A toric face ring, which generalizes both Stanley-Reisner rings and affine semigroup rings, is studied by Bruns, Römer and their coauthors recently. In this paper, under the “normality” assumption, we describe a dualizing complex of a toric face ring R in a very concise way. Since R is not a graded ring in general, the proof is not straightforward. We also develop the squarefree module theory o...

we investigate the relative cohomology and relative homology theories of $f$-gorenstein modules, consider the relations between classical and $f$-gorenstein (co)homology theories.

in this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $z_{2}$. we also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.

Motivated by their impact on homological algebra, the change of rings results have been the subject of several interesting works in Gorenstein homological algebra over Noetherian rings. In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, sec...

There is a well-known equivalence between the homotopy types of connected CW-spaces X with πnX=0 for n 6= 1, 2 and the quasi-isomorphism classes of crossed modules ∂ : M → P [16]. When the homotopy groups π1X and π2X are finite one can represent the homotopy type of X by a crossed module in which M and P are finite groups. We define the order of such a crossed module to be |∂| = |M | × |P |, an...

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.

We investigate the relative cohomology and relative homology theories of $F$-Gorenstein modules, consider the relations between classical and $F$-Gorenstein (co)homology theories.