Let be a prime ideal in a commutative noetherian ring R. It is proved that if an R-module M satisfies ${\rm Tor}_n^R $(k ( ), M) = 0 for some n ⩾ R , where k( ) is the residue field at , then ${\rm ...Tor}_i^R $(k ( ), M) = 0 holds for all i ⩾ n. Similar rigidity results concerning ${\rm Tor}_R^{\ast} $(k ( ), M) are proved, and applications to the theory of homological dimensions are explored.
Complete homology over associative rings Celikbas, Olgur; Christensen, Lars Winther; Liang, Li ...
Israel journal of mathematics,
09/2017, Letnik:
221, Številka:
1
Journal Article
Recenzirano
Odprti dostop
We compare two generalizations of Tate homology to the realm of associative rings: stable homology and the J-completion of Tor, also known as complete homology. For finitely generated modules, we ...show that the two theories agree over Artin algebras and over commutative noetherian rings that are Gorenstein, or local and complete.
For a commutative ring
R
and a faithfully flat
R
-algebra
S
we prove, under mild extra assumptions, that an
R
-module
M
is Gorenstein flat if and only if the left
S
-module
S
⊗
R
M
is Gorenstein ...flat, and that an
R
-module
N
is Gorenstein injective if and only if it is cotorsion and the left
S
-module Hom
R
(
S
,
N
) is Gorenstein injective. We apply these results to the study of Gorenstein homological dimensions of unbounded complexes. In particular, we prove two theorems on stability of these dimensions under faithfully flat (co-)base change.
Auslander's depth formula for pairs of Tor-independent modules over a regular local ring, depth(M⊗RN)=depthM+depthN−depthR, has been generalized in several directions; most significantly it has been ...shown to hold for pairs of Tor-independent modules over complete intersection rings.
In this paper we establish a depth formula that holds for every pair of Tate Tor-independent modules over a Gorenstein local ring. It subsumes previous generalizations of Auslander's formula and yields new results on vanishing of cohomology over certain Gorenstein rings.
A 2003 counterexample to a conjecture of Auslander brought attention to a family of rings—colloquially called AC rings—that satisfy a natural condition on vanishing of cohomology. Several results ...attest to the remarkable homological properties of AC rings, but their definition is barely operational, and it remains unknown if they form a class that is closed under typical constructions in ring theory. In this paper, we study transfer of the AC property along local homomorphisms of Cohen–Macaulay rings. In particular, we show that the AC property is preserved by standard procedures in local algebra. Our results also yield new examples of Cohen–Macaulay AC rings.
Let R be a commutative Noetherian ring. We study R--modules, and complexes of such, with excellent duality properties. While their common properties are strong enough to admit a rich theory, we count ...among them such, potentially, diverse objects as \emph{dualizing complexes} for R on one side, and on the other, the \emph{ring} itself. In several ways, these two examples constitute the extremes, and their well-understood properties serve as guidelines for our study; however, also the employment, in recent studies of ring homomorphisms, of complexes ``lying between'' these extremes is incentive.
It is proved that a module
M
over a commutative noetherian ring
R
is injective if
Ext
R
i
(
(
R
/
p
)
p
,
M
)
=
0
holds for every
i
⩾
1
and every prime ideal
p
in
R
. This leads to the following ...characterization of injective modules: If
F
is faithfully flat, then a module
M
such that
Hom
R
(
F
,
M
)
is injective and
Ext
R
i
(
F
,
M
)
=
0
for all
i
⩾
1
is injective. A limited version of this characterization is also proved for certain non-noetherian rings.
Artinian quotients R of the local ring Q=kx,y,z are classified by multiplicative structures on A=Tor⁎Q(R,k); in particular, R is Gorenstein if and only if A is a Poincaré duality algebra while R is ...Golod if and only if all products in A⩾1 are trivial. There is empirical evidence that generic quotient rings with small socle ranks fall on a spectrum between Golod and Gorenstein in a very precise sense: The algebra A breaks up as a direct sum of a Poincaré duality algebra P and a graded vector space V, on which P⩾1 acts trivially. That is, A is a trivial extension, A=P⋉V, and the extremes A=(k⊕Σk)⋉V and A=P correspond to R being Golod and Gorenstein, respectively.
We prove that this observed behavior is, indeed, the generic behavior for graded quotients R of socle rank 2, and we show that the rank of P is controlled by the difference between the order and the degree of the socle polynomial of R.
PDF
