**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)