Introduction. A module over a ring will be said to be locally projective if and only if every finitely generated submodule is projective. As will be shown (7.14), it readily follows from known facts that if M is a locally projective module over a regular ring R, then the set L(M, R) of all finitely generated submodules of M is a relatively complemented modular lattice. This paper is concerned w...

Let Λ be an Auslander’s 1-Gorenstein Artinian algebra with global dimension 2. If Λ admits a trivial maximal 1-orthogonal subcategory of modΛ, then for any indecomposable module M ∈ modΛ, we have that the projective dimension of M is equal to 1 if and only if so is its injective dimension and that M is injective if the projective dimension of M is equal to 2, which implies that Λ is almost here...

We will introduce the notion of Gorenstein category as the convenient setup for doing Gorenstein Homological Algebra in categories of sheaves, or in general in categories without enough projective objects. We will illustrate this notion by showing that the category of Qcoh(X) of quasi-coherent sheaves on a locally Gorenstein projective scheme fits into this setup. Then we will focus on the cate...

In this paper, we introduce a novel pictorial approach for solving problems in n-dimensional Euclidean spaces called the n-dimensional projective approach. The projective approach is based on a hierarchical and modular architecture, where its ground module is rooted on geometrical concepts. The result is an effective and consistent spatial approach able to solve problems in ndimensional spaces ...

In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results est...

We give a short and simple argument that proves, in a uniform way, the Auslander-Buchsbaum formula, relating depth and projective dimension, and the Auslander-Bridger formula, relating depth and G-dimension. Moreover, the same type of argument quickly reproves the fact that, in the degrees above the codepth, the syzygy modules of a finite module over a commutative local ring have no free summan...

We show that every tilting module of projective dimension one over a ring R is associated in a natural way to the universal localization R → RU at a set U of finitely presented modules of projective dimension one. We then investigate tilting modules of the form RU ⊕ RU/R. Furthermore, we discuss the relationship between universal localization and the localization R → QG given by a perfect Gabri...

2.1. U̇(sl2) and its representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 2.1.1. Algebra U̇(sl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 2.1.2. Representations of U̇(sl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 2.2. Temperley-Lieb algebra . . . . . . . . ...

Let M be a left module over a ring R and I an ideal of R. We call (P,f ) a projective I -cover of M if f is an epimorphism from P to M , P is projective, Kerf ⊆ IP , and whenever P = Kerf + X, then there exists a summand Y of P in Kerf such that P = Y +X. This definition generalizes projective covers and projective δ-covers. Similar to semiregular and semiperfect rings, we characterize I -semir...

Bhargava defined p-orderings of subsets of Dedekind domains and with them studied polynomials which take integer values on those subsets. In analogy with this construction for subsets of Z(p) and p-local integer-valued polynomials in one variable, we define projective p-orderings of subsets of Z(p). With such a projective p-ordering for Z(p) we construct a basis for the module of homogeneous, p...