Modules Of Projective Dimension Research Articles

Ext modules related to syzygies of the residue field

Let R be a commutative noetherian ring. In this paper, we find out close relationships between the module M being embedded in a module of projective dimension at most n and the (n+1)-torsionfreeness of the nth syzygy of M. As an application, when R is local with residue field k, we compute the dimensions as k-vector spaces of Ext modules related to syzygies of k.

$\tau$-rigid Modules over Auslander Algebras

We give a characterization of $\tau$-rigid modules over Auslander algebras in terms of projective dimension of modules. Moreover, we show that for an Auslander algebra $\Lambda$ admitting finite number of non-isomorphic basic tilting $\Lambda$-modules and tilting $\Lambda^{\operatorname{op}}$-modules, if all indecomposable $\tau$-rigid $\Lambda$-modules of projective dimension $2$ are of grade $2$, then $\Lambda$ is $\tau$-tilting finite.

On the Cotorsion Pair (𝒫1, 𝒟)

We consider the perfect cotorsion pair (𝒫1, 𝒟) over a commutative ring (often over a domain) R consisting of modules of projective dimension ≤1 and of divisible modules, respectively. The kernel modules in this cotorsion pair are well kown (Theorem 2.1), and those domains are identified whose injectives are kernel modules (Theorem 2.3). The kernel modules always generate the cotorsion pair (𝒫1, 𝒟); they also cogenerate it if the global dimension of the domain R is finite (Theorems 3.1, 3.2). An analogue of the well-known Faith–Walker theorem on injective modules is proved: an integral domain R is noetherian of Krull dimension 1 if there is a cardinal number λ such that every weak-injective (or divisible) R-module is a direct sum of modules of cardinalities ≤ λ (Theorem 4.4). The proof relies on our Theorem 4.1 which generalizes the Faith–Walker theorem as well as a theorem by Guil and Herzog in [14]. Theorem 4.2 provides a general sufficient criterion for a complete cotorsion pair to be Σ-cotorsion; here Σ-cotorsion means that both classes in the cotorsion pair are closed under direct sums.

Relative parametrization of linear multidimensional systems

In the last chapter of his book “The Algebraic Theory of Modular Systems” published in 1916, F. S. Macaulay developped specific techniques for dealing with “unmixed polynomial ideals” by introducing what he called “inverse systems”. The purpose of this paper is to extend such a point of view to differential modules defined by linear multidimensional systems, that is by linear systems of ordinary differential or partial differential equations of any order, with any number of independent variables, any number of unknowns and even with variable coefficients. The first and main idea is to replace unmixed polynomial ideals by pure differential modules. The second idea is to notice that a module is \(0\)-pure if and only if it is torsion-free and thus if and only if it admits an “absolute parametrization” by means of arbitrary potential like functions, or, equivalently, if it can be embedded into a free module by means of an “absolute localization”. The third idea is to refer to a difficult theorem of algebraic analysis saying that an \(r\)-pure module can be embedded into a module of projective dimension \(r\), that is a module admitting a projective resolution with exactly \(r\) operators. The fourth and final idea is to establish a link between the use of extension modules for such a purpose and specific formal properties of the underlying multidimensional system through the use of “involution” and a ‘relative localization” leading to a “relative parametrization”. The paper is written in a rather effective self-contained way and we provide many explicit examples that should become test examples for a future use of computer algebra.

Categories of modules for elementary abelian p -groups and generalized Beilinson algebras

In this paper, we approach the study of modules of constant Jordan type and equal images modules over elementary abelian p-groups Er of rank r⩾2 by exploiting a functor from the module category of a generalized Beilinson algebra B(n, r), n⩽p, to mod Er. We define analogues of the above-mentioned properties in mod B(n, r) and give a homological characterization of the resulting subcategories via a ℙr−1-family of B(n, r)-modules of projective dimension 1. This enables us to apply general methods from Auslander–Reiten theory and thereby arrive at results that, in particular, contrast the findings for equal images modules of Loewy length 2 over E2 by Carlson, Friedlander and Suslin [Commentarii Math. Helv. 86 (2011) 609–657] with the case r>2. Moreover, we give a generalization of the W-modules introduced by the aforementioned authors.

One Dimensional Tilting Modules are of Finite Type

We prove that every tilting module of projective dimension at most one is of finite type, namely that its associated tilting class is the Ext-orthogonal of a family of finitely presented modules of projective dimension at most one.

THE COHOMOLOGY OF THE KOSZUL COMPLEXES ASSOCIATED TO THE TENSOR PRODUCT OF TWO FREE MODULES#

Let E and G be free modules of rank e and g, respectively, over a commutative noetherian ring R. The identity map on E* ⊗ G induces the Koszul complex and its dual Let H 𝒩(m, n, p) and H ℳ (m, n, p) be the homology of the above complexes at respectively. In this paper, we investigate the modules H 𝒩(m, n, p) and H ℳ(m, n, p) when −e ≤ m − n ≤ g. We record the fact, already implicitly calculated by Bruns and Guerrieri, that H 𝒩(m, n, p) ≅ H ℳ (m′, n′, p′), provided m + m′ = g − 1, n + n′ = e − 1, p + p′ = (e − 1)(g − 1), and 1 − e ≤ m − n ≤ g − 1. If m − n is equal to either g or −e , then we prove that the only nonzero modules of the form H 𝒩(m, n, p) and H ℳ (m, n, p) appear in one of the split exact sequences where p + p′ = (e − 1)(g − 1) − 1. The modules that we study are not always free modules. Indeed, if m = n, then the module H 𝒩(m, n, p) is equal to a homogeneous summand of the graded module (T, R), where P is a polynomial ring in eg variables over R and T is the determinantal ring defined by the 2×2 minors of the corresponding e×g matrix of indeterminates. Hashimoto's work shows that if e and g are both at least five, then H 𝒩(2, 2, 3) is not a free module when R is ℤ, and when R is a field, the rank of this module depends on the characteristic of R . When the modules H ℳ(m, n, p) are free, they are summands of the resolution of the universal ring for finite length modules of projective dimension two.

Equivalent Presentations of Modules Over Prüfer Domains

AbstractIf F and F′ are free R-modules, then M ≅ F/H and M ≅ F′/H′ are viewed as equivalent presentations of the R-module M if there is an isomorphism F → F′ which carries the submodule H onto H′. We study when presentations of modules of projective dimension 1 over Prüfer domains of finite character are necessarily equivalent.

Q-Torsion freeness of symmetric powers

Letq be an integer ≥1, we study theq-torsion freeness of the symmetric powers of a module of projective dimension ≥2, by using the approximation complex of the module.