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)