Owing to the difference in K-theory, an example by Dugger and Shipley implies that equivalence of stable categories Gorenstein projective modules should not be a Quillen equivalence. We give sufficient necessary condition for Frobenius pair faithful functors between two abelian equivalence, which is also equivalent induce mutually inverse equivalences objects. show category objects Waldhausen c...

Let G be a finite subgroup of SL(r,C). In dimensions r = 2 and r = 3, McKay correspondence provides a natural bijection between the set of irreducible representations of G and a cohomology-ring basis of the overlying space of a projective, crepant desingularization of C/G. For r = 2 this desingularization is unique and is known to be determined by the Hilbert scheme of the Gorbits. Similar stat...

For a finite quiver without sources or sinks, we prove that the homotopy category of acyclic complexes of injective modules over the corresponding finite-dimensional algebra with radical square zero is triangle equivalent to the derived category of the Leavitt path algebra viewed as a differential graded algebra with trivial differential, which is further triangle equivalent to the stable categ...

The stringy E-function for normal irreducible complex varieties with at worst log terminal singularities was introduced by Batyrev. It is defined by data from a log resolution. If the variety is projective and Gorenstein and the stringy E-function is a polynomial, Batyrev also defined the stringy Hodge numbers as a generalization of the Hodge numbers of nonsingular projective varieties, and con...

It is well-known that for a big class of local rings of positive characteristic, including complete intersection rings, the Frobenius endomorphism can be used as a test for finite projective dimension. In this paper, we exploit this property to study the structure of such rings. One of our results states that the class groups cannot have any p-torsion, thus providing a purely algebraic proof of...

The stringy E-function for normal irreducible complex varieties with at worst log terminal singularities was introduced by Batyrev. It is defined by data from a log resolution. If the variety is projective and Gorenstein and the stringy E-function is a polynomial, Batyrev also defined the stringy Hodge numbers as a generalization of the Hodge numbers of nonsingular projective varieties, and con...

We study the Zariski tangent cone TX π −→ X to an affine variety X and the closure TX of π−1(Reg(X)) in TX . We focus on the comparison between TX and TX , giving sufficient conditions on X in order that TX = TX . One aspect of the results is to understand when this equality takes place in the presence of the reducedness of the Zariski tangent cone. Our other interest is to consider conditions ...

Batyrev has defined the stringy E-function for complex varieties with at most log terminal singularities. It is a rational function in two variables if the singularities are Gorenstein. Furthermore, if the variety is projective and its stringy E-function is a polynomial, Batyrev defined its stringy Hodge numbers essentially as the coefficients of this E-function, generalizing the usual notion o...