We prove that if M , N are finite modules over a Gorenstein local ring R of codimension at most 4, then the vanishing of Ext R (M,N) for n ≫ 0 is equivalent to the vanishing of Ext R (N,M) for n ≫ 0. Furthermore, if b R has no embedded deformation, then such vanishing occurs if and only if M or N has finite projective dimension.

We classify the smooth projective symmetric G-varieties with Picard number one (and G semisimple). Moreover we prove a criterion for the smoothness of the simple (normal) symmetric varieties whose closed orbit is complete. In particular we prove that, given a such variety X which is not exceptional, then X is smooth if and only if an appropriate toric variety contained in X is smooth. keywords:...

In his technical report~\cite[sec. 6]{barrontech}, Barron states that the de Bruijn's identity for Gaussian perturbations holds for any RV having a finite variance. In this report, we follow Barron's steps as we prove the existence of $J_{\alpha}\left(X + \sqrt[\alpha]{\eta}N\right)$, $\eta>0$ for any Radom Variable (RV) $X \in \mathcal{L}$ where \begin{equation*} \mathcal{L} = \left\{ \text{RV...

A compact metric space $(X, \rho)$ is given. Let $\mu$ be a Borel measure on $X$. By $r$-cluster we mean a measurable subset of $X$ with diameter at most $r$. A family of $k$ $2r$-clusters is called a $r$-cluster structure of order $k$ if any two clusters from the family are separated by a distance at least $r$. By measure of a cluster structure we mean a sum of clusters measures from the clust...

In [Pas15], we described the Minimal Model Program in the family of Q-Gorenstein projective horospherical varieties, by studying a family of polytopes defined from the moment polytope of an ample Q-Cartier Q-divisor of the variety we begin with. Here, we summarize the results of [Pas15] and we explain how to generalize them in order to describe the Log Minimal Model Program for pairs (X,∆) wher...

Let X be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by F a smooth model of a generic irreducible element in fibers of φ1 and so F is a curve or a smooth surface. The main result is that there is a computable constant K independent of X such that g(F ) ≤ 647 or pg(F ) ≤ 38 whenever pg(X) ≥ K.

The category gp ( Λ ) of Gorenstein-projective modules over tensor algebra = A ⊗ k B can be described as the monomorphism mon , . In particular, Λ-modules are monic. this paper, we find similar relation between semi-Gorenstein-projective and -modules, via monic modules, namely, ⊥ ∩ Using this, it is proved that if weakly Gorenstein, then Gorenstein only each monic; Q with a finite acyclic quive...

Let R be a right GF-closed ring with finite left and right Gorenstein global dimension. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorensntein flat dimensnion of R/I as a right R-module and the Gorensntein injective dimensnnion of R/I as a left R-module are identical. In particular, we show that for a simple module S over a commutative Gorensntein ring R, ...