Hannah Altmann, Sean K. Sather-Wagstaff
Gerko proves that if an artinian local ring $(R,\mathfrak{m}_R)$ possesses a sequence of strongly Tor-independent modules of length $n$, then $\mathfrak{m}_R^n\neq 0$. This generalizes readily to Cohen-Macaulay rings. We present a version of this result for non-Cohen-Macaulay rings.
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Cohen-Macaulay dimension for modules over a commutative noetherian local ring has been defined by A. A. Gerko. That is a homological invariant sharing many properties with projective dimension and Gorenstein dimension. The main purpose of this paper is to extend the notion of Cohen-Macaulay dimension for modules over commutative noetherian local rings to that for bounded complexes over non-commutative noetherian rings.
Through a study of torsion functors of local cohomology modules we improve some non-finiteness results on the top non-zero local cohomology modules with respect to an ideal.
We construct an example where a local cohomology module of a hypersurface has p-torsion elements for every prime integer p, and consequently has infinitely many associated prime ideals. We also answer a related question of Hochster.
Tensor products usually have nonzero torsion. This is a central theme of Auslander's paper "Modules over unramified regular local rings"; the theme continues in the work of Huneke and Wiegand. The main focus in this note is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modules N with the property that its tensor product with any module M has torsion, unless M is very special. Important examples of such modules N are the Frobenius power of a ring, viewed as a module over itself, that is a complete intersection domain of positive characteristic.
In this paper, we study Serre's condition $(S_n)$ for tensor products of modules over a commutative noetherian local ring. The paper aims to show the following. Let $M$ and $N$ be finitely generated module over a commutative noetherian local ring $R$, either of which is $(n+1)$-Tor-rigid. If the tensor product $M \otimes_R N$ satisfies $(S_{n+1})$, then under some assumptions $\mathrm{Tor}_{i}^R(M, N) = 0$ for all $i \ge 1$. The key role is played by $(n+1)$-Tor-rigidity of modules. As applications, we will show that the result recovers several known results.
Let A be a Noetherian local ring with canonical module K. We characterize A when K is a torsionless, reflexive, or q-torsionfree module. If A is a Cohen-Macaulay ring, H.-B. Foxby proved in 1974 that the A-module K is q-torsionfree if and only if the ring A is q-Gorenstein. With mild assumptions, we provide a generalization of Foxby's result to arbitrary Noetherian local rings admitting the canonical module. In particular, since the reflexivity of the canonical module is closely related to the ring being Gorenstein in low codimension, we also explore quasi-normal rings, introduced by W. V. Vas…
Let $M, N$ be finite modules over a Noetherian local ring $R$, and let $G$ be the associated graded ring of $R$. We show that the bigraded Hilbert series of $gr(Tor^R(M,N))$ is obtained from that of $Tor^G(gr(M),gr(N))$ by negative consecutive cancellations, thus extending a theorem of Rossi and Sharifan.
Let R be a commutative noetherian ring. We consider the question of when n-syzygy modules over R are n-torsionfree in the sense of Auslander and Bridger. Our tools include Serre's condition and certain conditions on the local Gorenstein property of R. Our main result implies the converse of a celebrated theorem of Evans and Griffith.
Lyubeznik conjectured that local cohomology modules of regular rings have finitely many associated primes. We examine this conjecture for polynomial rings over the integers, and record some equational identities that arise from studying special cases.
For finitely generated modules M and N over a complete intersection R, the vanishing of Tor_i^R(M,N) for all i> 0 gives a tight relationship among depth properties of M, N and their tensor product. Here we concentrate on the converse and show, under mild conditions, that the tensor product of M and N being torsion-free (or satisfying higher Serre conditions) forces vanishing of Tor. Special attention is paid to the case of tensor powers of a single module.
