Cofiniteness with respect to two ideals and local cohomology (Q1737236)

Let $A$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $A$, and $n$ a non-negative integer. In this paper, the authors say that $A$ admits $\text{P}_n(\mathfrak{a})$ if for any $A$-module $M$ with $\text{Supp}_A(M)\subseteq \text{V}(\mathfrak{a})$, finiteness of modules $\text{Ext}^i_A(A/\mathfrak{a}, M)$ for all $i\leq n$ implies that $M$ is $\mathfrak{a}$-cofinite. Recall that an $A$-module $M$ is said to be $\mathfrak{a}$-cofinite if $\text{Supp}_A(M)\subseteq \text{V}(\mathfrak{a})$ and $\text{Ext}^i_A(A/\mathfrak{a}, M)$ is finite for all $i$. They prove that if $d:= \dim(A)\geq 1$ and $A$ admits $\text{P}_{d- 1}(\mathfrak{a})$ for all ideals $\mathfrak{a}$ of $A$ with $\dim(A/\mathfrak{a})\leq d- 1$, then $A$ admits $\text{P}_{d- 1}(\mathfrak{a})$ for all ideals $\mathfrak{a}$ of $A$. As a consequence, $A$ satisfies $\text{P}_{1}(\mathfrak{a})$ (resp. $\text{P}_{2}(\mathfrak{a})$) for all ideals $\mathfrak{a}$ of $A$ if $\dim(A)= 2$ (resp. $A$ is local and $\dim(A)= 3$). In the case that $\dim(A)\leq 3$ (resp. $A$ is local and $\dim(A/\mathfrak{a})= 2$), it is shown that the full subcategory of $\mathfrak{a}$-cofinite $A$-modules of the category of $A$-modules is Abelian if and only if $(0:_{\text{coker}f}\mathfrak{a})$ (resp. $\text{Ext}^2_A(A/\mathfrak{a}, \text{ker}f)$) is finite for any homomorphism of $\mathfrak{a}$-cofinite $A$-modules $f$. Let $M$ be an $A$-module and $\mathfrak{b}$ an ideal of $A$ with $\mathfrak{b}\subseteq \mathfrak{a}$. The authors prove that $\text{Ext}^i_A(A/\mathfrak{a}, M)$ is finite for all $i\leq n$ when $\text{Ext}^i_A(A/\mathfrak{b}, M)$ is $\mathfrak{a}$-cofinite for all $i\leq n$. In particular, $M$ is $\mathfrak{a}$-cofinite whenever $A$ admits $\text{P}_n(\mathfrak{a})$, $\text{Supp}_A(M)\subseteq \text{V}(\mathfrak{a})$, and $\text{Ext}^i_A(A/\mathfrak{b}, M)$ is $\mathfrak{a}$-cofinite for all $i\leq n$. When $\text{Ext}^i_A(A/\mathfrak{a}, M)$ is finite for all $i\leq n+ 1$, they show that if $\dim(A/\mathfrak{a})= 1$, then $\text{H}^i_{\mathfrak{a}}(M)$ is $\mathfrak{a}$-cofinite for all $i\leq n$ and if $\dim(A)\leq 3$ or $A$ is local with $\dim(A/\mathfrak{a})= 2$, then $\text{H}^i_{\mathfrak{a}}(M)$ is $\mathfrak{a}$-cofinite for all $i< n$ if and only if $\text{Hom}_A(A/\mathfrak{a}, \text{H}^i_{\mathfrak{a}}(M))$ is finite for all $i\leq n$. It is also shown that $\text{H}^{\dim_A(M)}_{\mathfrak{a}}(M)$ is Artinian whenever $(0:_{\text{H}^{\dim_A(M)}_{\mathfrak{a}}(M)} \mathfrak{a})$ is finite.