = i′, so that τ is a permutation.

We see that T_{V}^{a} looks like:

(Ext_{a}^{1}(V_{i}, V_{j}))

where

Ext_{a}^{1}(V_{i}, V_{j}) = \begin{cases}0 & if j ≠ τ(i) \\ k & if j = τ(i) and some Ext_{a}^{1}(V_{i}, V_{j}) ≠ 0 \end{cases}

Corollary 3.5. With the assumptions of the theorem above, we find that

T_{H_{0}} = \{(α_{ij})| \begin{cases}α_{ij} ∈ k, α_{ij} = 0 if j ≠ τ(i) \\ α_{ii} ∈ Ext_{A}^{1}(V_{i}, V_{i}) ∀ϕ_{ji} ∈ End_{A}(V_{i}), ϕ_{ij}α_{ii} = α_{jj}ϕ_{ji}, if i=j \end{cases} \}

Remark 3.3. Consider again a not necessarily finite swarm V = {V_{i}}_{i=1}^{ℵ} of noetherian A-modules. The k^{r}-algebra

End_{A}(V)_{r} := (Hom_{A}(V_{i}, V_{j})), i, j ⩽ r

acts on

(Ext_{A}^{1}(V_{i}, V_{j})), i, j ⩽ r,

in the way described above. Suppose that the modules V_{i} are non-isomorphic, indecomposables, and that for each i, End_{A}(V_{i}) is a local ring with maximal ideal \underline{m}_{i}. Suppose moreover that any iterated extension is a direct sum of such V′_{i}s. This is obviously the case when V = {V_{i}}_{i=1}^{ℵ} is the family of all indecomposible A-modules, but holds in many other interesting cases, see [14].

Let H := H(V) and Ṽ be the prorepresentable hull and the formal versal family, as defined in §2. For every quotient R of H in \underline{a}_{ℵ}, such that

dim_{k}Rad(R)/Rad(R)^{2} < ∞

we consider the image Ṽ (R) ∈ Def_{V}(R) of Ṽ. Denote by L_{i}(R) the i^{th.}-line of Ṽ (R). L_{i}(R) is an A-module and a finite iterated extension of the V_{j}, therefore a finite sum of indecomposables {L_{i}(R, p)}, from our family. Obviously there is a canonical surjection,

L_{i}(R) → V_{i},

and a homomorphism of k-algebras,

ι : H → End_{A}(Ṽ (R)),

defined by left multiplication. Any element,

r_{i,j} ∈ R_{i,j} ⊂ R

defines a homomorphism of right A-modules,

r_{i,j}_{∗} : L_{j}(R) → L_{i}(R).

In particular, if r_{i,j} is in the socle of R, this morphism induces a homomorphism of right A-modules,

r_{i,j}_{∗} : V_{j} → L_{i}(R).

Using this, we may consider different quiver-structures on the set of indecomposible modules, {V_{i}}_{i=1}^{ℵ}. The Auslander-Reiten quiver, see [14], is obtained by picking R = H_{0}/Rad(H_{0})^{2}, a basis {h_{i,j}} of Rad(H_{0})/Rad(H_{0})^{2}, the dual tangent space of H_{0} and letting the arrows arriving at an indecomposible V_{i} be the compositions,

L_{i}(R, p) → L_{i}(R) → V_{i},

and the arrows leaving an indecomposible V_{j} be the compositions,

V_{j} →^{h_{i,j}}→ L_{i}(R) → L_{i}(R, p).

For an arbitrary quotient R of H, we may construct another quiver containing more information than the Auslander-Reiten quiver. Consider representatives

{r_{i,j}} ∈ Rad(R)

of a basis of the dual tangent space Rad(R)/Rad(R)^{2} of R, and let the arrows of the quiver be the compositions

L_{j}(R, q) → L_{j}(R) →^{r_{i,j}_{∗}}→ L_{i}(R) → L_{j}(R, p).

There is a ring homomorphism,

R → End_{A}(Ṽ) = (Hom_{A}(L_{j}, L_{i})) = (Hom_{A}(V_{q}, V_{p})^{n_{p,q}}).

If this homomorphism is surjective, or an isomorphism, we find that the arrows of the quiver generate, in an obvious way, all morphisms of the full sub-category of Amodules defined by the family of indecomposibles V = {V_{i}}. In both case, it is easy to see that the relations in the quiver correspond to non-trivial cup and Massey products of Ext_{a}^{∗}(V_{i}, V_{j}). When A is artinian, and the family {V_{i}} generates the category of A-modules, it turns out that Ṽ is a projective generator. H defines a quiver, with vertices corresponding to the indecomposible projectives, and

H = End_{A}(Ṽ)

is Morita equivalent to A. Moreover, H is determined by the quiver (with relations). In particular, if {V_{i}} is the family of simple A-modules, we shall see in the next paragraph that,

A → (H_{i,j} ⊗ Hom_{k}(V_{i}, V_{j}))

is an isomorphism, and that

L_{i} = ⊕_{j=1,2,···}H_{i,j} ⊗ V_{j}

is a projective A-module, for i = 1, 2, ..r. Since therefore

H ≃ End_{A}(Ṽ) = (Hom_{A}(L_{j}, L_{i}))

is Morita-equivalent to A, the quiver of (projective) summands of Ṽ determines the Morita-equivalence class of A.

