We study the coinduction functor on the category of FI-modules and its variants. Using the coinduction functor, we give new and simpler proofs of (generalizations of) various results on homological properties of FI-modules. We also prove that any finitely generated projective VI-module over a field of characteristic 0 is injective.

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.

Earlier, for an action of a finite group$G$on a germ of an analytic variety, an equivariant$G$-Poincaré series of a multi-index filtration in the ring of germs of functions on the variety was defined as an element of the Grothendieck ring of$G$-sets with an additional structure. We discuss to which extent the$G$-Poincaré series of a filtration defined by a set of curve or divisorial valuations on the ring of germs of analytic functions in two variables determines the (equivariant) topology of the curve or of the set of divisors.

Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about the artinianness of some special local cohomology modules in the graded case.

Let$X$⊂$V$be a closed embedding, with$V$∖$X$nonsingular. We define a constructible function$ψ$_{$X$,$V$}on$X$, agreeing with Verdier’s specialization of the constant function$1$_{$V$}when$X$is the zero-locus of a function on$V$. Our definition is given in terms of an embedded resolution of$X$; the independence of the choice of resolution is obtained as a consequence of the weak factorization theorem of Abramovich–Karu–Matsuki–Włodarczyk. The main property of$ψ$_{$X$,$V$}is a compatibility with the specialization of the Chern class of the complement$V$∖$X$. With the definition adopted here, this is an easy consequence of standard intersection theory. It recovers Verdier’s result when$X$is the zero-locus of a function on$V$.