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Rigid complexes via DG algebras


Authors: Amnon Yekutieli and James J. Zhang
Journal: Trans. Amer. Math. Soc. 360 (2008), 3211-3248
MSC (2000): Primary 18E30; Secondary 18G10, 16E45, 18G15
DOI: https://doi.org/10.1090/S0002-9947-08-04465-6
Published electronically: January 30, 2008
MathSciNet review: 2379794
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A$ be a commutative ring, $ B$ a commutative $ A$-algebra and $ M$ a complex of $ B$-modules. We begin by constructing the square $ \operatorname{Sq}_{B / A} M$, which is also a complex of $ B$-modules. The squaring operation is a quadratic functor, and its construction requires differential graded (DG) algebras. If there exists an isomorphism $ \rho : M \xrightarrow{\simeq} \operatorname{Sq}_{B / A} M$, then the pair $ (M, \rho)$ is called a rigid complex over $ B$ relative to $ A$ (there are some finiteness conditions). There is an obvious notion of rigid morphism between rigid complexes.

We establish several properties of rigid complexes, including their uniqueness, existence (under some extra hypothesis), and formation of pullbacks $ f^{\flat}(M, \rho)$ (resp. $ f^{\sharp}(M, \rho)$) along a finite (resp. essentially smooth) ring homomorphism $ f^* : B \to C$.

In the subsequent paper, Rigid Dualizing Complexes over Commutative Rings, we consider rigid dualizing complexes over commutative rings, building on the results of the present paper. The project culminates in our paper Rigid Dualizing Complexes and Perverse Sheaves on Schemes, where we give a comprehensive version of Grothendieck duality for schemes.

The idea of rigid complexes originates in noncommutative algebraic geometry, and is due to Van den Bergh (1997).


References [Enhancements On Off] (What's this?)

  • [AFH] L. Avramov, H.-B. Foxby and S. Halperin, Differential graded homological algebra, in preparation.
  • [Be] K. Behrend, Differential Graded Schemes I: Perfect Resolving Algebras, eprint math.AG/0212225 at http://arxiv.org.
  • [DGI] W. Dwyer, J. P. C. Greenlees and S. Iyengar, Duality in algebra and topology, Adv . Math. 200 (2006), 357-402. MR 2200850 (2006k:55017)
  • [Dr] V. Drinfeld, DG quotients of DG categories, J. Algebra 272, Number 2 (2004), 643-691. MR 2028075 (2006e:18018)
  • [EGA] A. Grothendieck and J. Dieudonné, ``Éléments de Géometrie Algébrique.'' Chapitre $ 0{}_{\mathrm{IV}}$, Publ. Math. IHES 20 (1964); Chapitre IV, Publ. Math. IHES 32 (1967). MR 0173675 (30:3885)
  • [FIJ] A. Frankild, S. Iyengar and P. Jørgensen, Dualizing Differential Graded Modules and Gorenstein Differential Graded Algebras, J. London Math. Soc. (2) 68 (2003), 288-306. MR 1994683 (2004f:16013)
  • [Hi] V. Hinich, Homological algebra of homotopy algebras, Comm. Algebra 25 (1997), no. 10, 3291-3323. MR 1465117 (99b:18017)
  • [Ke] B. Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63-102. MR 1258406 (95e:18010)
  • [KM] I. Kriz and J.P. May, ``Operads, Algebras, Modules and Motives,'' Astérisque 233 (1995). MR 1361938 (96j:18006)
  • [KS] M. Kashiwara and P. Schapira, ``Sheaves on Manifolds,'' Springer-Verlag, Berlin, 1990. MR 1074006 (92a:58132)
  • [ML] S. Maclane, ``Homology,'' Springer-Verlag, 1994 (reprint). MR 1344215 (96d:18001)
  • [Ne] A. Neeman, The Grothendieck duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205-236. MR 1308405 (96c:18006)
  • [Qu] D. Quillen, On the cohomology of commutative rings, Proc. Symp. Pure Math. (AMS) 17 (1970), pp. 65-87. MR 0257068 (41:1722)
  • [RD] R. Hartshorne, ``Residues and Duality,'' Lecture Notes in Math. 20, Springer-Verlag, Berlin, 1966. MR 0222093 (36:5145)
  • [Sp] N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121-154. MR 932640 (89m:18013)
  • [Ta] G. Tabuada, Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories, C. R. Acad. Sci. Paris, Ser. I 340 (2005), 15-19. MR 2112034 (2005h:18033)
  • [VdB] M. Van den Bergh, Existence theorems for dualizing complexes over non-commutative graded and filtered rings, J. Algebra 195 (1997), no. 2, 662-679. MR 1469646 (99b:16010)
  • [Ye1] A. Yekutieli, Dualizing Complexes over Noncommutative Graded Algebras, J. Algebra 153 no. 1 (1992), 41-84. MR 1195406 (94a:16077)
  • [Ye2] A. Yekutieli, ``An Explicit Construction of the Grothendieck Residue Complex'' (with an appendix by P. Sastry), Astérisque 208 (1992). MR 1213064 (94e:14026)
  • [Ye3] A. Yekutieli, Smooth formal embeddings and the residue complex, Canadian J. Math. 50 (1998), 863-896. MR 1638635 (99i:14004)
  • [Ye4] A. Yekutieli, Rigid Dualizing Complexes and Perverse Coherent Sheaves on Schemes, in preparation.
  • [YZ1] A. Yekutieli and J.J. Zhang, Rings with Auslander dualizing complexes, J. Algebra 213 (1999), no. 1, 1-51. MR 1674648 (2000f:16012)
  • [YZ2] A. Yekutieli and J.J. Zhang, Residue complexes over noncommutative rings, J. Algebra 259 (2003) no. 2, 451-493. MR 1955528 (2004a:16010)
  • [YZ3] A. Yekutieli and J.J. Zhang, Dualizing complexes and perverse modules over differential algebras, Compositio Math. 141 (2005), 620-654. MR 2135281 (2006c:16014)
  • [YZ4] A. Yekutieli and J.J. Zhang, Rigid Dualizing Complexes over Commutative Rings, to appear in Algebr. Represent. Theory, eprint math.AG/0601654 at http://arxiv.org.

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Additional Information

Amnon Yekutieli
Affiliation: Department of Mathematics, Ben Gurion University, Be’er Sheva 84105, Israel
Email: amyekut@math.bgu.ac.il

James J. Zhang
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: zhang@math.washington.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04465-6
Keywords: Commutative rings, DG algebras, derived categories, rigid complexes.
Received by editor(s): June 22, 2006
Published electronically: January 30, 2008
Additional Notes: This research was supported by the US-Israel Binational Science Foundation. The second author was partially supported by the US National Science Foundation.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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