We shall end this paragraph by proving the following easy result, see [3] for the notions of Frobenius extension and Frobenius bi-module.

Proposition 3.2. Suppose the following conditions hold:

(i) The family V = {V_{i}} of right A-modules are either finite dimensional as k-vector spaces, or such that,

Ext_{A}^{p}(V_{i}, V_{j}) = Ext_{A}^{p}(V_{j}^{∗}, V_{i}^{∗}).

(ii) The hull of the noncommutative deformation functor, H(V) = (H_{i,j}) is a finite dimensional k-vectorspace.

(iii) For each i, the projective cover of V_{i} has a (finite) filtration with graded components contained in the family V.

Then

η : A → O(V)

is a Frobenius extension.

Proof. The assumption 1. implies that the versal family P = Ṽ as a left H and right A-module has the duality property, ^{∗}P = P^{∗}. The assumption 2. implies that P as left H-module is finite projective, and the assumption 3. garantees that P, as right A-module, is finite projective, therefore a Frobenius bi-module.

§4. The generalized Burnside theorem In §2 we proved the following result,

Corollary 3.6. Suppose the k-algebra A is of finite dimension and assume the swarm V = {V_{i}}_{i=1}^{r} contains all simple A-modules, then the natural k-algebra homomorphism

η : A → O(V) = (H_{ij} ⊗_{k} Hom_{k}(V_{i}, V_{j}))

is injective.

Recall also the classical Burnside-Wedderburn-Malcev theorems, see [4], and [1].

Theorem (Burnside). Let V be a finite dimensional k-vectorspace. Assume k is algebraically closed and let A be a subalgebra of End_{k}(V). If V is a simple A-module, then A = End_{k}(V).

Theorem (Wedderburn). Let A be a ring, and let V be a simple faithfull Amodule. Put D = End_{A}(V) and assume V is a finite dimensional D-vector space. Then A ≃ End_{D}(V).

Theorem (Wedderburn-Malcev). Let A be a finite dimensional k-algebra, kany field. Let 𝔯 be the radical of A, and suppose the residue class algebra A/𝔯 is separable. Then there exists a semi-simple subalgebra S of A such that A is the semidirect sum of S and 𝔯. If S_{1} and S_{2} are subalgebras such that A = S_{i} ⊕ 𝔯, i = 1, 2, then there exists an element n ∈ 𝔯, such that S_{1} = (1 − n) · S_{2} · (1 − n)^{−1}.

In this § we shall prove a generalization of the theorem of Burnside. In fact, assuming the field k is algebraically closed and that V = {V_{i}}_{i=1}^{r} is the family of all simple A-modules we shall prove that the homomorphism η of the above Corollary (2.10), is an isomorphism.

When A is semi-simple we know that Ext_{A}^{1}(V_{i}, V_{j}) = 0 for all i, j = 1, …, r, therefore the formal moduli H of V is isomorphic to k^{r}. This implies that

End_{H}(Ṽ) = ⨁_{i=1}^{r} End_{k}(V_{i}),

which is the classical extension of Burnsides theorem.

We shall need the following elementary lemma

Lemma 3.5. Let the k-algebra A be a direct sum of the right-A-modules V_{i}, i = 1, …, d of the family V = {V }_{i=1}^{r}. Then left multiplication with an element a ∈ A induces A-module homomorphisms

a_{ij} ∈ Hom_{A}(V_{i}, V_{j}), i, j = 1, …, d .

Moreover, any k-linear map x : A → A expressed as x = (x_{ij}) ∈ End_{k}(V) := (Hom_{k}(V_{i}, V_{j})), commuting with all φ = (φ_{ij}) ∈ End_{A}(V) := (Hom_{A}(V_{i}, V_{j})) is necessarily a right multiplication by some element x̃ ∈ A.

Proof. Trivial, since x commuting with all φ ∈ (Hom_{A}(V_{i}, V_{j})) commutes with all left-multiplications by a ∈ A, and therefore x(a) = a · x(1), and we may put x̃ = x(1).

