## Recursos de colección

#### Project Euclid (Hosted at Cornell University Library) (198.174 recursos)

Journal of Commutative Algebra

1. #### Postulation and reduction vectors of multigraded filtrations of ideals

Sarkar, Parangama; Verma, J.K.
We study the relationship between postulation and reduction vectors of admissible multigraded filtrations $\mathcal{F}= \{\mathcal{F} (\underline{n})\}_{\underline{n} \in \mathbb{Z} ^s}$ of ideals in Cohen-Macaulay local rings of dimension at most two. This is enabled by a suitable generalization of the Kirby-Mehran complex. An analysis of its homology leads to an analogue of Huneke's fundamental lemma which plays a crucial role in our investigations. We also clarify the relationship between the Cohen-Macaulay property of the multigraded Rees algebra of $\mathcal{F}$ and reduction vectors with respect to complete reductions of $\mathcal{F}$.

2. #### Postulation and reduction vectors of multigraded filtrations of ideals

Sarkar, Parangama; Verma, J.K.
We study the relationship between postulation and reduction vectors of admissible multigraded filtrations $\mathcal{F}= \{\mathcal{F} (\underline{n})\}_{\underline{n} \in \mathbb{Z} ^s}$ of ideals in Cohen-Macaulay local rings of dimension at most two. This is enabled by a suitable generalization of the Kirby-Mehran complex. An analysis of its homology leads to an analogue of Huneke's fundamental lemma which plays a crucial role in our investigations. We also clarify the relationship between the Cohen-Macaulay property of the multigraded Rees algebra of $\mathcal{F}$ and reduction vectors with respect to complete reductions of $\mathcal{F}$.

3. #### Graded version of local cohomology with respect to a pair of ideals

Lima, P.H.; Pérez, V.H. Jorge
In this paper, we prove some well-known results on local cohomology with respect to a pair of ideals in graded version, such as the Independence theorem, Lichtenbaum-Harshorne vanishing theorem, Basic finiteness and vanishing theorem, among others. In addition, we present a generalized version of the Melkersson theorem regarding the Artinianness of modules, and a result concerning Artinianness of local cohomology modules.

4. #### Graded version of local cohomology with respect to a pair of ideals

Lima, P.H.; Pérez, V.H. Jorge
In this paper, we prove some well-known results on local cohomology with respect to a pair of ideals in graded version, such as the Independence theorem, Lichtenbaum-Harshorne vanishing theorem, Basic finiteness and vanishing theorem, among others. In addition, we present a generalized version of the Melkersson theorem regarding the Artinianness of modules, and a result concerning Artinianness of local cohomology modules.

5. #### Quasi-Gorensteinness of extended Rees algebras

Kim, Youngsu
Let $R$ be a Noetherian local ring and $I$ an $R$-ideal. It is well known that, if the associated graded ring ${gr} _I(R)$ is Cohen-Macaulay (Gorenstein), then so is $R$, but in general, the converse is not true. In this paper, we investigate the Cohen-Macaulayness and Gorensteinness of the associated graded ring ${gr} _I(R)$ under the hypothesis that the extended Rees algebra $R[It,t^{-1}]$ is quasi-Gorenstein or the associated graded ring ${gr} _I(R)$ is a domain.

6. #### Quasi-Gorensteinness of extended Rees algebras

Kim, Youngsu
Let $R$ be a Noetherian local ring and $I$ an $R$-ideal. It is well known that, if the associated graded ring ${gr} _I(R)$ is Cohen-Macaulay (Gorenstein), then so is $R$, but in general, the converse is not true. In this paper, we investigate the Cohen-Macaulayness and Gorensteinness of the associated graded ring ${gr} _I(R)$ under the hypothesis that the extended Rees algebra $R[It,t^{-1}]$ is quasi-Gorenstein or the associated graded ring ${gr} _I(R)$ is a domain.

