Co-$t$-structures on derived categories of coherent sheaves and the cohomology of tilting modules

Abstract

We construct a co-t-structure on the derived category of coherent sheaves on the nilpotent cone, as well as on the derived category of coherent sheaves on any parabolic Springer resolution. These structures are employed to show that the push-forwards of the "exotic parity objects" (considered in [AHR1]), along the (classical) Springer resolution, give indecomposable objects inside the coheart of the co-t-structure on N . We also demonstrate how the various parabolic co-t-structures can be related by introducing an analogue to the usual translation functors.Additional applications include a proof of the scheme-theoretic formulation of Humphreys conjecture on support varieties of tilting modules in type A for p > h, as well as a verification of the conjecture in arbitrary type, (for p > h), over a large class of orbits, which includes all Richardson orbits. We also obtain new information regarding the structure of certain p-canonical cells and thick-tensor-ideals of tilting modules.

Silting complexes of coherent sheaves and the Humphreys conjecture

“…Section 3 contains preliminaries and notation related to the nilpotent cone, and Sections 4, 5, and 6 are devoted to constructing co-t-structures on coherent sheaves in increasingly difficult settings, culminating with the orbitwise co-t-structure on D b Coh GˆGm pN q, obtained in Theorem 6.8. In Section 7, we prove that this co-tstructure coincides with the supportive co-t-structure from [4], and we describe the combinatorics of the relationship between the two. Section 8 contains the proof of the relative Humphreys conjecture, and Section 9 discusses potential applications to the study of the p-canonical basis.…”

“…For an algebraic group H over an algebraically closed field k, let ReppHq be the category of finite-dimensional algebraic representations. If H is connected and reductive, then (as observed in, say, [4,Remark 2.6]), the category TiltpHq Ă ReppHq is the coheart of a co-t-structure. If H is disconnected, and if the characteristic of k does not divide |H{H ˝|, then ReppHq is again a highest-weight category (see [7,Theorem 3.7]), and TiltpHq is again the coheart of a co-t-structure.…”

“…In [4], the authors gave a rather different construction of a silting subcategory of D b Coh GˆGm pN q, called the supportive silting subcategory. Its indecomposable objects are denoted by S λ , where λ is a dominant weight for G. The second main result of this paper states that the orbitwise and supportive categories actually coincide.…”

“…In this paper, as in its predecessor [4], we use the term silting object to mean any object in a silting subcategory, not just a generator. See [4,Remark 2.2] for context.…”

“…In this paper, as in its predecessor [4], we use the term silting object to mean any object in a silting subcategory, not just a generator. See [4,Remark 2.2] for context. The notion of a silting subcategory is equivalent to that of a bounded co-t-structure, and we mainly use the latter notion in this paper.…”