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Published in:
Preprojective Algebras, Singularity Categories and Orthogonal Decompositions Read chapter Read first chapter

2013 | OriginalPaper | Chapter

(Contravariant) Koszul Duality for DG Algebras

Author : Luchezar L. Avramov

Published in: Algebras, Quivers and Representations

Publisher: Springer Berlin Heidelberg

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Abstract

A DG algebras A over a field k with H(A) connected and H<0(A)=0 has a unique up to isomorphism DG module K with H(K)≅k. It is proved that if \(\operatorname {H}(A)\) is degreewise finite, then \(\mathrm{RHom}_{A}(?,K)\colon \mathsf{D^{df}_{+}}(A)^{\mathsf{op}}\equiv \mathsf{D_{df}^{+}}(\mathrm{RHom}_{A}(K,K))\) is an exact equivalence of derived categories of DG modules with degreewise finite-dimensional homology. It induces an equivalences of \(\mathsf{D^{df}_{b}}(A)^{\mathsf{op}}\) and the category of perfect DG RHom A (K,K) modules, and vice-versa. Corresponding statements are proved also when H(A) is simply connected and H<0(A)=0.

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previous chapter Preprojective Algebras, Singularity Categories and Orthogonal Decompositions
next chapter The Fundamental Group of a Morphism in a Triangulated Category
Appendix
Available only for authorised users
Footnotes
1
The reference has been redirected to the bibliography in the present paper.
 
2
In order to keep displays readable we write | instead of ⊗. Brackets are placed under those https://static-content.springer.com/image/chp%3A10.1007%2F978-3-642-39485-0_2/316967_1_En_2_IEq26_HTML.gif that are modified at the given step. Thus, computations can be followed by checking for commutativity small diagrams involving only the selected terms.
 
3
This is the opposite algebra of the quadratic dual defined in [6, 2.8.1], which is constructed by using the canonical map V ∗⊗U ∗→(U⊗V)∗, see [6, 2.7], rather than the map ϖ UV from Appendix A.4(3).
 
Literature
1.
L. L. Avramov, Golod homomorphisms, Algebra, algebraic topology and their interactions (Stockholm, 1983), 59–78, Lecture Notes in Math. 1183, Springer, Berlin, 1986.
2.
L. L. Avramov, R.-O. Buchweitz, S. B. Iyengar, C. Miller, Homology of perfect complexes, Adv. Math. 225 (2010), 1731–1781. Corrigendum, Adv. Math. 223 (2010), 3576-3578. CrossRef
3.
L. L. Avramov, S. Halperin, Through the looking glass: a dictionary between rational homotopy theory and local algebra, Algebra, algebraic topology and their interactions (Stockholm, 1983), 1–27, Lecture Notes in Math. 1183, Springer, Berlin, 1986. CrossRef
4.
L. L. Avramov, D. Jorgensen, Reverse homological algebra over local rings, in preparation.
5.
6.
A. Beilinson, V. Ginzburg, W. Sorgel, Koszul duality patterns in representation theory, J. Am. Math. Soc. 9 (1996), 473–527. MATHCrossRef
7.
8.
9.
Y. Félix, S. Halperin, J.-C. Thomas, Adams’ cobar equivalence, Trans. Am. Math. Soc. 329 (1992), 531–549. MATH
10.
Y. Félix, S. Halperin, J.-C. Thomas, Rational homotopy theory, Graduate Texts in Math. 205, Springer, Berlin, 2001. MATHCrossRef
11.
E. S. Golod, Homologies of some local rings, Dokl. Akad. Nauk SSSR 144 (1962), 479–482. MathSciNet
12.
13.
14.
15.
D. Husemoller, J. C. Moore, J. D. Stasheff, Differential homological algebra and homogeneous spaces, J. Pure Appl. Algebra 5 (1974), 113–185. MathSciNetMATHCrossRef
16.
B. Keller, Deriving DG categories, Ann. Sci. Ec. Norm. Super. (4) 27 (1994), 63–104. MATH
17.
B. Keller, P. Nicolas, Weight structures and simple DG modules for positive DG algebras, Int. Math. Res. Not. 2013 (2013), 1028–1078. MathSciNet
18.
J.-M. Lemaire, Algèbres connexes et homologie des éspaces de lacets, Lecture Notes in Math. 422, Springer, Berlin, 1974. MATH
19.
J.-L. Loday, B. Vallette, Algebraic operads, Grundlehren Math. Wiss. 346, Springer, Berlin, 2012. MATHCrossRef
20.
C. Löfwall, On the subalgebra generated by the one-dimensional elements in the Yoneda Ext-algebra, Algebra, algebraic topology and their interactions (Stockholm, 1983), 291–338, Lecture Notes in Math. 1183, Springer, Berlin, 1986. CrossRef
21.
J. C. Moore, Differential homological algebra, Actes du congrès international des mathématiciens (Nice, 1970), Tome 1, 335–339, Gauthier-Villars, Paris, 1971.
22.
J. Neisendorfer, Algebraic methods in unstable homotopy theory, New Math. Monographs 12, Cambridge Univ. Press, Cambridge, 2010.
23.
Metadata
Title
(Contravariant) Koszul Duality for DG Algebras
Author
Luchezar L. Avramov
Copyright Year
2013
Publisher
Springer Berlin Heidelberg
Book
Algebras, Quivers and Representations

Print ISBN: 978-3-642-39484-3

Electronic ISBN: 978-3-642-39485-0

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-3-642-39485-0_2

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