Takesi Kawasaki
In the present article, the author shows that Faltings' annihilator theorem holds for any Noetherian ring $A$ if $A$ is universally catenary; all the formal fibers of all the localizations of $A$ are Cohen-Macaulay; and the Cohen-Macaulay locus of each finitely generated $A$-algebra is open.
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Let $\mathcal{Z}$ be a specialization closed subset of $\Spec R$ and $X$ a homologically left-bounded complex with finitely generated homologies. We establish Faltings' Local-global Principle and Annihilator Theorems for the local cohomology modules {$\H_{\mathcal{Z}^i(X).$ }} Our versions contain variations of results already known on these theorems.
Let $R$ be a commutative Noetherian ring and let $n$ be a non-negative integer. In this article, by using the theory of Gorenstein dimensions, it is shown that whenever $R$ is a homomorphic image of a Noetherian Gorenstein ring, then the invariants $\inf\{i\in\nat_0|\, {\dim\Supp}(\fb^tH_{\fa}^i(M))\geq n\text{for all} t\in\nat_0\}$ and $\inf\{λ_{\fa R_{\p}}^{\fb R_{\p}}(M_{\p})|\,\p\in {\rm Spec} \, R \text{and} \dim R/ \p\geq n\}$ are equal, for every finitely generated $R$-module $M$ and for every ideals $\frak a, \frak b$ of $R$ with $\frak b\subseteq \frak a$. This generalizes the Falting…
In this paper, we prove Faltings' annihilator theorem for complexes over a CM-excellent ring. As an application, we give a complete classification of the t-structures of the bounded derived category of finitely generated modules over a CM-excellent ring of finite Krull dimension.
Faltings' annihilator theorem is an important result in local cohomology theory. Recently, Doustimehr and Naghipour generalized the Falitings' annihilator theorem. They proved that if $R$ is a homomorphic image of a Gorenstein ring, then $f_\mathfrak{a}^\mathfrak{b}(M)_n = λ_\mathfrak{a}^\mathfrak{b}(M)_n$, where $f_\mathfrak{a}^\mathfrak{b}(M)_n := \inf\{i \in \mathbb{N} \mid \operatorname{dim}{\operatorname{Supp}(\mathfrak{b}^t H_\mathfrak{a}^i(M))} \geq n \text{ for all } t\in \mathbb{N}\}$ and $λ_\mathfrak{a}^\mathfrak{b}(M)_n := \inf\{λ_{\mathfrak{a} R_\mathfrak{p}}^{\mathfrak{b} R_\mathf…
Let $R$ be a commutative Noetherian ring, $M$ a finitely generated $R$-module and $n$ be a non-negative integer. In this article, it is shown that there is a finitely generated submodule $N_i$ of $H_{\frak a}^i(M)$ such that $\dim{\rm Supp } H_{\frak a}^i(M)/N_i<n$ for all $i<t$ if and only if there is a finitely generated submodule $N_{i,{\frak p}}$ of $H_{{\frak a} R_{\frak p}}^i(M_{\frak p})$ such that $\dim{\rm Supp } H_{{\frak a} R_{\frak p}}^i(M_{\frak p})/N_{i,{\frak p}}<n$ for all $i<t$. This generalizes Faltings' Local-global Principle for the finiteness of local cohomology modules (F…
This paper investigates when local cohomology modules have an annihilator that does not depend on the choice of an ideal. Takahashi classified the dominant resolving subcategories of the category of finitely generated modules over a commutative Noetherian ring. We show that his classification theorem describes annihilation results of local cohomology modules over a finite-dimensional ring with certain assumptions or a Cohen-Macaulay ring.
Let $R$ be a commutative Noetherian ring and $\fa$ an ideal of $R$. We intend to establish the dual of two Faltings' Theorems for local homology modules of an Artinian module. As a consequence of this, we show that, if $A$ is an Artinian module over semi-local complete ring $R$ and $j$ is an integer such that $H_i^{\fa}(A)$ is Artinian for all $i<j$, then the set $\Coass_R(H_j^{\fa}(A))$ is finite. We also introduce the notion of the $n$th Artinianness dimension $g_n^\fa(A)=\inf\{g^{\fa R_{\fp}}(^\fp A): \fp\in\Spec(R) \ \ \text{and} \ \ \dim R/\fp\geq n\}$, for all $n\in\mathbb{N}_{0}$ and pr…
Let \frak a be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. It is shown that {\rm Ann}_R(H_{\frak a}^{{\dim M}({\frak a}, M)}(M))= {\rm Ann}_R(M/T_R({\frak a}, M)), where T_R({\frak a}, M) is the largest submodule of M such that {\rm cd}({\frak a}, T_R({\frak a}, M))< {\rm cd}({\frak a}, M). Several applications of this result are given. Among other things, it is shown that there exists an ideal \frak b of R such that {\rm Ann}_R(H_{\frak a}^{\dim M}(M))={\rm Ann}_R(M/H_{\frak b}^{0}(M)). Using this, we show that if H_{\frak a}^{\dim R}(R)=0, then {\rm Att}_…
The first effective versions of Faltings product theorem are given by J.H. Evertse and R. Ferretti. Their results are improved quantitatively by P. Philippon, thanks to Samuel's multiplicity instead of the length in his zeros estimate. This version is a generalization of the usuel Roth's lemma on powers of the additive group. Here following this approach with some modifications, we obtain an analogue of the theorem for any product of commutative algebraic groups.
This paper studies Frobenius powers of parameter ideals in a commutative Noetherian local ring $R$ of prime characteristic $p$. For a given ideal $\fa$ of $R$, there is a power $Q$ of $p$, depending on $\fa$, such that the $Q$-th Frobenius power of the Frobenius closure of $\fa$ is equal to the $Q$-th Frobenius power of $\fa$. The paper addresses the question as to whether there exists a {\em uniform} $Q_0$ which `works' in this context for all parameter ideals of $R$ simultaneously. In a recent paper, Katzman and Sharp proved that there does exists such a uniform $Q_0$ when $R$ is Cohen--Maca…
Structure of modules stably annihilated by a fixed ideal
本文引入由稳定零化子包含固定理想I的有限生成模构成的子范畴，研究其结构，并将Dey-Liu定理从Gorenstein情形推广到Cohen-Macaulay情形。
A generalization of the dimension and radius of a subcategory of modules and its applications
本文从有限生成模范畴满子范畴的维数与半径有限性出发，研究奇异轨迹及 Tor、Ext 零化子，并恢复及推广相关定理。
Let R be a commutative noetherian local ring. As an analogue of the notion of the dimension of a triangulated category defined by Rouquier, the notion of the dimension of a subcategory of finitely generated R-modules is introduced in this paper. We found evidence that certain categories over nice singularities have small dimensions. When R is Cohen-Macaulay, under a mild assumption it is proved that finiteness of the dimension of the full subcategory consisting of maximal Cohen-Macaulay modules which are locally free on the punctured spectrum is equivalent to saying that R is an isolated singu…
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