Coherent functors and asymptotic stability
type: Item
instance of: scholarly article
title: Coherent functors and asymptotic stability
zbMATH Open document ID: 1409.13005
author: Adson Banda
author: Leif Melkersson
published in: Journal of Algebra
publication date: 2019-01-22T00:00:00Z
review text: Let $A$ denote a commutative Noetherian ring. On the category of $A$-modules an $A$-linear functor $F$ is called coherent if there is an exact sequence $h_Y \to h_X \to F \to 0$, where $h_Z = \text{Hom}(Z,-)$ for an $A$-module $Z$. The notion of coherent functors was introduced by \textit{M. Auslander} [in: Proc. Conf. Categor. Algebra, La Jolla 1965, 189--231 (1966; Zbl 0192.10902)] and also studied by \textit{R. Hartshorne} [Adv. Math. 140, No. 1, 44--94 (1998; Zbl 0921.13010)]. \par In the present paper, the authors continues with the study of coherent functors $F$ and apply their result to the study of asymptotic behaviour for an ideal $\mathfrak{a} \subset A$. At first they simplify an argument of \textit{T. Se} [J. Algebra 484, 247--264 (2017; Zbl 1386.13004)] who showed that the set of associated prime ideals $\text{Ass}_A (F(M/\mathfrak{a}^nM))$ and $\text{Ass}_A (F(\mathfrak{a}^nM/\mathfrak{a}^{n+1}M))$ are independent of $n$ for $n \gg 0$ for $M$ a finitely generated $A$-module. Moreover, they prove the dual result for the set of attached prime ideals $\text{Att}_A (F(0:_N \mathfrak{a}^n))$ and $\text{Att}_A (F(0:_N \mathfrak{a}^n/0:_N \mathfrak{a}^{n-1}))$ for an Artinian $A$-module $N$. \par These are far reaching generalizations of results of \textit{M. Brodmann} [Proc. Am. Math. Soc. 74, 16--18 (1979; Zbl 0395.13008)], \textit{R. Y. Sharp} [J. Lond. Math. Soc., II. Ser. 34, 212--218 (1986; Zbl 0605.13002)] resp. the second author and the reviewer (see [Proc. Am. Math. Soc. 117, No. 4, 935--938 (1993; Zbl 0780.13003)]. \par With the previous notations, suppose that $F(M/\mathfrak{a}^nM)$ resp. $F(0:_N \mathfrak{a}^n)$ are modules of finite length for all $n$ resp. for $n=1$. Then the authors prove the existence of polynomials counting the length of the corresponding functions for $n \gg 0$. As a further application of their technique the following is proved: When A is local it is shown that, the Betti numbers $\beta_i(F(M/\mathfrak{a}^nM))$ and the Bass…
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 7004741
zbMATH Keywords: asymptotic prime ideal
zbMATH Keywords: coherent functor
zbMATH Keywords: Hilbert polynomial
zbMATH Keywords: Betti number
zbMATH Keywords: Bass number
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2018.11.035
OpenAlex ID: W2905170280
cites work: Q5586467
cites work: Asymptotic Stability of Ass(M/I n M)
cites work: The asymptotic nature of the analytic spread
cites work: Coherent functors
cites work: ARTINIAN MODULES AND HILBERT POLYNOMIALS
cites work: Homological Invariants of Powers of an Ideal
cites work: Q2770573
cites work: On asymptotic stability for sets of prime ideals connected with the powers of an ideal
cites work: Modules cofinite with respect to an ideal
cites work: Asymptotic Prime Ideals Related to Derived Functors
cites work: On prime divisors of \(I^n\), \(n\) large
cites work: An introduction to homological algebra
cites work: Covariant functors and asymptotic stability
cites work: Asymptotic Behaviour of Certain Sets of Attached Prime Ideals
DOI: 10.1016/J.JALGEBRA.2018.11.035
Recommended article: Covariant functors and asymptotic stability
Recommended article: Coherent functors in stable homotopy theory
Recommended article: Coherent functors
Recommended article: Coinduction functor in representation stability theory
Recommended article: Coherent functors and covariantly finite subcategories
Recommended article: Q4276458
Recommended article: Q4307253
Recommended article: Q3731744
Recommended article: Stability and certain \(\mathbb{P}^n\)-functors
Recommended article: Q4361636
description: scientific article; zbMATH DE number 7004741