This main purpose of this article is the unification of the Galois theory of algebraic differential equations by Umemura and the Galois theory of algebraic difference equations by Morikawa-Umemura in a common framework using Artinian simple D-module algebras, where D is a bialgebra. We construct the Galois hull of an extension of Artinian simple D-module algebras and define its Galois group, which consists of infinitesimal coordinate transformations fulfilling certain partial differential equations and which we call Umemura functor. We eliminate the restriction to characteristic 0 from the above mentioned theories and remove the limitation to field extensions in the theory of Morikawa-Umemura, allowing also direct products of fields, which is essential in the theory of difference equations. In order to compare our theory with the Picard-Vessiot theory of Artinian simple D-module algebras due to Amano and Masuoka, we first slightly generalize the definition and some results about them in order to encompass as well non-inversive difference rings. Finally, we give equivalent characterizations for smooth Picard-Vessiot extensions, describe their Galois hull and show that their Umemura functor becomes isomorphic to the formal scheme associated to the classical Galois group scheme after a finite étale base extension.
Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.
We study two subcategories of the category of artinian modules, a wide subcategory and a Serre subcategory. We prove that all wide subcategories of artinian modules are Serre subcategories. We also provide the bijection between the set of Serre subcategories and the set of specialization closed subsets of the set of closed prime ideals of some completed ring.
Let $(R,\mathfrak{m})$ be a commutative local noetherian ring. For an artinian $R$-module $M$, we show the equality $$\mathrm{cosupp}_RM=\mathrm{Cosupp}_RM$$ using the semi-discrete linearly compactness of $R$-module $\mathrm{Hom}_R(R_\mathfrak{p},M)$ where $\mathfrak{p}$ is a prime ideal of $R$.
Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. In this article, we examine the question of whether an arbitrary top local cohomology module, $\operatorname{H}^{\operatorname{cd}(I,M)}_I(M)$, is Artinian, or not. Several results related to this question are obtained; in particular, we prove that over a Noetherian local unique factorization domain $R$ of dimension three, for a finitely generated faithful module $M$, a top local cohomology module is Artinian if and only if $\operatorname{cd}(I,M)= 3$.
In this paper, considering the difference between the finiteness dimension and cohomological dimension for a finitely generated module, we investigate the asymptotic behavior of grades of components of graded local cohomology modules with respect to irrelevant ideal; as long as we study some artinian and tameness property of such modules.
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.
In this paper, we prove that a finitely embedded $R$-module $M$ is Artinian if and only if for every prime ideal $\mathfrak{p}$ of $R$ with $(0:_RM)\subseteq \mathfrak{p}$, there exists a submodule $N^\mathfrak{p}$ of $M$ such that $M/N^\mathfrak{p}$ is finitely embedded and $M[\mathfrak{p}]\subseteq N^\mathfrak{p}\subseteq (0:_M\mathfrak{p})$.
We describe the positive cone generated by bigraded Betti diagrams of artinian modules of codimension two, whose resolutions become pure of a given type when taking total degrees. If the differences of these total degrees, p and q, are relatively prime, the extremal rays are parametrised by order ideals in N^2 contained in the region px + qy < (p-1)(q-1). We also consider some examples concerning artinian modules of codimension three.
