Perverse coherent sheaves (after Deligne)

By R. Bezrukavnikov. 2000

69 citations recorded by the source corpus.

This note is mostly an exposition of an unpublished result of Deligne, which introduces an analogue of perverse $t$-structure on the derived category of coherent sheaves on a Noetherian scheme with a dualizing complex. Construction extends to the category of coherent sheaves equivariant under an action of an algebraic group; though proof of the general statement in this case does not require new ideas, it provides examples (such as sheaves on the nilpotent cone of a semi-simple group equivariant under the adjoint action) where construction of coherent "intersection cohomology" sheaves works.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c699a048573ae1c795f8e8327dd308782c9a761f

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Citations in the corpus

Koszul duality and modular representations of semi-simple Lie algebras (2008, background): “e of situation (a) (taking Yb to be a the restriction of Yto Zb). The notations are compatible. Recall the construction of resolutions by injective G m-equivariant quasicoherent sheaves on Y(see e.g. [B1]): if Fis an injective object of QCoh(Y), then Av(F) := a∗p∗ YFis injective in QCoh Gm(Y), where”
Perverse coherent sheaves (2009, background): “He thanks IAS for excellent work conditions and Leonid Positselski for his participation in the early stages of the work on [ 3 ].”
Dualizing complexes and perverse sheaves on noncommutative ringed schemes (2002, background): “Remark 4.16. In [BBD] the authors considered t-structures on certain subcategories of D(Mod kX), for a topological space X and a constant sheaf of rings kX. Perverse t-structures on Dbc(Mod A) as above where A is “quasi-coherent” have only been considered recently; see [ Bz ], [Br], [Ka] and [YZ4].”
Perverse sheaves on affine flags and nilpotent cone of the langlands dual group (2002, unknown): “If w 2 f W but w 62 f W f then for some simple root �, � 6 �0 we have Lw = � ! (L 0 ); here �0 is the affine simple root,�� : F` ! F`(�) is the projection (P 1 fibration) to the corresponding partial affine flag variety, and L 0 is an Iequivariant constructible complex on F`(�) (actually, L 0 [ 1 ] ”
On tensor categories attached to cells in affine Weyl groups, III (2000, background): “Definition 1. Let A be a monoidal category, and B be a tensor (symmetric monoidal) category.”
Polynomial Bridgeland stability conditions and the large volume limit (2007, methodology): “nt for our stability conditions are t-structures of perverse coherent sheaves. The theory of perverse coherent sheaves is apparently originally due to Deligne, and has been developed by Bezrukavnikov [Bez00] and Kashiwara [Kas04]. We will need only a special case of perverse coherent sheaves, which ”
Compactly generated t-structures on the derived category of a Noetherian ring (2007, background): “sheaves on a Noetherian scheme under certain hypothesis, namely the existence of a dualizing complex and of global locally free resolutions. His work did not appear until the e-print by Bezrukavnikov [Be]. He gave a collection of tstructures that agree essentially with the ones characterized by us i”
Perverse sheaves on affine flags and langlands dual group (2002, unknown): “˜ af). For any F ∈Hot0(CohGˇ×Tˇ fr ( ˆ˜ Naf)) the corresponding object of Db(CohGˇ×Tˇ(Nˆ˜ af) can be represented by a ﬁnite complex of coherent sheaves set-theoretically supported on ∂ ˆ˜ N (cf. e.g. [B2], Lemma 3(b)). If Cis such a complex, then for some λ∈Λ+ we have Bλ⊗idC= 0, hence Kλ⊗C∼= M i F ⊗”
Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves (2004, unknown): “In this paper we provide a reasonably explicit description of H(M) when M is a tilting object of Uq-mod 0 . To state this description we have to recall the category P Coh of G-equivariant coherent perverse sheaves on N with respect to the middle perversity, see [ B1 ].”
Curve counting theories via stable objects I. DT/PT correspondence (2009, background): “ry Coh†(X) to be the tilting with respect to (22), i.e. Coh†(X) = hCoh ≥2(X)[1],Coh≤1(X)iex. Remark 3.3. The category Coh†(X) is one of the hearts of perverse t-structures introduced by Bezrukavnikov [5] and Kashiwara [19]. Remark 3.4. It is easy to see that the subcategory Coh≤1(X) ⊂ Coh†(X) is clo”
Constant families of t-structures on derived categories of coherent sheaves (2006, methodology): “For completeness we supply a proof (sketched in [ 4 ]).”