We consider relative Tor functors built from resolutions described by a semidualizing module C over a commutative noetherian ring R. We show that the bifunctors Tor^{F_CM}_i (-,-) and Tor^{P_CM}_i (-,-), defined using flat-like and projective-like resolutions, are isomorphic. We show how the vanishing of these functors characterizes the finiteness of the homological dimension F_C-pd, and we use this to give a relation between the F_C-pd of a given module and that of a pure submodule. On the other hand, we show that other relations that one may expect to hold similarly, fail in general. In fact…
In this article we establish equivariant isomorphisms of Ext and Tor modules over different relative complete intersections. More precisely, for a commutative ring $Q$, this paper investigates how $Ext_{Q/(\boldsymbol{f})}^*(M,N)$ and $Tor^{Q/(\boldsymbol{f})}_*(M,N)$ change when one varies $\boldsymbol{f}$ among all Koszul-regular sequences of a fixed length such that $\boldsymbol{f} M=0$ and $\boldsymbol{f} N=0$. Of notable interest is how the theory of perturbations is used to establish isomorphisms of certain DG modules.
A commutative ring is said to have ITI with respect to an ideal a if the a-torsion functor preserves injectivity of modules. Classes of rings with ITI or without ITI with respect to certain sets of ideals are identified. Behaviour of ITI under formation of rings of fractions, tensor products and idealisation is studied. Applications to local cohomology over non-noetherian rings are given.
Cotorsion modules relative to a hereditary torsion theory
本文针对遗传挠理论τ_w，引入Tor对、余挠对及强平坦模，以此刻画完全环等多种已知环类。
On torsion-free modules and semi-hereditary rings
本文研究了半遗传环的结构理论，重点探讨其与无挠模平坦性的关系，并考虑了Shimomoto关于Frobenius映射平坦性的问题。
Multiple Tor modules: rigidity and Mayer-Vietoris spectral sequences
本文研究多重理想下的托模块性质扩展，提出新的复形并探讨其同调关系，揭示托独立性在多格下的支撑区域关联。
For any commutative ring $A$ we introduce a generalization of $S$--artinian rings using a hereditary torsion theory $σ$ instead of a multiplicative closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally $σ$--artinian ring, then $σ$ must be of finite type, and $A$ is totally $σ$--noetherian.
The long-standing Auslander and Reiten Conjecture states that a finitely generated module over a finite-dimensional algebra is projective if certain Ext-groups vanish. Several authors, including Avramov, Buchweitz, Iyengar, Jorgensen, Nasseh, Sather-Wagstaff, and Şega, have studied a possible counterpart of the conjecture, or question, for commutative rings in terms of vanishing of Tor. This has led to the notion of Tor-persistent rings. Our main result shows that the class of Tor-persistent local rings is closed under a number of standard procedures in ring theory.
Let $R$ a commutative ring, $\mathfrak{a} \subset R$ an ideal, $I$ an injective $R$-module and $S \subset R$ a multiplicatively closed set. When $R$ is Noetherian it is well-known that the $\mathfrak{a}$-torsion sub-module $Γ_{\mathfrak{a}}(I)$, the factor module $I/Γ_{\mathfrak{a}}(I)$ and the localization $I_S$ are again injective $R$-modules. We investigate these properties in the case of a commutative ring $R$ by means of a notion of relatively-$\mathfrak{a}$-injective $R$-modules. In particular we get another characterization of weakly pro-regular sequences in terms of relatively injectiv…
We study two properties of modules over a local hypersurface $R$: decency and rigidity. We show that the vanishing of Hochster's function $θ^R(M,N)$, known to imply decent intersection, also implies rigidity. We investigate the vanishing of $θ^R(M,N)$ to obtain new results about decency and rigidity over hypersurfaces. We employ a mixture of techniques from Commutative Algebra and Intersection Theory of algebraic cycles.
Let $\frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\V(\fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{\fa}_i(M)\neq 0$. We show that $M$ is $\fa$-cofinite if and only if the $R$-module $\Tor^R_i(R/\fa,M)$ is finitely generated for every $0\leq i\leq n$. This provides a hands-on and computable finitely-many-steps criterion to examine $\mathfrak{a}$-confiniteness. Our approach relies heavily on the theory of local homology which demonstrates the effectiveness and indispensability of this tool.
