اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدCharacterizing certain semidualizing complexes via their Betti and Bass numbers
113
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
It is known that the numerical invariants Betti numbers and Bass numbers are worthwhile tools for decoding a large amount of information about modules over commutative rings. We highlight this fact, further, by establishing some criteria for certain semidualizing complexes via their Betti and Bass numbers. Two distinguished types of semidualizing complexes are the shifts of the underlying rings and dualizing complexes. Let $C$ be a semidualizing complex for an analytically irreducible local ring $R$ and set $n:=sup C$ and $d:=dim_RC$. We show that $C$ is quasi-isomorphic to a shift of $R$ if and only if the $n$th Betti number of $C$ is one. Also, we show that $C$ is a dualizing complex for $R$ if and only if the $d$th Bass number of $C$ is one.
قيم البحث
اقرأ أيضاً
Let $(A, m, k)$ be a Gorenstein local ring of dimension $ dgeq 1.$ Let $I$ be an ideal of $A$ with $htt(I) geq d-1.$ We prove that the numerical function [ n mapsto ell(ext_A^i(k, A/I^{n+1}))] is given by a polynomial of degree $d-1 $ in the case w
hen $ i geq d+1 $ and $curv(I^n) > 1$ for all $n geq 1.$ We prove a similar result for the numerical function [ n mapsto ell(Tor_i^A(k, A/I^{n+1}))] under the assumption that $A$ is a CM ~ local ring. oindent We note that there are many examples of ideals satisfying the condition $curv(I^n) > 1,$ for all $ n geq 1.$ We also consider more general functions $n mapsto ell(Tor_i^A(M, A/I_n)$ for a filtration ${I_n }$ of ideals in $A.$ We prove similar results in the case when $M$ is a maximal CM ~ $A$-module and ${I_n=overline{I^n} }$ is the integral closure filtration, $I$ an $m$-primary ideal in $A.$
We introduce and study a class of objects that encompasses Christensen and Foxbys semidualizing modules and complexes and Kubiks quasi-dualizing modules: the class of $mathfrak{a}$-adic semidualizing modules and complexes. We give examples and equiva
lent characterizations of these objects, including a characterization in terms of the more familiar semidualizing property. As an application, we give a proof of the existence of dualizing complexes over complete local rings that does not use the Cohen Structure Theorem.
A result of Foxby states that if there exists a complex with finite depth, finite flat dimension, and finite injective dimension over a local ring $R$, then $R$ is Gorenstein. In this paper we investigate some homological dimensions involving a semid
ualizing complex and improve on Foxbys result by answering a question of Takahashi and White. In particular, we prove for a semidualizing complex $C$, if there exists a complex with finite depth, finite $mathcal{F}_C$-projective dimension, and finite $mathcal{I}_C$-injective dimension over a local ring $R$, then $R$ is Gorenstein.
We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered
as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.
We study homological properties of random quadratic monomial ideals in a polynomial ring $R = {mathbb K}[x_1, dots x_n]$, utilizing methods from the Erd{o}s-R{e}nyi model of random graphs. Here for a graph $G sim G(n, p)$ we consider the `coedge idea
l $I_G$ corresponding to the missing edges of $G$, and study Betti numbers of $R/I_G$ as $n$ tends to infinity. Our main results involve fixing the edge probability $p = p(n)$ so that asymptotically almost surely the Krull dimension of $R/I_G$ is fixed. Under these conditions we establish various properties regarding the Betti table of $R/I_G$, including sharp bounds on regularity and projective dimension, and distribution of nonzero normalized Betti numbers. These results extend work of Erman and Yang, who studied such ideals in the context of conjectured phenomena in the nonvanishing of asymptotic syzygies. Along the way we establish results regarding subcomplexes of random clique complexes as well as notions of higher-dimensional vertex $k$-connectivity that may be of independent interest.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
