Tilting exercises

By A. Beilinson, R. Bezrukavnikov, I. Mirkovic. 2003

105 citations recorded by the source corpus.

This is a geometry-oriented review of the basic formalism of tilting objects (originally due to Ringel, see [Ri], §5). In the first section we explain that tilting extensions form a natural framework for the gluing construction from [B1] and [MV]. We show that in case of a stratification with contractible strata, the homotopy category of complexes of tilting perverse sheaves is equivalent to the derived category of sheaves smooth along the stratification. Thus tilting objects play the role similar to projective or injective ones (with advantage of being self-dual and having local origin). In the second section we discuss tilting perverse sheaves smooth along the Schubert stratification of the flag space (or, equivalently, tilting objects in the Bernstein-Gelfand-Gelfand category O). In this case a Radon transform interchanges tilting, projective, and injective modules. As a corollary, we give a short proof of Soergel’s Struktursatz [S1], and describe the Serre functor for D(O) (as conjectured by M. Kapranov).

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/aaaafb240b7eba92b7ea9cf7e41df7783c58f568

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Koszul duality for Kac–Moody groups and characters of tilting modules (2017, background): “s result we need some preliminary lemmas. Lemma 3.9. The functor V′ : Tilt(U )G/B,k) → R∨-ModZ introduced in the proof of Proposition 3.7 is faithful. Proof. The argument for this proof is taken from [BBM]. By construction of the functor V′, to prove the lemma it suﬃces to prove that the image of an”
Projective-injective modules, Serre functors and symmetric algebras (2005, result): “The original conjecture has recently been proved by Beilinson, Bezrukavnikov and Mirkovic in [8].”
Parity Sheaves (2009, background): “5Warning: the notion of tilting sheaf used here is not the same as that of [BBM04].”
Tilting modules and the p-canonical basis (2015, methodology): “tilting objects as in Theorem 1.2, and a character formula for these tilting objects as in Corollary 1.4. In 10 SIMON RICHE AND GEORDIE WILLIAMSON case (1) one can show (using a Radon transform as in [BBM], or an algebraic analogue) that this formula is equivalent to the Kazhdan–Lusztig conjecture p”
Character D-modules via Drinfeld center of Harish-Chandra bimodules (2009, background): “By the result of [4], the functor Iw0 is the Serre functor for the derived category DN (G/B).”
On functors associated to a simple root (2004, background): “r instance • the Kazhdan-Lusztig combinatorics of the category O (see e.g. [KL]); • the Serre functor for the bounded derived category of the category of perverse sheaves on the ﬂag variety (see e.g. [BBM]); • structure and characters of tilting modules (see e.g. [So4]); • ﬁnding functorial invarian”
Knot invariants and higher representation theory II: the categorification of quantum knot invariants (2010, background): “ strands between that and the (i+1)-st rightward to the immediate left of the (i+2)-nd. 1.2. Serre functors. It is a well-supported principle (see, for example, Beilinson, Bezrukavnikov and Mirkovic´ [BBM04] or Mazorchuk and Stroppel [MSb]) that for any suitable braid group action on a category, the”
Schubert calculus and torsion explosion (2013, background): “However a proof may be obtained by adapting ideas of [BBM04].”