Let $(R,\mathfrak{m})$ be a commutative noetherian local ring with $\mathfrak{m}$-adic topology, $I$ an ideal of $R$. We investigate coartinianess of $\mathrm{Ext}$ and $\mathrm{Tor}$, show that the $R$-module $\mathrm{Ext}_{R}^{i}(N,M)$ is $I$-coartinian if $M$ is a linearly compact $I$-coartinian $R$-module and $N$ is an $I$-cofinite $R$-module of dimension at most $1$; the $R$-module $\mathrm{Tor}_{i}^{R}(N,M)$ is $I$-coartinian in the case $M$ is semi-discrete linearly compact $I$-coartinian and $N$ is finitely generated with dimension at most $2$.
Given finitely generated modules $M$ and $N$ over a local ring $R$, the tensor product $M\otimes_RN$ typically has nonzero torsion. Indeed, the assumption that the tensor product is torsion-free influences the structure and vanishing of the modules $\Tor^R_i(M,N)$ for all $i\geq 1$. In turn, the vanishing of $\Tor^R_i(M,N)$ imposes restrictions on the depth properties of the modules $M$ and $N$. These connections made their first appearance in Auslander's 1961 paper "Modules over unramified regular local rings". We will survey the literature on these topics, with emphasis on progress during th…
Test properties of some Cohen-Macaulay modules and criteria for local rings via finite vanishing of Ext or Tor
本文通过Ext或Tor的有限消失性质，测试Cohen-Macaulay模的特性，并用于刻画局部环，验证了Auslander-Reiten猜想的扩展结果。
In analogy with the classical, affine toric rings, we define a local toric ring as the quotient of a regular local ring modulo an ideal generated by binomials in a regular system of parameters with unit coefficients; if the coefficients are just $\pm1$, we call the ring purely toric. We prove the following results on the existence of MCM's (=maximal Cohen-Macaulay modules): (EQUI$\mathstrut_p$) we construct certain families of local toric rings satisfying Hochster's small MCM conjecture in positive characteristic; (EQUI$\mathstrut_0$) provided Hochster's small MCM conjecture holds in positive …
Our main result states that for a toric surface in $P^4$ canonical module is always Cohen-Macaulay.
Let R be a commutative noetherian ring. Denote by mod R the category of finitely generated R-modules. In this paper, we study n-torsionfree modules in the sense of Auslander and Bridger, by comparing them with n-syzygy modules, and modules satisfying Serre's condition (S_n). We mainly investigate closedness properties of the full subcategories of mod R consisting of those modules.
Let S be a toric algebra over a field K of characteristic 0 and let I be a monomial ideal of S. We show that the local cohomology modules H^i_I(S) are of finite length over the ring of differential operators D(S;K), generalizing the classical case of a polynomial algebra S. As an application, we compute the characteristic cycles of some local cohomology modules.
We study the difference between the lengths of the torsion of the differential modules of the local ring R of an algebroid curve and its first quadratic transform. The conjecture is that this difference should be positive if R is not regular. This is shown to be true in some cases, especially for stable complete intersections and for semigroup rings which are complete intersections.
We introduce the notion of Betti category for graded modules over suitably graded polynomial rings, and more generally for modules over certain small categories. Our categorical approach allows us to treat simultaneously many important cases, such as monomial ideals and toric rings. We prove that in these cases the Betti category is a finite combinatorial object that completely determines the structure of the minimal free resolution. For monomial ideals, the Betti category is the same as the Betti poset that we studied in a previous article. We describe in detail and with examples how the theo…
Generalizing the concepts of Stanley-Reisner and affine monoid algebras, one can associate to a rational pointed fan the toric face ring. Assuming that this ring is Cohen-Macaulay, the main result of this paper is to characterize the situation when its canonical module is isomorphic to a fine graded ideal of the toric face ring. From this result several algebraic and combinatorial consequences are deduced in the situations where the fan may be related to a manifold with non-empty boundary, or the fan is a shellable fan.
Torsion pairs in subcategories of modules over a commutative ring
本文针对交换诺特环$R$上有限生成$R$-模范畴的特定全子范畴，研究其中挠对的分类问题。
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