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Transactions of the American Mathematical Society

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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
Published electronically: January 30, 2008
MathSciNet review: 2379794
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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).

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Additional Information

Amnon Yekutieli
Affiliation: Department of Mathematics, Ben Gurion University, Be’er Sheva 84105, Israel

James J. Zhang
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195

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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