123 research outputs found
Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology
Classical definitions of locally complete intersection (l.c.i.) homomorphisms
of commutative rings are limited to maps that are essentially of finite type,
or flat. The concept introduced in this paper is meaningful for homomorphisms
phi : R \longrightarrow S of commutative noetherian rings. It is defined in
terms of the structure of phi in a formal neighborhood of each point of Spec S.
We characterize the l.c.i. property by different conditions on the vanishing of
the Andr\'e-Quillen homology of the R-algebra S. One of these descriptions
establishes a very general form of a conjecture of Quillen that was open even
for homomorphisms of finite type: If S has a finite resolution by flat
R-modules and the cotangent complex \cot SR is quasi-isomorphic to a bounded
complex of flat S-modules, then phi is l.c.i. The proof uses a mixture of
methods from commutative algebra, differential graded homological algebra, and
homotopy theory. The l.c.i. property is shown to be stable under a variety of
operations, including composition, decomposition, flat base change,
localization, and completion. The present framework allows for the results to
be stated in proper generality; many of them are new even with classical
assumptions. For instance, the stability of l.c.i. homomorphisms under
decomposition settles an open case in Fulton's treatment of orientations of
morphisms of schemes.Comment: 33 pages, published versio
A cohomological study of local rings of embedding codepth 3
The generating series of the Bass numbers of local rings with residue field are computed
in closed rational form, in case the embedding dimension of and its
depth satisfy . For each such it is proved that there is a
real number , such that holds for
all , except for in two explicitly described cases, where
. New restrictions are obtained on the
multiplicative structures of minimal free resolutions of length 3 over regular
local rings.Comment: In version 2 numerous typos have been corrected, details have been
added in a few places, and local rearrangements have been made. To appear in
JPAA. 24 page
Stable cohomology over local rings
The focus of this paper is on a poorly understood invariant of a commutative
noetherian local ring with residue field : the stable cohomology modules
, defined for each by Benson and
Carlson, Mislin, and Vogel; it coincides with Tate cohomology when is
Gorenstein. It is proved that important properties of , such as being
regular, complete intersection, or Gorenstein, are detected by the -rank of
for an arbitrary . Such numerical
characterizations are made possible by results on the structure of
-graded -algebra carried by . It is proved
that in many cases this algebra is determined by the absolute cohomology
algebra through a canonical homomorphism
.Comment: Final version, to appear in Adv. Math. Major reorganization of the
presentation. Many minor correction
Cohomology over complete intersections via exterior algebras
A general method for establishing results over a commutative complete
intersection local ring by passing to differential graded modules over a graded
exterior algebra is described. It is used to deduce, in a uniform way, results
on the growth of resolutions of complexes over such local rings.Comment: 18 pages; to appear in "Triangulated categories (Leeds, 2006)", LMS
lecture notes series
Constructing modules with prescribed cohomological support
A cohomological support, Supp_A(M), is defined for finitely generated modules
M over an left noetherian ring R, with respect to a ring A of central
cohomology operations on the derived category of R-modules. It is proved that
if the A-module Ext^R(M,M) is noetherian and Ext_i^R(M,R)=0 for i>>0, then
every closed subset of Supp_A(M) is the support of some finitely generated
R-module. This theorem specializes to known realizability results for varieties
of modules over group algebras, over local complete intersections, and over
finite dimensional algebras over a field. The theorem is also used to produce
large families of finitely generated modules of finite projective dimension
over commutative local noetherian rings.Comment: To appear in the Illinois Journal of Mathematics, the issue honoring
Phillip Griffith. Revised version has 18 pages. A word (the first one) has
been added to the title and the material has been reorganized into seven
sections, in place of the original six. There are, however, no changes of any
substanc
Homology over local homomorphisms
The notions of Betti numbers and of Bass numbers of a finite module N over a
local ring R are extended to modules that are only assumed to be finite over S,
for some local homomorphism f: R --> S. Various techniques are developed to
study the new invariants and to establish their basic properties. In several
cases they are computed in closed form. Applications go in several directions.
One is to identify new classes of finite R-modules whose classical Betti
numbers or Bass numbers have extremal growth. Another is to transfer ring
theoretical properties between R and S in situations where S may have infinite
flat dimension over R. A third is to obtain criteria for a ring equipped with a
`contracting' endomorphism -- such as the Frobenius endomorphism -- to be
regular or complete intersection; these results represent broad generalizations
of Kunz's characterization of regularity in prime characteristic.Comment: To appear in the American Journal of Mathematics; new version has
minor changes in the presentation; table of content removed; 52 page
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