Derived categories and Morita theory
type: Item
instance of: scholarly article
title: Derived categories and Morita theory
zbMATH Open document ID: 0604.16025
DOI: 10.1016/0021-8693(86)90224-3
author: Brian J. Parshall
author: Leonard L. Scott
author: Edward Cline
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: The authors develop a Morita theory for derived categories using tilting modules. Let A be an arbitrary ring, \({\mathcal A}=mod\)-A the category of right A-modules and T a right A-module, which is a tilting module; i.e., (i) T has a finite, finitely generated projective resolution of length f, (ii) \(Ext^ n_ A(T,T)=0\) for all positive integers n, (iii) A has a finite resolution \(0\to A\to T^ 0\to...\to T^ g\to 0\), with \(T^ i\in add(T)\). Let \(B=End_ A(T)\), and \({\mathcal B}=mod\)-B, \({\mathbb{F}}: {\mathcal A}\to {\mathcal B}\) is \(Hom_ A(T\),-) and \({\mathbb{G}}: {\mathcal B}\to {\mathcal A}\) is \(-\otimes_ BT\). \(D({\mathcal A})\) and \(D^ b({\mathcal A})\) is the derived category resp. the category of bounded complexes, similarly for \({\mathcal B}.\) The main result is Theorem: (a) The right derived functor \(R^{+,b}{\mathbb{F}}: D^ b({\mathcal A})\to D({\mathcal B})\) has image in \(D^ b({\mathcal B})\). In fact, \(\dim (R^{+,b})\leq f\). (b) Similarly \(L^{- ,b}{\mathbb{G}}: D^ b({\mathcal B})\to D({\mathcal A})\) has image in \(D^ b({\mathcal A})\) with \(\dim (L^{-,b})\leq g\). (c) The functors \(R^{+,b}{\mathbb{F}}\) and \(L^{-,b}{\mathbb{G}}\) induce mutually inverse equivalences between \(D^ b({\mathcal A})\) and \(D^ b({\mathcal B})\). (d) \(A\cong End_ B(T)\), and T satisfies the left analogues of (i), (ii) and (iii) for B. Moreover, the projective dimension of T as B-module is at most g, and in (iii) one may choose \(g=p\dim_ B(T).\) \textit{D. Happel} [Triangulated categories in representation theory of finite dimensional algebras (Preprint Jan. 85, to appear in Comment. Math. Helv.)] has obtained the equivalence between the derived categories in case A is a finite dimensional algebra of finite global dimension, using the notion of a tilting module only for \(f=g=1\). \textit{Y. Miyashita} [Math. Z. 193, 113-146 (1986; Zbl 0578.16015)] considers tilting modules over arbitrary rings - in the above generality, and also obtains (d) of the ab…
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16B50
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16E10
zbMATH DE Number: 3975153
zbMATH Keywords: Morita theory for derived categories
zbMATH Keywords: tilting modules
zbMATH Keywords: projective resolution
zbMATH Keywords: right derived functor
zbMATH Keywords: equivalences
zbMATH Keywords: projective dimension
zbMATH Keywords: finite dimensional algebra
zbMATH Keywords: global dimension
zbMATH Keywords: triangulated categories
zbMATH Keywords: homotopy categories
Wikidata QID: Q61795041
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90224-3
OpenAlex ID: W1966584552
cites work: Q5562621
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cites work: Kazhdan-Lusztig conjecture and holonomic systems
cites work: Q3890859
cites work: Q3889229
cites work: Affine duality and cofiniteness
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cites work: TILTED ORDERS AND ORDERS OF FINITE GLOBAL DIMENSION
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description: scientific article; zbMATH DE number 3975153