Corollary 3.7. Assume that the family of right A-modules V = {V_{i}}_{i=1}^{r} is such that

A ≃ ⨁_{i=1}^{m} V_{i}^{n_{i}}  (i)

Hom_{A}(V_{i}, V_{j}) = 0 for i ≠ j  (ii)

Then the canonical morphism of k-algebras

η : A → ⨁_{i=1}^{n_{i}} End_{k}(V_{i})

is injective. Moreover, η induces an isomorphism

A ≃ ⨁_{i=1}^{n_{i}} End_{D_{i}}(V_{i})

where D_{i} = End_{A}(V_{i}).

This, in particular, implies the Wedderburn theorem for semisimple k-algebras A.

Theorem 3.4 (A generalized Burnside theorem). Let A be a finite dimensional k-algebra, k an algebraically closed field. Consider the family V = {V_{i}}_{i=1}^{r} of simple A-modules, then

A ≃ O(V) = (H_{i,j} ⊗ Hom_{k}(V_{i}, V_{j}))

Proof. We know that the canonical map

η : A → O(V)

is injective. Since Rad(A)^{n} = 0 for some n, we know that Â = A. The theorem therefore follows from the following lemmas.

Lemma 3.6. Let A and B be finite type k-algebras and let φ : A → B be a homomorphism of k-algebras such that the induced morphism

φ_{2} : A → B/Rad(B)^{2}

is surjective, then

φ̂: Â → B̂

is surjective.

Proof. Well-known.

Lemma 3.7. Let A be a finite dimensional k-algebra, k an algebraically closed field. Let V = {V_{i}}_{i=1}^{r} be the family of simple A-modules. Then the homomorphism

η : A → O(V)

induces an isomorphism

Rad(A)/Rad(A)^{2} ≃ (Ext_{A}^{1}(V_{i}, V_{j})^{∗} ⊗_{k} Hom_{k}(V_{i}, V_{j})).

Proof. The classical Burnside theorem implies that the canonical homomorphism of k-algebras

A →→ ⨁_{i=1}^{r} End_{k}(V_{i})

induces an isomorphism,

A/Rad(A) ≃ ⨁_{i=1}^{r} End_{k}(V_{i}).

According to the Wedderburn-Malcev theorem we may assume that A/Rad(A)^{2} is a semidirect sum,

A/Rad(A) ⊕ Rad(A)/Rad(A)^{2}.

Since Rad(A)/Rad(A)^{2} is both a left and a right ⨁_{i=1}^{r} End_{k}(V_{i})-module

Rad(A)/Rad(A)^{2} = (E_{ij})

each E_{ij} being an End_{k}(V_{i})^{op} ⊗_{k} End_{k}(V_{j})-module. This, however, means that

E_{ij} ≃ Hom_{k}(V_{i}, V_{j}) ⊗ k^{r_{ij}}

as a right End_{k}(V_{i})^{op} ⊗_{k} End_{k}(V_{j})-module. Since we already know that η is an injection, we must have,

E_{ij} ≃ Hom_{k}(V_{i}, V_{j}) ⊗ k^{r_{ij}} ⊂ Ext_{A}^{1}(V_{i}, V_{j}) ⊗ Hom_{k}(V_{i}, V_{j}).

We must show that this inclusion is an equality. Applying Hochschild cohomology as in §1, we find:

Ext_{A}^{1}(V_{i}, V_{j}) = HH^{1}(A, Hom_{k}(V_{i}, V_{j})) = Der_{k}(A, Hom_{k}(V_{i}, V_{j}))/im d^{∘}

where d^{∘} is the differential

Hom_{k}(V_{i}, V_{j}) → Der_{k}(A, Hom_{k}(V_{i}, V_{j})).

Clearly any derivation

ξ ∈ Der_{k}(A, Hom_{k}(V_{i}, V_{j}))

which is zero on Rad(A) induces a derivation

ξ_{0} ∈ Der_{k}(A/Rad(A), Hom_{k}(V_{i}, V_{j}))

which, since A/Rad(A) is semisimple, obviously is a coboundary, i.e. an element of im d^{∘}.

Moreover, any derivation ξ ∈ Der_{k}(A, Hom_{k}(V_{i}, V_{j})) induces the zero map on Rad(A)^{2} since ξ(r_{1} · r_{2}) = r_{i}ξ(r_{2}) + ξ(r_{1})r_{2} = 0 for r_{1}, r_{2} ∈ Rad(A), and any coboundary ν ∈ im d^{∘} must vanish on Rad(A) since ν(r) = φr − rφ, for some φ ∈ Hom_{k}(V_{i}, V_{j}). Now, every A^{op} ⊗_{k} A-linear map Rad(A)/Rad(A)^{2} → End_{k}(V_{i}, V_{j}) extends to a derivation of Der_{k}(A/Rad(A)^{2}, Hom_{k}(V_{i}, V_{j})). In fact, let φ be an A^{op} ⊗_{k} A-linear map

