Growth in the minimal injective resolution of a local ring
@article{Christensen2010GrowthIT, title={Growth in the minimal injective resolution of a local ring}, author={Lars Winther Christensen and Janet Striuli and Oana Veliche}, journal={Journal of the London Mathematical Society}, year={2010}, volume={81} }
Let R be a commutative noetherian local ring with residue field k and assume that it is not Gorenstein. In the minimal injective resolution of R, the injective envelope E of the residue field appears as a summand in every degree starting from the depth of R. The number of copies of E in degree i equals the k‐vector space dimension of the cohomology module ExtiR(k, R). These dimensions, known as Bass numbers, form an infinite sequence of invariants of R about which little is known. We prove that…
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References
SHOWING 1-10 OF 49 REFERENCES
Homology over local homomorphisms
- Mathematics
- 2003
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 φ: R → S.…
Injective dimension in Noetherian rings
- Mathematics
- 1962
Introduction. Among Noetherian rings quasi-Frobenius rings are those which are self injective [10, Theorem 18]. This paper is concerned primarily with Noetherian rings whose self injective dimension…
On the growth of the Betti sequence of the canonical module
- Mathematics
- 2006
We study the growth of the Betti sequence of the canonical module of a Cohen–Macaulay local ring. It is an open question whether this sequence grows exponentially whenever the ring is not Gorenstein.…
Two applications of change of rings theorems for Poincaré series
- Mathematics
- 1979
Let (R, m, k) be an artinian Gorenstein ring with dim* m/m2 > 2. If PR denotes the Poincare-series and Ä denotes the Bass-series of R, then <¡>K (PR l)x\l P'x2)' ' with R = A/0: m, see Proposition 1.…
Boundedness versus periodicity over commutative local rings
- Mathematics
- 1990
Over commutative graded local artinian rings, examples are constructed of periodic modules of arbitrary minimal period and modules with bounded Betti numbers, which are not eventually periodic. They…
Growth of Betti numbers of modules over local rings of small embedding codimension or small linkage number
- Mathematics
- 1994
Rings that are almost Gorenstein
- Mathematics
- 2006
We introduce classes of rings that are close to being Gorenstein and prove they arise naturally as specializations of rings of countable CM type. We study these rings in detail and, inter alia,…