Sergio Estrada

Projective Representations of Quivers

In the first part of this article, we describe the projective representations in the category of representations by modules of a quiver which does not contain any cycles and the quiver A ∞ as a subquiver, that is, the so-called rooted quivers. As a consequence of this, we show when the category of representations by modules of a quiver admits projective covers. In the second part, we develop a technique involving matrix computations for the quiver A ∞, which will allow us to characterize the projective representations of A ∞. This will improve some previous results and make more accurate the statement made in Benson (1991). We think this technique can be applied in many other general situations to provide information about the decomposition of a projective module.

Balance with unbounded complexes

Given a double complex X there are spectral sequences with the E2 terms being either HI (HII(X)) or HII(HI(X)). But if HI(X) = HII(X) = 0 both spectral sequences have all their terms 0. This can happen even though there is nonzero (co)homology of interest associated with X. This is frequently the case when dealing with Tate (co)homology. So in this situation the spectral sequences may not give any information about the (co)homology of interest. In this article we give a different way of constructing homology groups of X when HI(X) =HII(X) = 0. With this result we give a new and elementary proof of balance of Tate homology and cohomology.

Locally projective monoidal model structure for complexes of quasi-coherent sheaves on P1(k)

AbstractWe will generalize the projective model structure in the category of unbounded complexes of modules over a commutative ring to the category of unbounded complexes of quasi-coherent sheaves over the projective line. Concretely we will define a locally projective model structure in the category of complexes of quasi-coherent sheaves on the projective line. In this model structure the cofibrant objects are the dg-locally projective complexes. We also describe the fibrations of this model structure and show that the model structure is monoidal. We point out that this model structure is necessarily different from other known model structures such as the injective model structure and the locally free model structure.

GENERALIZED QUASI-COHERENT SHEAVES

The category of quasi-coherent sheaves on the projective line P1(k) (k is a field) is equivalent to certain representations of the quiver • → • ← •. Many of the techniques which are used to study these sheaves apply to more general categories. We give the definitions of these more general categories and then consider one particular such category in depth. In this particular category we prove that there are no (nonzero) projective representations but that the category admits flat covers (or, equivalently in this situation, torsion free covers) and cotorsion envelopes.