Rad(A)/Rad(A)^{2} → End_{k}(V_{i}, V_{j})

and define the map

ψ : A/Rad(A)^{2} = A/Rad(A) ⨁ Rad(A)/Rad(A)^{2} → End_{k}(V_{i}, V_{j})

by

ψ(s, r) = φ(r) + φ(ρ(s))

where ρ is the 1-Hochschild cochain on A/Rad(A) with values in Rad(A)/Rad(A)^{2} that, according to the Wedderburn-Malcev theorem, defines the semidirect sum refered to above. Then,

ψ((s_{1}, r_{1}) · (s_{2}, r_{2})) = ψ((s_{1} · s_{2}, s_{1}ρ(s_{2}) − ρ(s_{1} · s_{2}) + ρ(s_{1})s_{2} + s_{1}r_{2} + r_{1}s_{2})) = φ(s_{1}r_{2} + r_{1}s_{2} + s_{1}ρ(s_{2}) − ρ(s_{1} · s_{2}) + ρ(s_{1}) · s_{2}) + φ(ρ(s_{1} · s_{2})) = (s_{1}, r_{1})ψ((s_{2}, r_{2})) + ψ((s_{1}, r_{1}))(s_{2}, r_{2})

Therefore,

Ext_{A}^{1}(V_{i}, V_{j}) = Hom_{A^{op}⊗A}(Rad(A)/Rad(A)^{2}, Hom_{k}(V_{i}, V_{j}))
= {φ : Rad(A)/Rad(A)^{2} → Hom_{k}(V_{i}, V_{j})| ∀a ∈ A, r ∈ Rad(A), s.t.
φ(a · r) = a · φ(r) and φ(ra) = φ(r) · a}

Since Rad(A)/Rad(A)^{2} ≃ (E_{ij}) with

E_{ij} ≃ (V_{i}^{∗} ⊗ V_{j})^{r_{ij}}

it is clear that

Hom_{A^{op}⊗A}(Rad(A)/Rad(A)^{2}, Hom_{k}(V_{i}, V_{j}))
≃ Hom_{End_{k}(V_{i})^{op}⊗_{k}End_{k}(V_{j})}((V_{i}^{∗} ⊗ V_{j})^{r_{ij}}, (V_{i}^{∗} ⊗ V_{j}))
≃ k^{r_{ij}}

which means that

E_{ij} ≃ Ext_{A}^{1}(V_{i}, V_{j})^{∗} ⊗_{k} Hom_{k}(V_{i}, V_{j}).

Now suppose, as above, that A is a finite dimensional k-algebra, and let V_{A} = {V_{i}}_{i=1}^{r} be any family of finite dimensional A-modules. Obviously

dim_{k}Ext_{A}^{p}(V_{i}, V_{j}) < ∞

for all p = 0, 1, 2, … and therefore the endomorphism ring

O(V_{A}) := End_{H}(Ṽ)

is a k-algebra such that

O(V)/Rad(O) = ⨁_{i=1}^{r} End_{k}(V_{i}).

This implies that V = {V_{i}}_{i=1}^{r} is the family of all simple O(V)-modules. The generalized Burnside theorem applies also in this case, showing that the operation

(A, V) ↦ (O(V), V)

is a closure operation. Moreover, we have the following,

Proposition 3.3. Let τ : A → B be any homomorphism of finite dimensional k-algebras. Consider a family V_{B} = {V_{i}}_{i=1}^{r} of finite dimensional B-modules and let V_{A} be the corresponding family of A-modules. Suppose moreover that V_{B} is the family of all simple B-modules. Then there exists an, up to isomorphisms, unique homomorphism of k-algebras

O(τ) : O(V_{A}) → O(V_{B}) ≃ B

extending τ.

Proof. There is an obvious forgetful functor defining a morphism of functors on \underline{a}_{r},

τ^{∗} : Def_{V_{B}} → Def_{V_{A}}

which in its turn induces a k-algebra homomorphism

η : H(V_{B}) → H(V_{A})

unique up to isomorphisms, and therefore a k-algebra homomorphism

O(V_{A}) := (H(V_{A})_{i,j} ⊗ Hom_{k}(V_{i}, V_{j})) → (H(V_{B})_{i,j} ⊗ Hom_{k}(V_{i}, V_{j})) =: O(V_{B})

obviously extending τ. By the generalized Burnside theorem, O(V_{B}) ≃ B, and the Proposition follows.

Remark 3.4. Up to now we have only considered finite families of A-modules such that

dim_{k} Ext_{A}^{p}(V_{i}, V_{j}) < ∞, p = 1, 2.