Let A be a standard graded Artinian algebra over a field of characteristic zero and let z be a linear form in A. We define the central simple modules for each such pair (A, z). Assume that A is Gorenstein. Then we prove that A has the strong Lefschetz property if and only if there exists a linear form z in A such that all central simple modules of the pair (A,z) have the strong Lefschetz property. In the course of proof we need to extend the definition of the strong Lefschetz property to finite graded modules over graded Artinian algebra, which previously was defined only for standard graded A…
Let $M$ be a finite module over a commutative noetherian ring $R$. For ideals $\fa$ and $\fb$ of $R$, the relations between cohomological dimensions of $M$ with respect to $\fa, \fb$, $\fa\cap\fb$ and $\fa+ \fb$ are studied. When $R$ is local, it is shown that $M$ is generalized Cohen-Macaulay if there exists an ideal $\fa$ such that all local cohomology modules of $M$ with respect to $\fa$ have finite lengths. Also, when $r$ is an integer such that $0\leq r< \dim_R(M)$, any maximal element $\fq$ of the non-empty set of ideals $\{\fa$ : $\H_\fa^i(M)$ is not artinian for some $i$, $i\geq r$$\}$…
Let $(R,\mathfrak m)$ be a Noetherian local ring, $I$ an ideal of $R$ and $M$ a weakly finite or a coatomic $R$-module of dimension $n$. In this article, we resolve the Artinianness and non-Artinianness of top local cohomology modules, $H^{cd(I,M)}_I(M)$, in all cases except in the case $cd(I,M)=n-1$ and $dim(R/{I+Ann(M)})>1$ for which we have some sorter results under certain conditions.
An $R$-module $M$ is Hopfian (co-Hopfian) if any epic (monic) endomorphism of $M$ is an automorphism. If $R$ is commutative Noetherian, we characterize the co-Hopfian injective $R$-modules, and the Hopfian injectives in the case that $R$ is also reduced. For a commutative Artinian principal ideal ring, we show that $M$ is Hopfian (co-Hopfian) if and only if $M$ is finitely generated if and only if its injective envelope $E(M)$ is Hopfian (co-Hopfian) if and only if $E(M)$ is finitely generated. We identify the obstacle to generalizing this result to arbitrary Artinian principal ideal rings.
We propose an Artinian version of Berger's Conjecture for curves, concerning the module of Kähler differentials of an algebra. Our version implies Berger's Conjecture in characteristic 0. We establish our Artinian Berger Conjecture in a number of cases, and prove that Berger's Conjecture holds for curve singularities whose conductor ideal contains the cube of a maximal ideal.
We study the linear space generated by the multigraded Betti diagrams of Z^n-graded artinian modules of codimension n whose resolutions become pure of a given type when taking total degrees. We show that the multigraded Betti diagram of the equivariant resolution constructed by D.Eisenbud, J.Weyman, and the author, and all its twists, form a basis for this linear space.
Let R be a commutative local noetherian ring, and let L and L' be R-modules. We investigate the properties of the functors Tor_i^R(L,-) and Ext^i_R(L,-). For instance, we show the following: (a) if L is artinian and L' is noetherian, then Hom_R(L,L') has finite length; (b) if L and L' are artinian, then the tensor product L \otimes_R L' has finite length; (c) if L and L' are artinian, then Tor_i^R(L,L') is artinian, and Ext^i_R(L,L') is noetherian over the completion \hat R; and (d) if L is artinian and L' is Matlis reflexive, then Ext^i_R(L,L'), Ext^i_R(L',L), and Tor_i^R(L,L') are Matlis ref…
Let $R$ be a commutative Noetherian local ring of prime characteristic. The purpose of this paper is to provide a short proof of G. Lyubeznik's extension of a result of R. Hartshorne and R. Speiser about a module over the skew polynomial ring $R[x,f]$ (associated to $R$ and the Frobenius homomorphism $f$, in the indeterminate $x$) that is both $x$-torsion and Artinian over $R$.
Let $R$ be a commutative Noetherian ring. It is shown that $R$ is Artinian if and only if every $R$-module is good, if and only if every $R$-module is representable. As a result, it follows that every nonzero submodule of any representable $R$-module is representable if and only if $R$ is Artinian. This provides an answer to a question which is investigated in [{\bf 1}].
