Fourier-Mukai functors in the supported case
type: Item
instance of: scholarly article
zbMATH Open document ID: 1326.14039
DOI: 10.1112/S0010437X13007872
author: Alberto Canonaco
author: Paolo Stellari
published in: Compositio Mathematica
publication date: 2014-09-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.0798
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 6349697
zbMATH Keywords: derived categories
zbMATH Keywords: supported sheaves
zbMATH Keywords: Fourier-Mukai functors
MaRDI profile type: Publication
arXiv ID: 1010.0798
cites work: The Hall algebra of a spherical object
cites work: Chain complexes and stable categories
cites work: The homotopy theory of dg-categories and derived Morita theory
cites work: Equivalences of derived categories and \(K3\) surfaces
cites work: Equivalences of derived categories of sheaves on smooth stacks
cites work: Categories and Sheaves
cites work: Autoequivalences of derived categories on the minimal resolutions of \(A_n\)-singularities on surfaces
cites work: Twisted Fourier-Mukai functors
cites work: Stability conditions on \(A_n\)-singularities
cites work: Does full imply faithful?
cites work: Stability conditions on triangulated categories
cites work: The Ringel-Hall Lie algebra of a spherical object
cites work: DG quotients of DG categories.
cites work: Q5693333
cites work: Non-uniqueness of Fourier-Mukai kernels
cites work: The space of stability conditions on the local projective plane
cites work: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability
cites work: Uniqueness of enhancement for triangulated categories
title: Fourier-Mukai functors in the supported case
review text: \textit{D. O. Orlov} [J. Math. Sci., New York 84, No. 5, 1361--1381 (1997; Zbl 0938.14019)] proved that for smooth projective varieties \(X_1\) and \(X_2\) over a field \(k\) any full and faithful exact functor \(\mathsf{F} \colon {D}^b(X_1)\to {D}^b(X_2)\) can be represented by an integral transform functor with kernel \(\mathcal{E} \in {D}^b(X_1 \times X_2)\), that is, \(\mathsf{F}(-) \cong \mathsf{R}p_{2\,*}(\mathcal{E} \otimes (p_1)^*(-))\), where \(p_i\) denotes the projection onto \(X_i\) for \(i \in \{1, 2\}\).NEWLINENEWLINEIn the literature there have been several variants, in particular the category \({D}^b(X)\) is replaced by \(\mathbf{Perf}(X)\), its subcategory of perfect complexes. This is the approach taken in [\textit{V. A. Lunts} and \textit{D. O. Orlov}, J. Am. Math. Soc. 23, No. 3, 853--908 (2010; Zbl 1197.14014)]. It was shown that a fully faithful functor between the corresponding categories of perfect complexes is represented by an integral transform, thus generalizing the previous result to possibly singular schemes. The main ingredient in their proof is the existence of enhancements, i.e., nice DG-models of the derived categories.NEWLINENEWLINEIn this paper, the authors keep pursuing Orlov's question about whether all exact functors between the categories of perfect complexes on projective schemes are of Fourier--Mukai type, focusing in the supported situation. Specifically, let \(Z \subset X\) be a closed subset and consider the full subcategory \(\mathbf{Perf}_Z(X)\) of \(\mathbf{Perf}(X)\) formed by complexes supported at \(Z\). The main result is the following. Let \(X_1\) be a quasi-projective scheme over a field \(k\) and \(Z_1\) be a projective subscheme such that all of its infinitesimal embeddings have a structure sheaf that is perfect as an object in \({D}(\mathbf{Qco}(X_1))\). Let \(X_2\) be a separated scheme of finite type over \(k\) and let \(Z_2\) be a closed subscheme of \(X_2\). An exact functor NEWLINE\[NEWLINE \mathsf{F} \co…
reviewed by: Leovigildo Alonso-Tarrío
description: scientific article; zbMATH DE number 6349697