Neither of these conditions are essential. Introducing natural topologies we may, as in [6], treat general families of finite type A-modules. Notice also that if r_{1} ⩽ r_{2}, there is an obvious canonical morphism

\underline{a}_{r_{1}} → \underline{a}_{r_{2}}

inducing a restriction morphism of functors

Def_{V(2)} → Def_{V(1)}

where V(1) = {V_{i}}_{i=1}^{r_{1}}, V(2) = {V_{i}}_{i=1}^{r_{2}}. Therefore we obtain an up to isomorphisms unique k-algebra homomorphism

r_{2,1} : H_{A,V (2)} → H_{A,V (1)}.

However, this restriction morphism is not, in general, unique. The resulting problems will be dealt with later.

Filtered modules and iterated extensions

Let as above V = {V_{i}}_{i=1}^{r} be a family of right A-modules, and let E_{i_{1},...,i_{s}} : E_{s} ⊂ E_{s−1} ⊂ ⋯ ⊂ E_{1} = E be a filtered module such that E_{k}/E_{k+1} ≃ V_{i_{k}}. We shall, as before, refer to any such filtered module as an iterated extension of V. Notice that for every p there is an extension,

ξ_{p,p+1} ∈ Ext_{A}^{1}(V_{i_{p}}, V_{i_{p+1}})

given by the exact sequence.

0 → E_{p+1}/E_{p+2} → E_{p}/E_{p+2} → E_{p}/E_{p+1} → 0.

Corresponding to the iterated extension E_{i_{1},...,i_{s}} we shall associate two directed graphs, Γ(\underline{i}) and Γ(E\underline{i}). The first is gotten as the graph with nodes in bijection with the modules of the family V, and with arrows ϵ(i_{p}, i_{p+1}) connecting the node i_{p} with the node i_{p+1}. The second, the extension type of the iterated extension, is obtained from the first identifying two arrows ϵ(i_{p}, i_{p+1}) and ϵ(i_{q}, i_{q+1}) if the corresponding extensions ξ_{p,p+1} and ξ_{q,q+1} coincide. The corresponding k-algebra is the object of \underline{a}_{r} generated by the arrows ϵ(i_{p}, i_{p+1}) with relations given by composable monomials of the form

ϵ(j_{1}, j_{2})ϵ(j_{3}, j_{4}) … ϵ(j_{w}, j_{w+1})

where the sequence

{j_{1}, j_{2}, j_{3}, …, j_{w+1}}

is not contained in the sequence

{i_{1}, i_{2}, i_{3}, …, i_{s}}.

Example 3.1. Let us draw up all extension types for r, s ⩽ 3.

[figure: r = 3, s = 3 :; 1; 2; 3; r = 2, s = 3 :; 1; 2; 1; 2; 1; 2; r = 1, s = 3 :; 1; 1; 2]

The last example is a Γ(E\underline{i}) corresponding to the situation, i_{1} = i_{2} = i_{3} = 1, and ξ_{i_{1},i_{2}} = ξ_{i_{2},i_{3}} The associated k-algebras are, respectively, the matrix algebras,

(\begin{matrix}k & k & k \\ 0 & k & k \\ 0 & 0 & k \end{matrix}) (\begin{matrix}k[ϵ] & k[ϵ] \\ 0 & k \end{matrix}) (\begin{matrix}k & k[ϵ] \\ 0 & k[ϵ] \end{matrix}) (\begin{matrix}k[t_{1,2}t_{2,1}] & kt_{1,2} \\ kt_{2,1} & k \end{matrix})

with the obvious relations, and the k-algebras,

k{t, u}/(t^{2}, u^{2}, ut), k[t]/(t^{3})

Lemma 3.8. Let H be any object of ˆ \underline{a}_{r}, and let R ∈ \underline{a}_{r}. Then Mor(H,R) has a natural structure of an affine algebraic scheme Mor(H, R) = Spec(A(H, R)), and there is a universal morphism,

ϕ̃: H ⟶ A(H, R) ⊗_{k} R

Proof. Put (E_{i,j}) = rad(H)/rad(H)^{2}, and consider the affine space,

A^{N} = ∏_{i,j} E_{i,j}^{∗} ⊗_{k} R_{i,j}

with coordinates z_{i,j}(l, m) = t_{i,j}(l) ⊗ x_{i,j}(m), where {t_{i,j}(l)}_{i,j} is a basis of E_{i,j} and {r_{i,j}(m)}_{i,j} is a basis, and {x_{i,j}(m)}_{i,j} is a dual basis of R_{i,j}. An element

(α_{i,j}(l, m)) ∈ A^{N}

corresponds to a morphism ϕ ∈ Mor(H, R) if, and only if, the corresponding map

t_{i,j}(l) ⟼ ∑_{m} α_{i,j}(l, m)r_{i,j}(m) ∈ R_{i,j}

satisfies the relations of H. Let these, modulo a high enough power of the radical, be polynomials in the generators t_{i,j}(l)_{i,j} of the form

f_{p}(t_{i,j}(l)) = 0, p = 1, … s ,

and let the relations of R be expressed in terms of,

r_{i,j}(m)r_{j,k}(n) = ∑_{p} β_{i,j,k}^{p}(m, n)r_{i,k}(p), i, j = 1, … r.

