Rigid complexes via DG algebras

Yekutieli, Amnon; Zhang, James

doi:10.1090/S0002-9947-08-04465-6

Rigid complexes via DG algebras
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by Amnon Yekutieli and James J. Zhang PDF

Trans. Amer. Math. Soc. 360 (2008), 3211-3248 Request permission

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

L. Avramov, H.-B. Foxby and S. Halperin, Differential graded homological algebra, in preparation.
K. Behrend, Differential Graded Schemes I: Perfect Resolving Algebras, eprint math.AG/0212225 at http://arxiv.org.
W. G. Dwyer, J. P. C. Greenlees, and S. Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006), no. 2, 357–402. MR 2200850, DOI 10.1016/j.aim.2005.11.004
Vladimir Drinfeld, DG quotients of DG categories, J. Algebra 272 (2004), no. 2, 643–691. MR 2028075, DOI 10.1016/j.jalgebra.2003.05.001
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 173675
Anders Frankild, Srikanth Iyengar, and Peter Jørgensen, Dualizing differential graded modules and Gorenstein differential graded algebras, J. London Math. Soc. (2) 68 (2003), no. 2, 288–306. MR 1994683, DOI 10.1112/S0024610703004496
Vladimir Hinich, Homological algebra of homotopy algebras, Comm. Algebra 25 (1997), no. 10, 3291–3323. MR 1465117, DOI 10.1080/00927879708826055
Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102. MR 1258406
Igor Kříž and J. P. May, Operads, algebras, modules and motives, Astérisque 233 (1995), iv+145pp (English, with English and French summaries). MR 1361938
Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel. MR 1074006, DOI 10.1007/978-3-662-02661-8
Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215
Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236. MR 1308405, DOI 10.1090/S0894-0347-96-00174-9
Daniel Quillen, On the (co-) homology of commutative rings, Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 65–87. MR 0257068
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093
N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154. MR 932640
Goncalo Tabuada, Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories, C. R. Math. Acad. Sci. Paris 340 (2005), no. 1, 15–19 (French, with English and French summaries). MR 2112034, DOI 10.1016/j.crma.2004.11.007
Michel 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, DOI 10.1006/jabr.1997.7052
Amnon Yekutieli, Dualizing complexes over noncommutative graded algebras, J. Algebra 153 (1992), no. 1, 41–84. MR 1195406, DOI 10.1016/0021-8693(92)90148-F
Amnon Yekutieli, An explicit construction of the Grothendieck residue complex, Astérisque 208 (1992), 127 (English, with French summary). With an appendix by Pramathanath Sastry. MR 1213064
Amnon Yekutieli, Smooth formal embeddings and the residue complex, Canad. J. Math. 50 (1998), no. 4, 863–896. MR 1638635, DOI 10.4153/CJM-1998-046-1
A. Yekutieli, Rigid Dualizing Complexes and Perverse Coherent Sheaves on Schemes, in preparation.
Amnon Yekutieli and James J. Zhang, Rings with Auslander dualizing complexes, J. Algebra 213 (1999), no. 1, 1–51. MR 1674648, DOI 10.1006/jabr.1998.7657
Amnon Yekutieli and James J. Zhang, Residue complexes over noncommutative rings, J. Algebra 259 (2003), no. 2, 451–493. MR 1955528, DOI 10.1016/S0021-8693(02)00579-3
Amnon Yekutieli and James J. Zhang, Dualizing complexes and perverse modules over differential algebras, Compos. Math. 141 (2005), no. 3, 620–654. MR 2135281, DOI 10.1112/S0010437X04001307
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.