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Derived categories of Grassmannians over integers and modular representation theory

Alexander I. Efimov

TL;DR

This work constructs a characteristic-free description of the derived category of Grassmannians over $\mathbb{Z}$ by producing a semi-orthogonal decomposition with blocks equivalent to derived categories of modular representations of $GL_k$, and proves the existence of a full exceptional collection refining Kapranov's result. It introduces a Koszul-duality framework for strict polynomial functors that interchanges the two dual decompositions and yields inverse equivalences between corresponding blocks. A tilting bundle $\mathcal{E}(k,n)$ is constructed, whose endomorphism algebra $\mathcal{B}(k,n)$ admits two natural split quasi-hereditary structures, relating standard/costandard objects to Schur and Weyl functors; this provides a robust bridge between geometric and representation-theoretic descriptions. All results are stable under base change to arbitrary commutative base rings and extend to perfect complexes; analogous results over fields of arbitrary characteristic were obtained independently in BLVdB.

Abstract

In this paper we study the derived categories of coherent sheaves on Grassmannians $\operatorname{Gr}(k,n),$ defined over the ring of integers. We prove that the category $D^b(\operatorname{Gr}(k,n))$ has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of $GL_k.$ This in particular implies existence of a full exceptional collection, which is a refinement of Kapranov's collection \cite{Kap}, which was constructed over a field of characteristic zero. We also describe the right dual semi-orthogonal decomposition which has a similar form, and its components are full subcategories of the derived category of representations of $GL_{n-k}.$ The resulting equivalences between the components of the two decompositions are given by a version of Koszul duality for strict polynomial functors. We also construct a tilting vector bundle on $\operatorname{Gr}(k,n).$ We show that its endomorphism algebra has two natural structures of a split quasi-hereditary algebra over $\mathbb{Z},$ and we identify the objects of $D^b(\operatorname{Gr}(k,n)),$ which correspond to the standard and costandard modules in both structures. All the results automatically extend to the case of arbitrary commutative base ring and the category of perfect complexes on the Grassmannian, by extension of scalars (base change). Similar results over fields of arbitrary characteristic were obtained independently in \cite{BLVdB}, by different methods.

Derived categories of Grassmannians over integers and modular representation theory

TL;DR

This work constructs a characteristic-free description of the derived category of Grassmannians over by producing a semi-orthogonal decomposition with blocks equivalent to derived categories of modular representations of , and proves the existence of a full exceptional collection refining Kapranov's result. It introduces a Koszul-duality framework for strict polynomial functors that interchanges the two dual decompositions and yields inverse equivalences between corresponding blocks. A tilting bundle is constructed, whose endomorphism algebra admits two natural split quasi-hereditary structures, relating standard/costandard objects to Schur and Weyl functors; this provides a robust bridge between geometric and representation-theoretic descriptions. All results are stable under base change to arbitrary commutative base rings and extend to perfect complexes; analogous results over fields of arbitrary characteristic were obtained independently in BLVdB.

Abstract

In this paper we study the derived categories of coherent sheaves on Grassmannians defined over the ring of integers. We prove that the category has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of This in particular implies existence of a full exceptional collection, which is a refinement of Kapranov's collection \cite{Kap}, which was constructed over a field of characteristic zero. We also describe the right dual semi-orthogonal decomposition which has a similar form, and its components are full subcategories of the derived category of representations of The resulting equivalences between the components of the two decompositions are given by a version of Koszul duality for strict polynomial functors. We also construct a tilting vector bundle on We show that its endomorphism algebra has two natural structures of a split quasi-hereditary algebra over and we identify the objects of which correspond to the standard and costandard modules in both structures. All the results automatically extend to the case of arbitrary commutative base ring and the category of perfect complexes on the Grassmannian, by extension of scalars (base change). Similar results over fields of arbitrary characteristic were obtained independently in \cite{BLVdB}, by different methods.

Paper Structure

This paper contains 18 sections, 48 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Triangulated and DG categories.
  3. Semi-orthogonal decompositions
  4. Smoothness and properness
  5. Exceptional collections and tilting objects
  6. Highest weight categories and split quasi-hereditary algebras
  7. Definitions and basic properties
  8. Gluing of split quasi-hereditary algebras and highest weight categories
  9. Strict polynomial functors, representations of $GL_n$ and Koszul duality
  10. Strict polynomial functors
  11. Schur algebras.
  12. Koszul duality
  13. Representations of $GL_n$
  14. Base change
  15. Derived categories of Grassmannians
  16. ...and 3 more sections

Key Result

Theorem 1.3

(Kap) Suppose that $K$ is a field of characteristic zero. Then the category $D^b(X)$ has full strong exceptional collection $\{S_{\lambda}({\mathcal{F}})\}_{\lambda\in{\mathcal{P}}(n-k,k)}.$ Its right dual exceptional collection is $\{S_{\mu}({\mathcal{Q}})[-|\mu|]\}_{\mu\in{\mathcal{P}}(k,n-k)}.$ I

Theorems & Definitions (120)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 110 more