Abstract view
Computing Noncommutative Deformations of Presheaves and Sheaves of Modules


Published:20091204
Printed: Jun 2010
Abstract
We describe a noncommutative deformation theory for presheaves and
sheaves of modules that generalizes the commutative deformation
theory of these global algebraic structures and the noncommutative
deformation theory of modules over algebras due to Laudal.
In the first part of the paper, we describe a noncommutative
deformation functor for presheaves of modules on a small category and
an obstruction theory for this functor in terms of global Hochschild
cohomology. An important feature of this obstruction theory is that it
can be computed in concrete terms in many interesting cases.
In the last part of the paper, we describe a noncommutative deformation
functor for quasicoherent sheaves of modules on a ringed space
$(X,\mathcal{A})$. We show that for any good $\mathcal{A}$affine open cover $\mathsf{U}$ of
$X$, the forgetful functor $\mathsf{QCoh}\mathcal{A} \to \mathsf{PreSh}(\mathsf{U}, \mathcal{A})$ induces
an isomorphism of noncommutative deformation functors.
\emph{Applications.} We consider noncommutative deformations of
quasicoherent $\mathcal{A}$modules on $X$ when
$(X, \mathcal{A}) = (X, \mathcal{O}_X)$ is
a scheme or $(X, \mathcal{A}) = (X, \mathcal{D})$ is a Dscheme in the sense of
Beilinson and Bernstein. In these cases, we may use any open affine
cover of $X$ closed under finite intersections to compute
noncommutative deformations in concrete terms using presheaf
methods. We compute the noncommutative deformations of the left
$\mathcal{D}_X$module $\mathcal{D}_X$ when $X$ is an elliptic curve as an
example.