We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.

In this paper we give an explicit algorithm to construct the ordinary quiver of a finite EI category for which the endomorphism groups of all objects have orders invertible in the field k. We classify all finite EI categories with hereditary category algebras, characterizing them as free EI categories (in a sense which we define) for which all endomorphism groups of objects have invertible orders. Some applications on the representation types of finite EI categories are derived.

For any unital separable simple infinite-dimensional nuclear$C$^{∗}-algebra with finitely many extremal traces, we prove that $$ \mathcal{Z} $$ -absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear$C$^{∗}-algebra with tracial rank zero is approximately divisible, and hence is $$ \mathcal{Z} $$ -absorbing.

Let $\Lambda$ be a finite dimensional algebra and $G$ be a finite group whose elements act on $\Lambda$ as algebra automorphisms. Under the assumption that $\Lambda$ has a complete set $E$ of primitive orthogonal idempotents, closed under the action of a Sylow $p$-subgroup $S \leqslant G$, we show that the skew group algebra $\Lambda G$ and $\Lambda$ have the same finitistic dimension and the same strong global dimension if the action of $S$ on $E$ is free. Using a homological characterization of piecewise hereditary algebras proved by Happel and Zacharia, we deduce that $\Lambda G$ is piecewise hereditary if and only if $S$ acts freely on $E$ and $\Lambda$ is piecewise hereditary as well.

Starting with a commutative ring $R$ and an ideal $I$ , it is possible to define a family of rings $R(I)_{a,b}$ , with $a,b \in R$ , as quotients of the Rees algebra $\oplus_{n \geq0} I^{n}t^{n}$ ; among the rings appearing in this family we find Nagata’s idealization and amalgamated duplication. Many properties of these rings depend only on $R$ and $I$ and not on $a$ , $b$ ; in this paper we show that the Gorenstein and the almost Gorenstein properties are independent of $a$ , $b$ . More precisely, we characterize when the rings in the family are Gorenstein, complete intersection, or almost Gorenstein and we find a formula for the type.