Let R be a commutative ring, and let L and L' be R-modules. We investigate finiteness conditions (e.g., noetherian, artinian, mini-max, Matlis reflexive) of the modules Ext^i_R(L,L') and Tor_i^R(L,L') when L and L' satisfy combinations of these finiteness conditions. For instance, if R is noetherian, then given R-modules M and M' such that M is Matlis reflexive and M' is mini-max (e.g., noetherian or artinian), we prove that Ext^i_R(M,M'), Ext^i_R(M',M), and Tor_i^R(M,M') are Matlis reflexive over R for all i\geq 0 and that Ext^i_R(M,M')^\vee\cong Tor_i^R(M,M'^\vee) and Ext^i_R(M',M)^\vee\cong…
We relax the definition of a string algebra to also include infinite-dimensional algebras such as k[x,y]/(xy). Using the functorial filtration method, which goes back to Gelfand and Ponomarev, we show that finitely generated and artinian modules (and more generally finitely controlled and pointwise artinian modules) are classified in terms of string and band modules. This subsumes the known classifications of finite-dimensional modules for string algebras and of finitely generated modules for k[x,y]/(xy). Unlike in the finite-dimensional case, the words parameterizing string modules may be inf…
In this paper we consider Artinian modules over power series rings endowed with a Frobenius map. We describe a method for finding the set of all prime annihilators of submodules which are preserved by the given Frobenius map and on which the Frobenius map is not nilpotent. This extends the algorithm by Karl Schwede and the first author, which solved this problem for submodules of the injective hull of the residue field. The Matlis dual of this problem asks for the radical annihilators of quotients of free modules by submodules preserved by a given Frobenius near-splitting, and the same method …
Let V be a finite-dimensional vector space over a field k and let W be a 1-dimensional k-vector space. Let < , >: V x V \to W be a symmetric bilinear form. Then < , > is called anisotropic if for all nonzero v \in V we have <v,v> \neq 0. Motivated by a problem in algebraic number theory, we come up with a generalization of the concept of anisotropy to symmetric bilinear forms on finitely generated modules over artinian principal ideal rings. We will give many equivalent definitions of this concept of anisotropy. One of the definitions shows that one can check if a form is anisotropic by checki…
Fix a pair of positive integers d and n. We create a ring R and a complex G of R-modules with the following universal property. Let P be a polynomial ring in d variables over a field and let I be a grade d Gorenstein ideal in P which is generated by homogeneous forms of degree n. If the resolution of P/I by free P-modules is linear, then there exists a ring homomorphism from R to P such that P tensor G is a minimal homogeneous resolution of P/I by free P-modules. Our construction is coordinate free.
We define the strong Lefschetz property for finite graded modules over graded Artinian algebras whose grading is not necessarily standard. We show that most results which have been obtained for Artinian algebras with standard grading can be extended for non-standard grading. Our results on the strong Lefschetz property for non-standard grading can be used to prove that certain Artinian complete intersections with standard grading have the strong Lefschetz property.
We show a number of properties of the commutator algebra of a nilpotent matrix over a field. In particular we determine the simple modules of the commutator algebra. Then the results are applied to prove that certain Artinian complete intersections have the strong Lefscehtz property.
