An example of noncommutative deformations

We compute the noncommutative deformations of a family of modules over the first Weyl algebra. This example shows some important properties of noncommutative deformation theory that separates it from commutative deformation theory.


Introduction
Let k be an algebraically closed field and let A be an associative k-algebra. For any left A-module M , there is a flat commutative deformation functor Def M : l → Sets defined on the category l of local Artinan commutative k-algebras with residue field k. We recall that for an object R ∈ l, a flat deformation of M over R is a pair (M R , τ ), where M R is an A-R bimodule (on which k acts centrally) that is R-flat, and τ : M R ⊗ R k → M is an isomorphism of left A-modules. Moreover, (M R , τ ) ∼ (M ′ R , τ ′ ) as deformations in Def M (R) if there is an isomorphism η : M R → M ′ R of A-R bimodules such that τ = τ ′ • (η ⊗ 1). Laudal introduced noncommutative deformations of modules in Laudal [2]. For any finite family M = {M 1 , . . . , M p } of left A-modules, there is a noncommutative deformation functor Def M : a p → Sets defined on the category a p of p-pointed Artinian k-algebras. We recall that an object R of a p is an Artinian ring R, together with a pair of structural ring homomorphisms f : k p → R and g : R → k p , such that g • f = id and the radical I(R) = ker(g) is nilpotent. The morphisms of a p are ring homomorphisms that commute with the structural morphisms.
A deformation of the family M over R is a (p + 1)-tuple (M R , τ 1 , . . . , τ p ), where M R is an A-R bimodule (on which k acts centrally) such that M R ∼ = (M i ⊗ k R ij ) as right R-modules, and τ i : with the natural right R-module structure, and k 1 , . . . , k p are the simple left R-modules of dimension one over k. Moreover, (M R , τ 1 , . . . , There is a cohomology theory and an obstruction calculus for Def M , see Laudal [2] and Eriksen [1]. We compute the noncommutative deformations of a family M = {M 1 , M 2 } of modules over the first Weyl algebra using the constructive methods described in Eriksen [1].

An example of noncommutative deformations of a family
Let k be an algebraically closed field of characteristic 0, let A = k[t], and let D = Diff(A) be the first Weyl algebra over k. We recall that D = k t, ∂ /(∂ t − t ∂ − 1). Let us consider the We shall compute the noncommutative deformations of the family M.
In this example, we use the methods described in Eriksen [1] to compute noncommutative deformations. In particular, we use the cohomology YH n (M j , M i ) of the Yoneda complex is a free resolution of M i , and an obstruction calculus based on these free resolutions. We recall that We use the free resolutions of M 1 and M 2 as left D-modules given by The base vector ξ ij is represented by the 1-cocycle given by D We conclude that Def M is unobstructed. Hence, in the notation of Eriksen [1], the prorepresenting hull H of Def M is given by where s 12 = H 11 · s 12 · H 22 and s 21 = H 22 · s 21 · H 11 . In order to describe the versal family M H of left D-modules defined over H, we use Mfree resolutions in the notation of Eriksen [1]. In fact, the D-H bimodule M H has an M-free resolution of the form where d H = (·∂) ⊗e i − (·1) ⊗s 12 − (·1) ⊗s 21 + (·t) ⊗e 2 . This means that for any P, Q ∈ D, we have that d H (P ⊗ e 1 ) = (P · ∂) ⊗e 1 − (P · 1) ⊗s 21 and d H (Q ⊗ e 2 ) = (Q · t) ⊗e 2 − (Q · 1) ⊗s 12 . We remark that there is a natural algebraization S of the pro-representation hull H, given by In other words, S is an associative k-algebra of finite type such that the J-adic completion S ∼ = H for the ideal J = (s 12 , s 21 ) ⊆ S. The corresponding algebraization M S of the versal family M H is given by the M-free resolution We shall determine the D-modules parameterized by the family M S over the noncommutative algebra S -this is much more complicated than in the commutative case. We consider the simple left S-modules as the points of the noncommutative algebra S, following Laudal [3], [4]. For any simple S-module T , we obtain a left D-module M T = M S ⊗ S T Therefore, we consider the problem of classifying simple S-modules of dimension n ≥ 1.
Any S-module of dimension n ≥ 1 is given by a ring homomorphism ρ : S → End k (T ), and we may identify End k (T ) ∼ = M n (k) by choosing a k-linear base {v 1 , . . . , v n } for T . We see that S is generated by e 1 , s 12 , s 21 as a k-algebra (since e 2 = 1 − e 1 ), and there are relations Any S-module of dimension n is therefore given by matrices E 1 , S 12 , S 21 ∈ M n (k) satisfying the matric equations The S-modules represented by (E 1 , S 12 , S 21 ) and (E ′ 1 , S ′ 12 , S ′ 21 ) are isomorphic if and only if there is an invertible matrix G ∈ M n (k) such that GE 1 G −1 = E ′ 1 , GS 12 G −1 = S ′ 12 , GS 21 G −1 = S ′ 21 . Using this characterization, it is a straight-forward but tedious task to classify all S-modules of dimension n up to isomorphism for a given integer n ≥ 1.
Let us first remark that for any S-module of dimension n = 1, ρ factorizes through the commutativization k 2 of S. It follows that there are exactly two non-isomorphic simple Smodules of dimension one, T 1,1 and T 1,2 , and the corresponding deformations of M are This reflects that M 1 and M 2 are rigid as left D-modules.
We obtain the following list of S-modules of dimension n = 2, up to isomorphism. We have used that, without loss of generality, we may assume that E 1 has Jordan form: We shall write T 2,1 -T 2,5 and T 2,6,a for the corresponding S-modules of dimension two. Notice that T 2,6,a is simple for all a ∈ k * , while T 2,1 -T 2,5 are extensions of simple S-modules of dimension one. The corresponding deformations of M are given by M 2,6,a = M S ⊗ S T 2,6,a for a ∈ k * In fact, one may show that M 2,6,a ∼ = D/D · (t∂ − a) for any a ∈ k * . In particular, M 2,6,a is a simple D-module if a ∈ Z, and in this case M 2,6,a ∼ = M 2,6,b if and only if a − b ∈ Z. Furthermore, M 2,6,−1 ∼ = D/D · ∂ t, M 2,6,n ∼ = M 1 for n = 1, 2, . . . , and M 2,6,−n ∼ = M 2 for n = 2, 3, . . . .
We obtain the following list of S-modules of dimension n = 3, up to isomorphism. We have used that, without loss of generality, we may assume that E 1 has Jordan form: We shall write T 3,1 -T 3,6 , T 3,7,b , T 3,8 -T 3,11 , and T 3,12,c for the corresponding S-modules of