7. #### $\star$-reductions of ideals and Prüfer $v$-multiplication domains

Houston, E.; Kabbaj, S.; Mimouni, A.
Let $R$ be a commutative ring and $I$ an ideal of $R$. An ideal $J\subseteq I$ is a reduction of $I$ if $JI^{n}=I^{n+1}$ for some positive integer~$n$. The ring~$R$ has the (finite) basic ideal property if (finitely generated) ideals of $R$ do not have proper reductions. Hays characterized (one-dimensional) Pr\"ufer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays's results to Pr\"ufer $v$-multiplication domains by replacing basic'' with $w$-basic,'' where $w$ is a particular star operation. We also investigate relations among $\star$-basic properties for certain star operations $\star$.

8. #### $\star$-reductions of ideals and Prüfer $v$-multiplication domains

Houston, E.; Kabbaj, S.; Mimouni, A.
Let $R$ be a commutative ring and $I$ an ideal of $R$. An ideal $J\subseteq I$ is a reduction of $I$ if $JI^{n}=I^{n+1}$ for some positive integer~$n$. The ring~$R$ has the (finite) basic ideal property if (finitely generated) ideals of $R$ do not have proper reductions. Hays characterized (one-dimensional) Pr\"ufer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays's results to Pr\"ufer $v$-multiplication domains by replacing basic'' with $w$-basic,'' where $w$ is a particular star operation. We also investigate relations among $\star$-basic properties for certain star operations $\star$.

9. #### Frobenius Betti numbers and modules of finite projective dimension

Stefani, Alessandro De; Huneke, Craig; Núñez-Betancourt, Luis
Let $(R,\mathfrak{m} ,K)$ be a local ring, and let $M$ be an $R$-module of finite length. We study asymptotic invariants, $\beta ^F_i(M,R)$, defined by twisting with Frobenius the free resolution of $M$. This family of invariants includes the Hilbert-Kunz multiplicity ($e_{HK}(\mathfrak{m} ,R)=\beta ^F_0(K,R)$). We discuss several properties of these numbers that resemble the behavior of the Hilbert-Kunz multiplicity. Furthermore, we study when the vanishing of $\beta ^F_i(M,R)$ implies that $M$ has finite projective dimension. In particular, we give a complete characterization of the vanishing of $\beta ^F_i(M,R)$ for one-dimensional rings. As a consequence of our methods we give conditions for...

10. #### Frobenius Betti numbers and modules of finite projective dimension

Stefani, Alessandro De; Huneke, Craig; Núñez-Betancourt, Luis
Let $(R,\mathfrak{m} ,K)$ be a local ring, and let $M$ be an $R$-module of finite length. We study asymptotic invariants, $\beta ^F_i(M,R)$, defined by twisting with Frobenius the free resolution of $M$. This family of invariants includes the Hilbert-Kunz multiplicity ($e_{HK}(\mathfrak{m} ,R)=\beta ^F_0(K,R)$). We discuss several properties of these numbers that resemble the behavior of the Hilbert-Kunz multiplicity. Furthermore, we study when the vanishing of $\beta ^F_i(M,R)$ implies that $M$ has finite projective dimension. In particular, we give a complete characterization of the vanishing of $\beta ^F_i(M,R)$ for one-dimensional rings. As a consequence of our methods we give conditions for...

11. #### Powers of edge ideals of regularity three bipartite graphs

Alilooee, Ali; Banerjee, Arindam
In this paper, we prove that, if $I(G)$ is the edge ideal of a connected bipartite graph with regularity 3, then, for all $s\geq 2$, the regularity of $I(G)^s$ is exactly $2s+1$.

12. #### Powers of edge ideals of regularity three bipartite graphs

Alilooee, Ali; Banerjee, Arindam
In this paper, we prove that, if $I(G)$ is the edge ideal of a connected bipartite graph with regularity 3, then, for all $s\geq 2$, the regularity of $I(G)^s$ is exactly $2s+1$.

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