Hernán de Alba, Daniel Duarte
We characterize the $k$-torsion freeness of the module of differentials of order $n$ of a point of a hypersurface in terms of the singular locus of the corresponding local ring.
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Let (R,m) be a local ring and U_R=Spec(R) -{m} be the punctured spectrum of R. Gabber conjectured that if R is a complete intersection of dimension 3, then the abelian group Pic(U_R) is torsion-free. In this note we prove Gabber's statement for the hypersurface case. We also point out certain connections between Gabber's Conjecture, Van den Bergh's notion of non-commutative crepant resolutions and some well-studied questions in homological algebra over local rings.
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.
We give an explicit presentation of the module of differentials of order $n$ of a finitely generated algebra via a higher-order Jacobian matrix. We use the presentation to study some aspects of this module in the case of hypersurfaces. More precisely, we prove higher-order versions of known results relating freness and torsion-freness of the module of differentials with the regularity and normality of the hypersurface. We also study its projective dimension.
If $(A,\mathfrak{m})$ is a hypersurface ring of dimension $d$ with $e(A)=3$. Let $M$ be an MCM $A$-module with $μ(M)=4$ then we prove that $\depth{G(M)}\geq d-3$.
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.
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.
The acquisition of the defining equations of Rees algebras is a natural way to study these algebras and allows certain invariants and properties to be deduced. In this paper, we consider Rees algebras of codimension 2 perfect ideals of hypersurface rings and produce a minimal generating set for their defining ideals. Then, using generic Bourbaki ideals, we study Rees algebras of modules with projective dimension one over hypersurface rings. We describe the defining ideal of such algebras and determine Cohen-Macaulayness and other invariants.
Let $A=Q/(f)$ where $(Q,\mathfrak{n})$ be a complete regular local ring of dimension $d+1$, $f\in \mathfrak{n}^i\setminus\mathfrak{n}^{i+1}$ for some $i\geq 2$ and $M$ an MCM $A-$module with $e(M)=μ(M)i(M)+1$ then we prove that depth $G(M)\geq d-1$. If $(A,\mathfrak{m})$ is a complete hypersurface ring of dimension $d$ with infinite residue field and $e(A)=3$, let $M$ be an MCM $A$-module with $μ(M)=2$ or $3$ then we prove that depth $G(M)\geq d-μ(M)+1$. Our paper is the first systematic study of depth of associated graded modules of MCM modules over hypersurface rings.
Let $(A,\mathfrak{m})$ be a hypersurface ring with dimension $d$, and $M$ a MCM $A-$module with red$(M)\leq 2$ and $μ(M)=2$ or $3$ then we have proved that depth $G(M)\geq d-μ(M)+1$. If $e(A)=3$ and $μ(M)=4$ then in this case we have proved that depth$G(M)\geq d-3$. Next we consider the case when $e(M)=μ(M)i(M)+1$ and prove that depth $G(M)\geq d-1$. When $A = Q/(f)$ where $Q = k[[X_1,\cdots, X_{d+1}]]$ then we give estimates for $\depth G(M)$ in terms of a minimal presentation of $M$. Our paper is the first systematic study of depth of associated graded modules of MCM modules over hypersurfac…
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.
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.
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.
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.
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。
This paper deals with the notion of grade of ideals with respect to torsion theories defined via some homological tools such as Ext-modules, Koszul cohomology modules, Čech and local cohomology modules over commutative rings which are not necessarily Noetherian. We also compare these approaches of grade.
Let $R$ be a Noetherian commutative ring and $M$ an $R$-module with $\operatorname{pd_R} M\le 1$ that has rank. Necessary and sufficient conditions were provided by Lebelt for an exterior power $\wedge^k M$ to be torsion free. When $M$ is an ideal of $R$ similar necessary and sufficient conditions were provided by Tchernev for a symmetric power $S_k M$ to be torsion free. We extend these results to a broad class of Schur modules $L_{λ/μ} M$. En route, for any map of finite free $R$ modules $ϕ\: F\rightarrow G$ we also study the general structure of the Schur complexes $L_{λ/μ}ϕ$, and provide n…
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.
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。
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…
Let $M$ be a finitely generated module over a Noetherian local ring. This paper gives, for a given parameter ideal $Q$ for $M$, bounds for the second Hilbert coefficients ${\mathrm{e}}_Q^2(M)$ in terms of the homological degrees and torsions of modules. We also report a criterion for a certain equality of the second Hilbert coefficients of parameters and the homological torsions of modules.
We calculate the Krull--Gabriel dimension of the functor category of the (stable) category of maximal Cohen--Macaulay modules over hypersurfaces of countable Cohen--Macaulay representation type. We show that the Krull--Gabriel dimension is $0$ if the hypersurface is of finite Cohen--Macaulay representation type and that is $2$ if the hypersurface is of countable but not finite Cohen--Macaulay representation type.
Let R be a commutative Noetherian domain, and let M and N be finitely generated R-modules. We give new criteria for determining when M tensor N has torsion. We also give constructive formulas for producing a module in the isomorphism class of the torsion submodule of M tensor N. In some cases we determine bounds on the length and minimal number of generators of this module. We focus on the case where R is a numerical semigroup ring with the goal of making progress on the Huneke-Wiegand Conjecture.
