Noncommutative enhancements of contractions

Abstract

Given a contraction of a variety X to a base Y , we enhance the locus in Y over which the contraction is not an isomorphism with a certain sheaf of noncommutative rings D, under mild assumptions which hold in the case of (1) crepant partial resolutions admitting a tilting bundle with trivial summand, and (2) all contractions with fibre dimension at most one. In all cases, this produces a global invariant. In the crepant setting, we then apply this to study derived autoequivalences of X. We work generally, dropping many of the usual restrictions, and so both extend and unify existing approaches. In full generality we construct a new endofunctor of the derived category of X by twisting over D, and then, under appropriate restrictions on singularities, give conditions for when it is an autoequivalence. We show that these conditions hold automatically when the non-isomorphism locus in Y has codimension 3 or more, which covers determinantal flops, and we also control the conditions when the non-isomorphism locus has codimension 2, which covers 3-fold divisor-to-curve contractions.

On the finiteness of the derived equivalence classes of some stable endomorphism rings

“…In this more general setup, Donovan-Wemyss introduce a more general invariant given by a sheaf of algebras [DW4]. As with the construction of the contraction algebra, the construction involves a vector bundle V := O X ⊕ V 0 on X satisfying…”

“…Although this bundle may not be tilting (as it is in the complete local case) there is a technical condition on V, detailed in [DW4,2.3], which ensures that for any choice of affine open Spec R in X con , the bundle V| f −1 (Spec R) is a tilting bundle.…”

“…With this bundle V, they define the sheaf of contraction algebras to be D := f * End X (V)/I where I is the ideal sheaf of local sections that at each stalk factor through a finitely generated projective O Xcon,v -module (see [DW4,2.8] for details).…”

“…Writing Z for the locus of points on X con above which f is not an isomorphism, [DW4,2.16] showed that the support of the sheaf D is precisely Z. In particular, in the setup of 4.17, the condition on X ensures that Z = {p 1 , .…”

“…Theorem 4.18 [DW4,. 2.24] The completion of D pi is morita equivalent to A i .As D pi is a finite length module over O Xcon,pi , there is an isomorphism D pi ∼ = D pi of O Xcon,pi -algebras, where D pi denotes the completion of D pi .…”