Gerstenhaber showed in 1961 that any commuting pair of n x n matrices over a field k generates a k-algebra A of k-dimension \leq n. A well-known example shows that the corresponding statement for 4 matrices is false. The question for 3 matrices is open. Gerstenhaber's result can be looked at as a statement about the relation between the length of a 2-generator finite-dimensional commutative k-algebra A and the lengths of faithful A-modules. Wadsworth generalized this result to a larger class of commutative rings than those generated by two elements over a field. We recover his generalization, …
The Jordan type of an element $\ell$ of the maximal ideal of an Artinian k-algebra A acting on an A-module M of k-dimension n, is the partition of n given by the Jordan block decomposition of the multiplication map $m_\ell$ on M. In general the Jordan type has more information than whether the pair $(\ell,M)$ is strong or weak Lefschetz. We develop basic properties of the Jordan type and their loci for modules over graded or local Artinian algebras. We as well study the relation of generic Jordan type of $A$ to the Hilbert function of $A$. We introduce and study a finer invariant, the Jordan d…
Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, $M$ and $N$ be two finitely generated $R$-modules. Let $t$ be a positive integer. We prove that if $R$ is local with maximal ideal $\fm$ and $ M\otimes_R N$ is of finite length then $H_{\fm}^t(M,N)$ is of finite length for all $t\geq 0$ and $l_R(H_{\fm}^t (M,N))\leq \sum_{i=0}^t l_R(\Ext_R^i(M,H_{\fm}^{t-i}(N)))$. This yields, $l_R(H_{\fm}^t(M,N))=l_R(\Ext_R^t(M,N))$. Additionally, we show that $\Ext_R^i(R/{\fa},N)$ is Artinian for all $ i\leq t$ if and only if $H_{\fa}^i(M,N)$ is Artinian for all $i\leq t$. Moreover, we show tha…
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 \fa be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_{\fa}(M), the finiteness dimension of M with respect to \fa, and, its dual notion q_{\fa}(M), the Artinianess dimension of M with respect to \fa. When (R,\fm) is local and r:=f_{\fa}(M) is less than f_{\fa}^{\fm}(M), the \fm-finiteness dimension of M relative to \fa, we prove that H^r_{\fa}(M) is not Artinian, and so the filter depth of \fa on M doesn't exceeds f_{\fa}(M). Also, we show that if M has finite dimension and H^i_{\fa}(M) is Artinian for all i>t, …
We study the homogeneous artinian ideals of the polynomial ring $K[x,y,z]$, generated by the homogenous polynomials of degree $d$ which are invariant under an action of the cyclic group $\mathbb Z/d\mathbb Z$, for any $d\geq 3$. We prove that they are all monomial Togliatti systems, and that they are minimal if the action is defined by a diagonal matrix having on the diagonal $(1, e, e^a)$, where $e$ is a primitive $d$-th root of the unity. We get a complete description when $d$ is prime or a power of a prime. We also establish the relation of these systems with linear Ceva configurations.
This is the third paper in a series of three papers. The first two papers in the series are called "Artinian Gorenstein algebras with linear resolutions", (arXiv:1306.2523) and "The explicit minimal resolution constructed from a Macaulay inverse system". In the present paper, we give the explicit minimal resolution of an Artinian Gorenstein algebra with a linear resolution. This minimal resolution is given in a polynomial manner in terms of the coefficients of the Macaulay inverse system for the Gorenstein algebra. Let k be a field, A a standard-graded Artinian Gorenstein k-algebra, S the stan…
Given positive integers e and s we consider Gorenstein Artinian local rings R of embedding dimension e whose maximal ideal $\mathfrak{m}$ satisfies $\mathfrak{m}^s\ne 0=\mathfrak{m}^{s+1}$. We say that R is a compressed Gorenstein local ring when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If $s\ne 3$, we prove that the Poincare series of all finitely generated modules over a compressed Gorenstein local ring are rational, sharing a common denominator. A formula for the denominator is given. When s is even this formula depends on…
Pink has given a qualitative answer to the Mumford-Tate conjecture for Drinfeld modules in the 90s. He showed that the image of the v-adic Galois representation is v-adically open in the motivic Galois group for any prime v. In contrast to this result, we provide a family of uniformizable Anderson t-modules for which the Galois representations of their t-adic Tate-modules are "far from" having t-adically open image in their motivic Galois groups. Nevertheless, the image is still Zariski-dense in the motivic Galois group which is in accordance to the Mumford-Tate conjecture. For the proof, we e…
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.
We introduce and study "quasidualizing" modules. An artinian R-module T is quasidualizing if the homothety map \hat R\rightarrow Hom(T,T) is an isomorphism and Ext_R^i(T,T)=0 for each integer i>0. Quasidualizing modules are associated to semidualizing modules via Matlis duality. We investigate the associations via Matlis duality between subclasses of the Auslander class and Bass class and subclasses of derived T-reflexive modules.
We prove a certain Riemann-Roch type formula for symmetric powers of Galois modules on Dedekind schemes which, in the number field or function field case, specializes to a formula of Burns and Chinburg for Cassou-Noguès-Taylor operations.
