Mostrando recursos 1 - 18 de 18

  1. Systems of parameters and the Cohen-Macaulay property

    Shultis, Katharine
    Let $R$ be a commutative, Noetherian, local ring and $M$ a finitely generated $R$-module. Consider the module of homomorphisms $Hom _R(R/\mathfrak{a} ,M/\mathfrak{b} M)$ where $\mathfrak{b} \subseteq \mathfrak{a} $ are parameter ideals of $M$. When $M=R$ and $R$ is Cohen-Macaulay, Rees showed that this module of homomorphisms is isomorphic to $R/\mathfrak{a} $, and in particular, a free module over $R/\mathfrak{a} $ of rank one. In this work, we study the structure of such modules of homomorphisms for a not necessarily Cohen-Macaulay $R$-module $M$.

  2. Tilting objects on some global quotient stacks

    Novaković, Saša
    We prove the existence of tilting objects on some global quotient stacks. As a consequence, we provide further evidence for a conjecture on the Rouquier dimension of derived categories formulated by Orlov.

  3. Dual $F$-signature of special Cohen-Macaulay modules over cyclic quotient surface singularities

    Nakajima, Yusuke
    The notion of $F$-signature was defined by Huneke and Leuschke and this numerical invariant characterizes some singularities. This notion is extended to finitely generated modules by Sannai and is called dual $F$-signature. In this paper, we determine the dual $F$-signature of a certain class of Cohen-Macaulay modules (so-called ``special") over cyclic quotient surface singularities. Also, we compare the dual $F$-signature of a special Cohen-Macaulay module with that of its Auslander-Reiten translation. This gives a new characterization of the Gorensteinness.

  4. Cohomology of finite modules over short Gorenstein rings

    Menning, Melissa C.; Şega, Liana M.
    Let $R$ be a Gorenstein local ring with maximal ideal $\mathfrak{m} $ satisfying $\mathfrak{m} ^3=0\ne \mathfrak{m} ^2$. Set $\mathfrak{k} =R/\mathfrak{m} $ and $e=rank _{\mathfrak{k} }(\mathfrak{m} /\mathfrak{m} ^2)$. If $e>2$ and $M$, $N$ are finitely generated $R$-modules, we show that the formal power series \[ \sum _{i=0}^\infty rank _{\mathfrak{k} }\left (Ext ^i_R(M,N)\otimes _R\mathfrak{k} \right )t^i \] and \[ \sum _{i=0}^\infty rank _{\mathfrak{k} }\left (Tor _i^R(M,N)\otimes _R \mathfrak{k} \right )t^i \] are rational, with denominator $1-et+t^2$.

  5. Discrete valuation overrings of a graded Noetherian domain

    Chang, Gyu Whan; Oh, Dong Yeol
    Let $R = \bigoplus _{\alpha \in \Gamma } R_{\alpha }$ be an integral domain graded by an arbitrary torsionless grading monoid $\Gamma $, $M$ a homogeneous maximal ideal of $R$ and $S(H) = R \setminus \bigcup _{P \in \text {h-}Spec (R)}P$. We show that $R$ is a graded Noetherian domain with $\text {h-}\dim (R) = 1$ if and only if $R_{S(H)}$ is a one-dimensional Noetherian domain. We then use this result to prove a graded Noetherian domain analogue of the Krull-Akizuki theorem. We prove that, if $R$ is a gr-valuation ring, then $R_M$ is a valuation domain, $\dim (R_M) =...

  6. Valuative and geometric characterizations of Cox sheaves

    Bechtold, Benjamin
    We give an intrinsic characterization of Cox sheaves on Krull schemes in terms of their valuative algebraic properties. We also provide a geometric characterization of their graded relative spectra in terms of good quotients of graded schemes, extending the existing theory on relative spectra of Cox sheaves on normal varieties. Moreover, we obtain an irredundant characterization of Cox rings which, in turn, produces a normality criterion for certain graded rings.

  7. 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} $.

  8. 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} $.

  9. 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.

  10. 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.

  11. 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.

  12. 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.

  13. $\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 $.

  14. $\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 $.

  15. 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...

  16. 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...

  17. 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 $2s+1$.

  18. 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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