Coherent sheaves on \({\mathbb{P}}^n\) and problems of linear algebra
type: Item
instance of: scholarly article
title: Coherent sheaves on \({\mathbb{P}}^n\) and problems of linear algebra
zbMATH Open document ID: 0424.14003
DOI: 10.1007/BF01681436
publication date: 1979-01-01T00:00:00Z
review text: In this classic paper the author studies the bounded derived category \(D^b ({\mathbb P}^n)\) of coherent sheaves on a projective space \({\mathbb P}^n\). The main result shows that this category is equivalent (as a triangulated category) to a certain subcategory of the category of graded modules over a symmetric algebra of \(V=H^0({\mathbb P}^n,{\mathcal O}(1))\). One can also replace the symmetric algebra by the exterior algebra of \(V\). This result was motivated by earlier results of Horrocks who introduced monads and it was the first example of computation of the derived category of coherent sheaves on an algebraic variety. Two other, although similar, descriptions of \(D^b ({\mathbb P}^n)\) were obtained at the same time by \textit{I. N. Bernstein, I. M. Gel'fand} and \textit{S. I. Gel'fand} [see Funkts. Anal. Prilozh. 12, 66--67 (1978; Zbl 0402.14005)] using a more algebraic approach. However, Beilinson's paper is very remarkable not only because of this result but also because it introduces a very useful method of describing the derived category of coherent sheaves on a variety \(X\) using a resolution of the structure sheaf of the diagonal \(\Delta \subset X\times X\). Implicitly, it also contains the Beilinson spectral sequences that proved to be extremely useful, e.g., when studying sheaves whose cohomology groups are known. In a nutshell, this small paper contains the striking concept that relevant geometric information is encoded in a seemingly highly abstract object, which has lead to a whole industry today, along the results of \textit{A. Bondal} and \textit{D. Orlov} [Compos. Math. 125, No.3, 327--344 (2001; Zbl 0994.18007)] that \(D^b_{coh}(X)\) indeed determines \(X\) for varieties with ample anticanonical (e.g., the Fano case) or ample canonical divisor, the theory of spherical objects and the associated spherical twists introduced by \textit{P. Seidel} and \textit{R. Thomas} [Duke Math. J. 108, No. 1, 37--108 (2001; Zbl 1092.14025)] or the homolo…
reviewed by: Adrian Langer
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 3659709
zbMATH Keywords: coherent sheaves
zbMATH Keywords: classification of vector bundle
zbMATH Keywords: exterior algebra
zbMATH Keywords: derived category
zbMATH Keywords: projective space
author: Alexander A. Beilinson
MaRDI profile type: Publication
Wikidata QID: Q56610723
cites work: Moduli of vector bundles on the projective plane
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3659709