Oslo University College, Postboks 4, St. Olavs plass, N-0130 Oslo, Norway, E-mail: [email protected]
Received date: December 12, 2007; Accepted date: March 27, 2008
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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.
Let k be an algebraically closed field and let A be an associative k-algebra. For any left A-module M, there is a commutative deformation functor defined on the category l of local Artinan commutative k-algebras with residue field k. We recall that for an object a deformation of M over R is a pair (MR, ), where MR is an A-R bimodule (on which k acts centrally) that is R-flat, and is an isomorphism of left A-modules. Moreover, as deformations in if there is an isomorphism of A-R bimodules such that
In , Laudal introduced noncommutative deformations of modules. For any finite family of left A-modules, there is a noncommutative deformation functor defined on the category ap of p-pointed Artinian k-algebras. We recall that an object R of ap is an Artinian ring R, together with a pair of structural ring homomorphisms and , such that and the radical I(R) = ker(g) is nilpotent. The morphisms of ap are ring homomorphisms that commute with the structural morphisms.
A deformation of the family over R is a (p + 1)-tuple where MR is an A-R bimodule (on which k acts centrally) such that as right R-modules, and is an isomorphism of left A-modules for By definition,
with the natural right R-module structure, and k1, . . . , kp are the simple left R-modules of dimension one over k. Moreover, as deformations in if there is an isomorphism bimodules such that
There is a cohomology theory and an obstruction calculus for see Laudal  and Eriksen . We compute the noncommutative deformations of a family of modules over the first Weyl algebra using the constructive methods described in Eriksen .
Let k be an algebraically closed field of characteristic 0, let and let D = Diff (A) be the first Weyl algebra over k. We recall that . Let us consider the family of left D-modules, where We shall compute the noncommutative deformations of the family .
In this example, we use the methods described in Eriksen  to compute noncommutative deformations. In particular, we use the cohomology of the Yoneda complex
for where is a free resolution of Mi, and an obstruction calculus based on these free resolutions. We recall that
Let us compute the cohomology We use the free resolutions of M1 and M2 as left D-modules given by
and the definition of the differentials in the Yoneda complex, and obtain
The base vector is represented by the 1-cocycle given by when for all i, j, it is clear that for
We conclude that is unobstructed. Hence, in the notation of Eriksen , the prorepresenting hull H of is given by
where is a basis of dual to the basis of for (i, j) = (1, 2) and (i, j) = (2, 1). We write and
In order to describe the versal family of left D-modules defined over H, we use Mfree resolutions in the notation of Eriksen . In fact, the D-H bimodule has an M-free resolution of the form
where This means that for any P,Q D, we have that and
In other words, S is an associative k-algebra of finite type such that the J-adic completion for the ideal The corresponding algebraization of the versal family is given by the M-free resolution
We shall determine the D-modules parameterized by the family 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 , . For any simple S-module T, we obtain a left D-module 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 and we may identify by choosing a k-linear base for T. We see that S is generated by e1, s12, s21 as a k-algebra (since e2 = 1 - e1), and there are relations
Any S-module of dimension n is therefore given by matrices satisfying the matric equations
The S-modules represented by and are isomorphic if and only if there is an invertible matrix such that 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, p factorizes through the commutativization of S. It follows that there are exactly two non-isomorphic simple S-modules of dimension one, T1,1 and T1,2, and the corresponding deformations of are
This reflects that M1 and M2 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 E1 has Jordan form:
We shall write for the corresponding S-modules of dimension two. Notice that is simple for all , while are extensions of simple S-modules of dimension one. In fact, are trivial extensions, while T2,4 is a non-trivial extension of T1,2 by T1,1 and T2,5 is a non-trivial extension of T1,1 by T1,2. The deformations of corresponding to the simple modules T2,6,α are given by M2,6,α for In fact, one may show that for any In particular, is a simple D-module if and in this case if and only if Furthermore, for 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 E1 has Jordan form:
We shall write T3,1 – T3,6, T3,7,b, T3,8 – T3,11, and T3,12,c for the corresponding S-modules of dimension three. Notice that all S-modules of dimension three are extensions of simple S-modules of dimension one and two, so there are no simple S-modules of dimension n = 3.
In fact, and are trivial extensions, while T3,6 is a non-trivial extension of T1,2 by T2,5 and T3,11 is a non-trivial extension of T1,1 by T2,4.
We remark that there are no simple S-modules of finite dimension n ≥ 3. In fact, if T is a simple S-module, then is a surjective ring homomorphism. This implies that can be generated by as a k-algebra. To see that this is impossible, notice that we may choose a k-base of T such that
where r)-matrix and Y is a (n − r) × r matrix. If r = 0 or r = n, then X = Y = 0, and this leads to a contradiction, because Mn(k) is not generated by diagonal matrices when n > 1. Moreover, r ≥ 2 leads to a contradiction, because is not generated by Ir and XY . Similarly leads to a contradiction, because is not generated by In−r and Y X. We conclude that n = r+(n−r) = 1+1 = 2, a contradiction.
Finally, we remark that the commutative deformation functor of the direct sum has pro-representing hull and an algebraization It is not difficult to find the family MS in this case. In fact, for any point the left D-module is given by
We see that we obtain exactly the same isomorphism classes of left D-modules as commutative deformations of as we obtained as noncommutative deformations of the family However, the points of the algebraization S of the pro-representing hull of the noncommutative deformation functor give a much better geometric picture of the local structure of the moduli space of left D-modules. In fact, the family of left D-modules parametrized by the points of S contains few isomorphic D-modules, and the simple S-modules have algebraic properties – such as extensions – that reflect the algebraic properties of the corresponding D-modules.