Then we obtain equations for Mor(H, R) given by (commutative) polynomial relations of the form,

F_{p}(z_{i,j}(l, m)) = 0, p = 1, … t.

But then

t_{i,j}(l) ⟼ ∑_{m} z_{i,j}(l, m)r_{i,j}(m) ∈ A(H, R) ⊗_{k} R_{i,j}

where the coordinates z_{i,j}(l, m) are subject to the conditions above, defines the universal morphism ϕ̃.

Proposition 3.4. Let A be any k-algebra, V = {V_{i}}_{i=1}^{r} any swarm of A-modules, i.e. such that,

dim_{k} Ext_{A}^{1}(V_{i}, V_{j}) < ∞ for all i, j = 1, …, r.

(i): Consider an iterated extension E of V, with directed graph Γ. Then there exists a morphism of k-algebras

ϕ : H(V) → k[Γ]

such that

E ≃ k[Γ]⊗_{ϕ}Ṽ

in the above sense.

(ii): The set of equivalence classes of iterated extensions of V with extension type Γ, is a quotient of set of closed points of the affine algebraic scheme

\underline{A}[Γ] = Mor(H(V), k[Γ])

(iii): There is a versal family Ṽ [Γ] of A-modules defined on A[Γ], containing as fibres all the isomorphism classes of iterated extensions of Vs with extension type Γ.

Proof. Any morphism φ : H → k[Γ] in \underline{a}_{r} correspond to an iterated extension of the V_{i}’s. This may be expressed in the following way. As vector spaces, we have an isomorphism,

k[Γ] ⊗_{φ} Ṽ ≃ V (Γ) ≃ V_{i_{1}} × V_{i_{2}} × ⋯ × V_{i_{s}}

An A-module structure on this vectorspace, corresponding to an iterated extension with extension type Γ, is given by a homomorphism of k-algebras,

ψ : A ⟶ End_{k}(V_{i_{1}} × V_{i_{2}} × ⋯ × V_{i_{s}})

inducing a family of linear maps

ψ_{p,p+1,...,p+q} : A ⟶ End_{k}(V_{i_{p}}, V_{i_{p+q})}

for 0 ⩽ p < p + q ⩽ s.

Consider these maps as 1-cochains in the Hochschild complex

HC^{∗}(A, Hom_{k}(V (Γ), V (Γ)))

The maps ψ_{p,p+1} correspond to the extensions ξ_{p,p+1} above, and must therefore be 1-cocycles, or derivations. To obtain an A-module structure, corresponding to an iterated extensions of V with extension type Γ, the conditions on these cochains are: For all a, b ∈ A,

ψ_{p,p+1}(a)ψ_{p+1,p+2}(b) = dψ_{p,p+1,p+2}(a, b)

ψ_{p,p+1}(a)ψ_{p+1,p+2,p+3}(b) + ψ_{p,p+1,p+2}(a)ψ_{p+2,p+3}(b) = dψ_{p,p+1,p+2,p+3}(a, b) …

∑_{m=2,...,s−1} ψ_{1,2,...,m}(a)ψ_{m,...,s}(b) = dψ_{1,2,...,s}(a, b),

which means that all Massey products of the form

< ξ_{i_{p},i_{p+1}}, ξ_{i_{p+1},i_{p+2}}, …, ξ_{i_{p+q−1},i_{p+q}} >

are defined and contain zero.

Now (i) follows from the very definition of H, generated as it is by a basis of the dual Ext^{1}’s, with relations exactly expressing the vanishing of the above Massey products. (ii) and (iii) then follows from deformation theory, together with the Lemma (4.9), above.

Example 3.2. Consider the extension E_{ijk} of length 3 given as the composite extension of ξ_{i,j} : 0 ← V_{i} ← E_{ij} ← V_{j} ← 0 and ξ_{i,j,k} : 0 ← E_{ij} ← E_{ijk} ← V_{k} ← 0. Take the pullback ξ_{i,k} of ξ_{i,j,k} via V_{j} → E_{ij} and consider the diagram

[figure: 0; 0; 0; 0; V_{i}; E_{ij}; V_{j}; 0; E_{ik}; E_{ijk}; E_{jk}; V_{k}; V_{k}; V_{k}; 0; 0; 0]

Let ψ_{i,j} ∈ Der_{k}(A, Hom_{k}(V_{i}, V_{j})) be a Hochschild cocycle representing the class ξ_{i,j}. The multiplication with a ∈ A on E_{ij}, identified with V_{j} × V_{i} as k-vectorspace, is given by

(v_{j}, v_{i})a = (v_{j}a + ψ_{i,j}(a, v_{i}), v_{i}a)

and the multiplication with a ∈ A on E_{ijk} identified with V_{k} × V_{j} × V_{i} as kvectorspace, must be such that,

(v_{k}, v_{j}, 0)a = (v_{k}a + ψ_{j,k}(a, v_{j}), v_{j}a, 0).

