Abstract
We propose a definition of Gorenstein Differential Graded Algebra. In order to give examples, we introduce the technical notion of Gorenstein morphism. This enables us to prove the following:
Theorem:Let A be a noetherian local commutative ring, let L be a bounded complex of finitely generated projective A-modules which is not homotopy equivalent to zero, and let ɛ=Hom A (L, L)be the endomorphism Differential Graded Algebra of L. Then ɛ is a Gorenstein Differential Graded Algebra if and only if A is a Gorenstein ring.
Theorem:Let A be a noetherian local commutative ring with a sequence of elements a=(a 1,…,a n )in the maximal ideal, and let K(a)be the Koszul complex of a.Then K(a)is a Gorenstein Differential Graded Algebra if and only if A is a Gorenstein ring.
Theorem:Let A be a noetherian local commutative ring containing a field k, and let X be a simply connected topological space with dim k H*(X;k)<∞,which has poincaré duality over k. Let C*(X;A)be the singular cochain Differential Graded Algebra of X with coefficients in A. Then C*(X; A)is a Gorenstein Differential Graded Algebra if and only if A is a Gorenstein ring.
The second of these theorems is a generalization of a result by Avramov and Golod from [4].
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Apassov,Homological dimensions over differential graded rings, inComplexes and Differential Graded Modules, Ph.D. thesis, Lund University, 1999, pp. 25–39.
L. L. Avramov and H.-B. Foxby,Locally Gorenstein homomorphisms, American Journal of Mathematics114 (1992), 1007–1047.
L. L. Avramov and H.-B. Foxby,Ring homomorphisms and finite Gorenstein dimension, Proceedings of the London Mathematical Society (3)75 (1997), 241–270.
L. L. Avramov and E. S. Golod,Homology algebra of the Koszul complex of a local Gorenstein ring, Mathematical Notes9 (1971), 30–32.
M. Bökstedt and A. Neeman,Homotopy limits in triangulated categories, Compositio Mathematica86 (1993), 209–234.
L. W. Christensen,Gorenstein Dimensions, Lecture Notes in Mathematics1747, Springer, Berlin, 2000.
W. G. Dwyer and J. P. C. Greenlees,Complete modules and torsion modules, American Journal of Mathematics124 (2002), 199–220.
W. G. Dwyer, J. P. C. Greenlees and S. Iyengar,Duality in algebra and topology, preprint, 2002.
Y. Félix, S. Halperin and J.-C. Thomas,Gorenstein spaces, Advances in Mathematics71 (1988), 92–112.
Y. Félix, S. Halperin and J.-C. Thomas,Rational Homotopy Theory, Graduate Texts in Mathematics, Vol. 205, Springer, Berlin, 2001.
A. Frankild, S. Iyengar and P. Jørgensen,Dualizing DG modules and Gorenstein DG algebras, to appear in Journal of the London Mathematical Society.
V. Hinich,Rings with approximation property admit a dualizing complex, Mathematische Nachrichten163 (1993), 289–296.
P. Jørgensen,Gorenstein homomorphisms of noncommutative rings, Journal of Algebra211 (1999), 240–267.
B. Keller,Deriving DG categories, Annales Scientifiques de l’École Normale Supérieure (4)27 (1994), 63–102.
B. Keller,On the cyclic homology of exact categories, Journal of Pure and Applied Algebra136 (1999), 1–56.
I. Kriz and J. P. May,Operads, algebras, modules and motives, Astérisque233 (1995).
H. Matsumura,Commutative Ring Theory, second edition, Cambridge Studies of Advanced Mathematics, Vol. 8, Cambridge University Press, Cambridge, 1989.
J. C. McConnell and J. C. Robson,Noncommutative Noetherian Rings, Wiley, New York, 1987.
N. Spaltenstein,Resolutions of unbounded complexes, Compositio Mathematica65 (1988), 121–154.
Q. S. Wu and J. J. Zhang,Some homological invariants of local PI algebras, Journal of Algebra225 (2000), 904–935.
A. Yekutieli,Dualizing complexes over noncommutative graded algebras, Jouranl of Algebra153 (1992), 41–84.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Frankild, A., Jørgensen, P. Gorenstein differential graded algebras. Isr. J. Math. 135, 327–353 (2003). https://doi.org/10.1007/BF02776063
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02776063