Derived categories of Grassmannians over integers and modular representation theory

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• 作者：Alexander I. Efimov^a ; ^b ; ¹ ; ^{efimov@mccme.ru" class="auth_mail" title="E-mail the corresponding author}
• 关键词：13D09 ; 14F05 ; 16G99
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：2 January 2017
• 年：2017
• 卷：304
• 期：Complete
• 页码：179-226
• 全文大小：684 K

文摘

In this paper we study the derived categories of coherent sheaves on Grassmannians Gr(k,n)$\mathrm{Gr}(k,n)$, defined over the ring of integers. We prove that the category D^b(Gr(k,n))$D^b(\mathrm{Gr}(k,n))$ has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of GL_k$G{L}_k$. This in particular implies existence of a full exceptional collection, which is a refinement of Kapranov's collection [13], 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$G{L}_{n-k}$. 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 Gr(k,n)$\mathrm{Gr}(k,n)$. We show that its endomorphism algebra has two natural structures of a split quasi-hereditary algebra over d22915a01d8c" title="Click to view the MathML source">Z$\mathbb{Z}$, and we identify the objects of D^b(Gr(k,n))$D^b(\mathrm{Gr}(k,n))$, 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 [7], by different methods.