If there exists an action of A on E_{ijk} consistent with the above, then one proves the existence of a Hochschild cochain

ψ_{i,j,k} ∈ Hom_{k}(A, Hom_{k}(V_{i}, V_{k}))

such that

dψ_{i,j,k}(a, b) = ψ_{i,j}(a)ψ_{j,k}(b).

From this follows that the cup-product ξ_{ij} ∪ ξ_{jk} is 0 in Ext_{A}^{2}(V_{i}, V_{k}). This is also the criterion for the existence of ξ_{i,j,k}. Moreover if ξ and ξ′ are two extensions

ξ : 0 ← E_{ij} ← E_{ijk} ← V_{k} ← 0
ξ′ : 0 ← E_{ij} ← E′_{ijk} ← V_{k} ← 0

with the same pullback ξ_{jk}, then there is an extension

ξ_{ik} : 0 ← V_{i} ← E_{ik} ← V_{k} ← 0

such that its pullback, via E_{ij} → V_{i}, is the difference ξ − ξ′.

Consider for the iterated extensions E_{ijk}, the extension diagram

Γ : [figure: i; j; k,]

the first one of the example (4.8) above, then the corresponding k-algebra is given by,

k[Γ] = ( \begin{matrix}& & i & j & k \\ & k & 0 & 0 & 0 & 0 \\ & ... & ⋯ & ⋯ & ⋯ & ⋮ \\ i & & k & k & k & 0 \\ & & ... & ⋯ & ⋯ & ⋮ \\ j & & & k & k & 0 \\ & & & ... & ⋯ & ⋮ \\ k & & & & k & 0 \\ & & & & ... & ⋯ \\ & & & & & k \end{matrix} )

Notice that E_{ijk} then corresponds to a morphism

ϕ : H → k[Γ]

determined, modulo Rad^{2}(k[Γ]) (i.e. the radical squared), by

ξ_{i,j} = ϕ|E_{i,j} → k, and ξ_{j,k} = ϕ|E_{j,k} → k

Here

Rad^{2}(k[Γ]) = ( \begin{matrix}& & i & j & k \\ & 0 & 0 & 0 & 0 & 0 \\ & ... & ⋯ & ⋯ & ⋯ & ⋮ \\ i & & 0 & 0 & k & 0 \\ & & ... & ⋯ & ⋯ & ⋮ \\ j & & & 0 & 0 & 0 \\ & & & ... & ⋯ & ⋮ \\ k & & & & 0 & 0 \\ & & & & ... & ⋯ \\ & & & & & 0 \end{matrix} )

and ϕ is, according to the analysis above, ”calibrated” by the morphism ϕ_{i,k} : E_{i,k} → k, i.e. by Ext_{A}^{1}(V_{i}, V_{k}), as it should.

Corollary 3.8. (i): Given any finitely generated module M on a Noetherian ring A, there is a finite set of primes E(M), containing the set of associated primes Ass(M), such that the module M is an iterated extensions of the corresponding modules A/𝔭 for 𝔭 ∈ E(M). The extension type of such an iterated extension is an ordered directed graph Γ(M) the nodes of which is E(M).

(ii): For any finite ordered directed graph Γ, with nodes corresponding to a set of primes P ⊂ Spec(A), there is an affine versal family of A-modules M̃ with extension type Γ, and E(M) = P.

Proof. Obvious

Example 3.3. Given any scheme \underline{H} = Spec(H), say the 2-dimensional affine space given by H = k[x_{1}, x_{2}]. We shall be interested in the noncommutative moduli space parametrizing subschemes of length 2 of \underline{H}. We may do this by simply considering a point in the space Spec(H) together with a tangent direction, i.e. the right Hmodule of the form,

V = k[x_{1}, x_{2}]/(x_{1}^{2}, x_{2}),

and compute the formal moduli of V.

