Parity Sheaves

By D. Juteau, C. Mautner, Geordie Williamson. 2009

127 citations recorded by the source corpus.

Given a stratified variety X with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness of"parity sheaves", which are objects in the constructible derived category of sheaves with coefficients in an arbitrary field or complete discrete valuation ring. This construction depends on the choice of a parity function on the strata. If X admits a resolution also satisfying a parity condition, then the direct image of the constant sheaf decomposes as a direct sum of parity sheaves, and the multiplicities of the indecomposable summands are encoded in certain refined intersection forms appearing in the work of de Cataldo and Migliorini. We give a criterion for the Decomposition Theorem to hold in the semi-small case. Our framework applies to many stratified varieties arising in representation theory such as generalised flag varieties, toric varieties, and nilpotent cones. Moreover, parity sheaves often correspond to interesting objects in representation theory. For example, on flag varieties we recover in a unified way several well-known complexes of sheaves. For one choice of parity function we obtain the indecomposable tilting perverse sheaves. For another, when using coefficients of characteristic zero, we recover the intersection cohomology sheaves and in arbitrary characteristic the special sheaves of Soergel, which are used by Fiebig in his proof of Lusztig's conjecture.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0f5f2628845b6319adc667bad24e80d22b2d1452

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Tilting modules and the p-canonical basis (2015, methodology): “e the subtleties related to the fact that G is not an ind-variety. 9.3. Parity complexes on ﬂag varieties. As in §9.2, we let K be a Noetherian commutative ring of ﬁnite global dimension. Recall from [JMW] the notion of parity complexes in Db BpX ,Kq, Db pBqpX ,Kq, D b BpX s,Kq or Db pBqpX s,Kq. (In”
Schubert calculus and torsion explosion (2013, background): “Under mild and explicit restrictions on the characteristic of k, Bw may be realised as the equivariant intersection cohomology of the indecomposable parity sheaf [JMW09] of the Schubert variety labelled by w in the flag variety of the complex group with root data (X,R,X, R) [Fie08, FW].”
Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties (2010, background): “[JMW09a]).”
Koszul duality for Kac–Moody groups and characters of tilting modules (2017, background): “AR3], where this category was deﬁned in terms of chain complexes over the additive category Parity(B)(G/B,k) of Bruhat-constructible parity complexes on G/B (in the sense of Juteau–Mautner–Williamson [JMW]). As explained in §1.3, the starting point of the Bezrukavnikov–Yun approach is the considerat”