Title: Rigid Complexes via DG Algebras
Authors: Amnon Yekutieli and James J. Zhang
Publication status: Trans. AMS 360 no. 6 (2008), 3211–3248

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
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 \iso 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 a subsequent paper we consider rigid dualizing complexes over
commutative rings, building on the results of the present paper.
The project culminates in a third paper, 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.

Electronic Preprint:
amslatex file
pdf file (acrobat)

journal pdf file

Warning. Some of the proofs in this paper have severe gaps in them. Still, almost all results in it are correct (!). Corrections are under preparation, and should be available early in 2015.

(updated 28 Dec 2014)