An artin algebra is called CM-free provided that all its finitely generated Gorenstein projective modules are projective. We show that a connected artin algebra with radical square zero is either self-injective or CM-free. As a consequence, we prove that a connected artin algebra with radical square zero is Gorenstein if and only if its valued quiver is either an oriented cycle with the trivial...

The Gorenstein projective modules are proved to form a precovering class in the module category of a ring which has a dualizing complex. 0. Introduction This paper proves over a wide class of rings that the Gorenstein projective modules form a precovering class in the module category. Let me explain this statement. There are two terms of mystery, “Gorenstein projective modules” and “precovering...

In 1966, Auslander introduced the notion of the G-dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of G-dime...

We show that there exists a positive real number $\delta>0$ such for any normal quasi-projective $\mathbb{Q}$-Gorenstein $3$-fold $X$, if $X$ has worse than canonical singularities, is, the minimal log discrepancy of is less $1$, then not greater $1-\delta$. As applications, we set all non-canonical klt Calabi-Yau $3$-folds are bounded modulo flops, and global indices from above.

‎let $mathcal a$ and $mathcal b$ be unital rings‎, ‎and $mathcal m$‎ ‎be an $(mathcal a‎, ‎mathcal b)$-bimodule‎, ‎which is faithful as a‎ ‎left $mathcal a$-module and also as a right $mathcal b$-module‎. ‎let ${mathcal u}=mbox{rm tri}(mathcal a‎, ‎mathcal m‎, ‎mathcal‎ ‎b)$ be the triangular ring and ${mathcal z}({mathcal u})$ its‎ ‎center‎. ‎assume that $f:{mathcal u}rightarrow{mathcal u}$ is...

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 ...

Let [Formula: see text] be an extriangulated category with a proper class of text]-triangles. We study complete cohomology objects in by applying text]-projective resolutions and text]-injective coresolutions constructed text]. Vanishing detects finite dimension dimension. As consequence, we obtain some criteria for the validity Wakamatsu tilting conjecture give necessary sufficient condition v...

Let $A$ be a right coherent ring and $\mathcal{X}$ contravariantly finite subcategory of ${\rm{mod}}\mbox{-}A$ containing projectives. In this paper, we construct recollement abelian categories $({\rm{mod}}_{0}\mbox{-}\mathcal{X}, {\rm{mod}}\mbox{-}\mathcal{X}, {\rm{mod}}\mbox{-}A)$, where ${\rm{mod}}_{0}\mbox{-}\mathcal{X}$ is full ${\rm{mod}}\mbox{-}\mathcal{X}$ consisting all functors vanish...

In this paper, we introduce and investigate the notions of ξ-strongly copure projective objects in a triangulated category. This extends Asadollahi’s notion of ξ-Gorenstein projective objects. Then we study the ξ-strongly copure projective dimension and investigate the existence of ξ-strongly copure projective precover.