Lemma 3.9. The formal moduli, H(V) of the H-module V = H/(x_{1}^{2}, x_{2}), is given as the completion of the k-algebra,

Ω = k{t_{1}, t_{2}, ω_{1}, ω_{2}}/(y_{1}, y_{2})

where

y_{1} = [t_{1}, t_{2}] − t_{1}[ω_{1}, ω_{2}] y_{2} = [t_{1}, ω_{2}] − [t_{2}, ω_{1}] − ω_{1}[ω_{1}, ω_{2}],

and where the family of left Ω-and right H-modules,

Ω ⊗_{k} k^{2}

is defined by the actions of x_{1} and x_{2}, given by,

x_{1} = (\begin{matrix}0 & t_{1} \\ 1 & ω_{1} \end{matrix}) , x_{2} = (\begin{matrix}t_{2} & t_{1}ω_{2} \\ ω_{2} & t_{2} + ω_{1}ω_{2} \end{matrix})

Proof. Consider the obvious free resolution of V := H/(x_{1}^{2}, x_{2}) as an H-module,

\xymatrix{V & H \ar[l]^{ρ} & H^{2} \ar[l]^{d_{0}} & H \ar[l]^{d_{1}} & 0 \ar[l]^{d_{2}}}

where we have,

d_{0} = (x_{1}^{2}, x_{2}), d_{1} = (\begin{matrix}x_{2} \\ −x_{1}^{2} \end{matrix}) .

Consider the Yoneda complex, and pick a basis

{t̂_{1}, t̂_{2}; ω̂_{1}, ω̂_{2}, }

of Ext_{H}^{1}(V, V) represented by the morphisms of the diagram,

\xymatrix{V & H \ar[l]_{ρ} & H^{2} \ar[dl]_{ω̂_{j}}^{t̂_{i}} \ar[l]_{d_{0}} & H \ar[dl]_{ω̂_{j}^{2}}^{t̂_{i}^{2}} \ar[l]_{d_{1}} \\ V & H \ar[l]_{ρ} & H^{2} \ar[dl]_{ω̂_{j}}^{t̂_{i}} \ar[l]_{d_{0}} & H \ar[dl]_{ω̂_{j}^{2}}^{t̂_{i}^{2}} \ar[l]_{d_{1}} & H \ar[l]_{d_{2}} & 0 \ar[l] \\ V & H \ar[l]^{ρ} & H^{3} \ar[l]^{d_{0}} & H^{3} \ar[l]^{d_{1}} & H \ar[l]^{d_{2}} & 0 \ar[l]}

Here,

t̂_{1} = (1, 0), t̂_{2} = (0, 1,);
ω̂_{1} = (x_{1}, 0), ω̂_{2} = (0, x_{1})

and,

t̂_{1}^{2} = (\begin{matrix}0 \\ 1 \end{matrix}) , t̂_{2}^{2} = (\begin{matrix}−1 \\ 0 \end{matrix}) ,

and finally,

ω̂_{1}^{2} = (\begin{matrix}0 \\ x_{1} \end{matrix}) , ω̂_{2}^{2} = (\begin{matrix}−x_{1} \\ 0 \end{matrix}) .

Using this it is easy to see that ,

t̂_{i} ∪ t̂_{i} = 0, t̂_{1} ∪ t̂_{2} = −t̂_{2} ∪ t̂_{1} = ŷ_{1},

and that

t̂_{1}ω̂_{2}^{2} = ω̂_{1}t_{2}^{2} = −ŷ_{2}, ω̂_{i}t̂_{i}^{2} = 0, ω̂_{i}ω̂_{j}^{2} = 0, t̂_{2}ω̂_{1}^{2} = ω̂_{2}t̂_{1}^{2} = ŷ_{2},

therefore

−y_{2} = t̂_{1} ∪ ω̂_{2} = ω̂_{1} ∪ t̂_{2} = −t̂_{2} ∪ ω̂_{1} = −ω̂_{2} ∪ t̂_{1}, ω̂_{i} ∪ ω̂_{j} = t̂_{i} ∪ t̂_{j} = 0.

Now, consider the dual basis {t_{1}, t_{2}; ω_{1}, ω_{2}} generating the hull of the deformation functor Def_{k[ϵ]}, we find after a simple computation of the 3. order Massey products the formulas we want.

Notice that we just have to compute the tangent situation and check that our formulas give us a lifting of the quadratic relations and of the corresponding Haction, to know that our result holds.

By a simple computation one checks that the k-points of Ω form an open dense part of Hilb^{2}A^{2} containing V. Hilb^{2}A^{2} is the blow-up of (A^{2} × A^{2})/Z_{2} along the diagonal. However, there are other simple representations of Ω. The homomrphism,

Ω → k|t_{1}, t_{2}, \frac{∂}{∂t_{1}} , \frac{∂}{∂t_{2}} ]

maping ω_{i} to \frac{∂}{∂t_{i}}, shows that k|t_{1}, t_{2}] is a simple representation of Ω.

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This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2002/n2a16/v4n2a16.(dvi,ps,pdf)

O. A. Laudal

University of Oslo