For any finitely generated module $M$ with non-zero rank over a commutative one dimensional Noetherian local domain, the numerical invariant $h(M)$ was introduced and studied in the author's previous work "Partial Trace Ideals and Berger's Conjecture". We establish a bound on it which helps capture information about the torsion submodule of $M$ when $M$ has rank one and it also generalizes the discussion in the mentioned previous article. We further study bounds and properties of $h(M)$ in the case when $M$ is the canonical module $ω_R$. This in turn helps in answering a question of S. Greco a…
A topological interpretation of Hochster's Theta pairing of two modules on a hypersurface ring is given in terms of linking numbers. This generalizes results of M. Hochster and proves a conjecture of J. Steenbrink. As a corollary we get that the Theta pairing vanishes for isolated hypersurface singularities in an odd number of variables, as was conjectured by H. Dao.
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 $2$.
In this article, we shall characterize torsionfreeness of modules with respect to a semidualizing module in terms of the Serre's condition (S_n). As its applications, we give a characterization of Cohen-Macaulay rings R such that R_p is Gorenstein for all prime ideals p of height less than n, and we will give a partial answer of Tachikawa conjecture and Auslander-Reiten conjecture.
Let R be a complete local hypersurface over an algebraically closed field of characteristic different from two, and suppose that R has countable Cohen-Macaulay representation type. In this paper, it is proved that the maximal Cohen-Macaulay R-modules which are locally free on the punctured spectrum are dominated by the maximal Cohen-Macaulay R-modules which are not locally free on the punctured spectrum. More precisely, there exists a single R-module X such that the indecomposable maximal Cohen-Macaulay R-modules not locally free on the punctured spectrum are X and its syzygy ΩX and that any o…
A submodule $W$ of a p-primary module $M$ of bounded order is known to be regular if $W$ and $M$ have simultaneous bases. In this paper we derive necessary and sufficient conditions for regularity of a submodule.
This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the special fiber to the multiplicity of the pair at the general fiber. It is as important to the study of multiplicities of modules as the basic theorem in ideal theory which relates the multiplicity of an ideal to the local degree of the map formed from the generators of a minimal reduction. In fact, as a corollary of the theorem, we show here that for M a subm…
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.
Let R be a Gorenstein local ring which is locally a hypersurface on the punctured spectrum. In this paper, we classify thick subcategories of the bounded derived category of finitely generated R-modules. Moreover, using this classification, we also classify thick subcategories of finitely generated R-modules, and find out the relationships with thick subcategories of Cohen-Macaulay R-modules.
Let $R$ be a hypersurface in an equicharacteristic or unramified regular local ring. For a pair of modules $(M,N)$ over $R$ we study applications of rigidity of $\Tor^R(M,N)$, based on ideas by Huneke, Wiegand and Jorgensen. We then focus on the hypersurfaces with isolated singularity and even dimension, and show that modules over such rings behave very much like those over regular local rings. Connections and applications to projective hypersurfaces such as intersection dimension of subvarieties and cohomological criterion for splitting of vector bundles are discussed.
There is a conjecture, that the torsionfreeness of the module of differentials in a point of an algebraic or algebroid curve should imply that the curve is non singular at that point. A report on the main results is given.
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。
We generalize known results on summands of completely decomposable and separable torsion-free abelian groups to modules over h-local Prüfer domains. Over such domains summands of completely decomposable torsion-free modules are again completely decomposable (Theorem 3.2) and summands of separable torsion-free modules are likewise separable (Theorem 4.2). In addition, a Pontryagin-Hill type theorem is established on countable chains of homogeneous completely decomposable modules over h-local Prüfer domains.
Let $(R,\mathfrak{m}_R,k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero. R. Berger conjectured that $R$ is regular if and only if the universally finite module of differentials $Ω_R$ is torsion free. When $R$ is a domain, we prove the conjecture in several cases. Our techniques are primarily reliant on making use of the valuation semi-group of $R$. First, we establish a method of verifying the conjecture by analyzing the valuation semi-group of $R$ and orders of units of the integral closure of $R$. We also prove the conjecture in the case when ce…
Projective length, phantom extensions, and the structure of torsion modules
本文刻画了可数戴德金整环上挠模的幻影扩张，证明射影长度至多$\alpha$等价于约化且Ulm长度至多$1+\alpha$，并给出有限挠模预层的余极限刻画。
Torsion-free Modules over Commutative Domains of Krull Dimension One
本研究探讨Krull维数1的交换整环上有限生成无挠模直和类的直和项封闭性，给出局部环、诺特环等情形的等价条件，分析h-局部环的局部化性质及逆命题成立的情形，并证明有限特征整环上可数生成模的同构类由亏格决定。
Torsion modules and differential operators in infinitely many variables
本文探讨了无限生成理想下的torsion模概念，提出微分算子的超限阶和局部阶定义，并证明任意序数均可作为无限多个变量多项式代数上微分算子的阶数。
Let M be a matrix whose entries are power series in several variables and determinant det(M) does not vanish identically. The equation det(M)=0 defines a hypersurface singularity and the (co)-kernel of M is a maximally Cohen-Macaulay module over the local ring of this singularity. Suppose the determinant det(M) is reducible, i.e. the hypersurface is locally reducible. A natural question is whether the matrix is equivalent to a block-diagonal or at least to an upper-block-triangular. (Or whether the corresponding module is decomposable or at least is an extension.) We give various necessary and…
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