It is proven that each indecomposable injective module over a valuation domain R is polyserial if and only if each maximal immediate extension R̂ of R is of finite rank over the completion R̃ of R in the R-topology. In this case, for each indecomposable injective module E, the following invariants are finite and equal: its Malcev rank, its Fleischer rank and its dual Goldie dimension. Similar res...

In analogy with the varietal case, we give an abstract characterization of those categories occurring as regular epireflective subcategories of presheaf categories such that the inclusion functor preserves small sums. MSC 2000 : 18A40, 18F20. The aim of this short note is to add one step to the nice parallelism between presheaf categories and algebraic categories. Presheaf categories can be abs...

In [9, 10], Hartshorne gives an account of the known vector bundles of "small" rank on P" and also on the punctured spectrum of a regular local ring. Specifically, he remarks on various constructions of rank two vector bundles on P", for n ^ 4, and Horrocks' examples [12] of rank three bundles on P as well. The result of Evans and Griffith [7] on syzygies shows that the existence of rank two in...

Let G be a semisimple, simply-connected algebraic group over an algebraically closed field of characteristic p > 0. We observe that the tensor product of the Steinberg module with a minuscule module is always indecomposable tilting. Although quite easy to prove, this fact does not seem to have been observed before. It has the following consequence: If p > 2h − 2 and a given tilting module has h...

Let A be a hereditary algebra over an algebraically closed field. We prove that an exact fundamental domain for the m-cluster category Cm(A) of A is the m-left part Lm(A (m)) of the m-replicated algebra of A. Moreover, we obtain a one-toone correspondence between the tilting objects in Cm(A) (that is, the m-clusters) and those tilting modules in modA(m) for which all non projective-injective di...

It is shown that each almost maximal valuation ring R, such that every indecomposable injective R-module is countably generated, satisfies the following condition (C): each fp-injective R-module is locally injective. The converse holds if R is a domain. Moreover, it is proved that a valuation ring R that satisfies this condition (C) is almost maximal. The converse holds if Spec(R) is countable....

We show that there is a reflection type bijection between the indecomposable summands of two multiplicity free tilting modules X and Y. This fixes common Y sends projective (resp., injective) exactly one module to non-projective non-injective) other. Moreover, this interchanges possible non-isomorphic complements an almost complete module.

Using Gröbner Basis, we introduce a general algorithm to determine the additive structure of a module, when we know about it using indirect information about its structure. We apply the algorithm to determine the additive structure of indecomposable modules over ZCp, where Cp is the cyclic group of order a prime number p, and the p−pullback {Z→ Zp ← Z} of Z⊕Z.

Let (R; m;k) be a local ring (commutative and Noetherian). We will discuss existence and uniqueness of direct-sum decompositions of nitely generated R-modules. One says that R has nite CM type provided there are only nitely many indecomposable maximal Cohen-Macaulay R-modules up to isomorphism. Among complete equicharacteristic hypersurface rings with k algebraically closed of characteristic 6 ...

An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely-generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely-generated Gorenstein-projective modules.