Coherent functors and asymptotic stability
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 numbers $\mu^i(F(M/\mathfrak{a}^nM))$ are given by polynomials in $n$ for large $n$. This extends results of \textit{V. Kodiyalam} [Proc. Am. Math. Soc. 118, No. 3, 757--764 (1993; Zbl 0780.13007)]. Further results concern the asymptotic grade of $F(M/\mathfrak{a}^nM)$.
scientific article; zbMATH DE number 7004741
scientific article