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Register a new userApplications and homological properties of local rings with decomposable maximal ideals
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We construct a local Cohen-Macaulay ring $R$ with a prime ideal $mathfrak{p}inspec(R)$ such that $R$ satisfies the uniform Auslander condition (UAC), but the localization $R_{mathfrak{p}}$ does not satisfy Auslanders condition (AC). Given any positive integer $n$, we also construct a local Cohen-Macaulay ring $R$ with a prime ideal $mathfrak{p}inspec(R)$ such that $R$ has exactly two non-isomorphic semidualizing modules, but the localization $R_{mathfrak{p}}$ has $2^n$ non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen-Macaulay fiber products of finite Cohen-Macaulay type.
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Let $(R,frak m)$ be a commutative noetherian local ring. In this paper, we prove that if $frak m$ is decomposable, then for any finitely generated $R$-module $M$ of infinite projective dimension $frak m$ is a direct summand of (a direct sum of) syzygies of $M$. Applying this result to the case where $frak m$ is quasi-decomposable, we obtain several classfications of subcategories, including a complete classification of the thick subcategories of the singularity category of $R$.
For a monomial ideal $I$, we consider the $i$th homological shift ideal of $I$, denoted by $text{HS}_i(I)$, that is, the ideal generated by the $i$th multigraded shifts of $I$. Some algebraic properties of this ideal are studied. It is shown that for any monomial ideal $I$ and any monomial prime ideal $P$, $text{HS}_i(I(P))subseteq text{HS}_i(I)(P)$ for all $i$, where $I(P)$ is the monomial localization of $I$. In particular, we consider the homological shift ideal of some families of monomial ideals with linear quotients. For any $textbf{c}$-bounded principal Borel ideal $I$ and for the edge ideal of complement of any path graph, it is proved that $text{HS}_i(I)$ has linear quotients for all $i$. As an example of $textbf{c}$-bounded principal Borel ideals, Veronese type ideals are considered and it is shown that the homological shift ideal of these ideals are polymatroidal. This implies that for any polymatroidal ideal which satisfies the strong exchange property, $text{HS}_j(I)$ is again a polymatroidal ideal for all $j$. Moreover, for any edge ideal with linear resolution, the ideal $text{HS}_j(I)$ is characterized and it is shown that $text{HS}_1(I)$ has linear quotients.
Let $mathcal{I}(R)$ be the set of all ideals of a ring $R$, $delta$ be an expansion function of $mathcal{I}(R)$. In this paper, the $delta$-$J$-ideal of a commutative ring is defined, that is, if $a, bin R$ and $abin Iin mathcal{I}(R)$, then $ain J(R)$ (the Jacobson radical of $R$) or $bin delta(I)$. Moreover, some properties of $delta$-$J$-ideals are discussed,such as localizations, homomorphic images, idealization and so on.
We introduce a notion of generalized local cohomology modules with respect to a pair of ideals $(I,J)$ which is a generalization of the concept of local cohomology modules with respect to $(I,J).$ We show that generalized local cohomology modules $H^i_{I,J}(M,N)$ can be computed by the v{C}ech cohomology modules. We also study the artinianness of generalized local cohomology modules $H^i_{I,J}(M,N).$
Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic $p$, we show that a similar result holds in equicharacteristic $0$ under the additional hypothesis that the symbolic Rees algebra of $I$ is noetherian.
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