Dan Zacharia

Approximations with modules having linear resolutions

Abstract Let Λ be a Koszul algebra over a field K . We study in this paper a class of modules closely related to the Koszul modules called weakly Koszul modules. It turns out that these modules have some special filtrations with modules having linear resolutions and therefore easy to describe minimal projective resolutions. We prove that if the Koszul dual of a finite-dimensional Koszul algebra is Noetherian then every finitely generated graded module has a weakly Koszul syzygy and as a consequence a rational Poincare series. If Λ is selfinjective Koszul, we prove that the stable part of each connected component of the graded Auslander–Reiten quiver containing a weakly Koszul module is of the form Z A ∞ , and if the Koszul dual of Λ is Noetherian, then every component has its stable part of the form Z A ∞ .

Minimal projective resolutions

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the Ext-algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the “no loop” conjecture.

On the cyclic homology of monomial relation algebras

Abstract A finite dimensional monomial relation algebra has finite global dimension if and only if the relative cyclic homology of its radical is equal to zero.

Auslander-Reiten components containing modules with bounded Betti numbers

Let R be a connected selfinjective Artin algebra, and M an indecomposable nonprojective R-module with bounded Betti numbers lying in a regular component of the Auslander-Reiten quiver of R. We prove that the Auslander-Reiten sequence ending at M has at most two indecomposable summands in the middle term. Furthermore we show that the component of the Auslander-Reiten quiver containing M is either a stable tube or of type ZA ∞ . We use these results to study modules with eventually constant Betti numbers, and modules with eventually periodic Betti numbers.

Finiteness of the strong global dimension of radical square zero algebras

The strong global dimension of a finite dimensional algebra A is the maximum of the width of indecomposable bounded differential complexes of finite dimensional projective A-modules. We prove that the strong global dimension of a finite dimensional radical square zero algebra A over an algebraically closed field is finite if and only if A is piecewise hereditary. Moreover, we discuss results concerning the finiteness of the strong global dimension of algebras and the related problem on the density of the push-down functors associated to the canonical Galois coverings of the trivial extensions of algebras by their repetitive algebras.

Auslander–Reiten sequences, locally free sheaves and Chebysheff polynomials

Let R be the exterior algebra in n + 1 variables, and let S denote the symmetric algebra in n + 1 variables. It is well known that R is a selfinjective Koszul algebra and S is its Koszul dual. By KR and KS we denote the categories of linear R-modules (S-modules respectively) where the morphisms are the degree zero homomorphisms. The Koszul duality can be then used to obtain mutually inverse dualities between the category of linear R-modules and that of the linear S-modules:

A note on sheaves without self-extensions on the projective n-space

Let

Algebras of Finite Global Dimension

{\bf P}^n

Representations of Algebras

be the projective

A homological characterization of piecewise hereditary algebras

n-