Let A be a graded algebra. In this paper we develop a generalized Koszul theory by assuming that A_0 is self-injective instead of semisimple and generalize many classical results. The application of this generalized theory to directed categories and finite EI categories is described.

Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FI_G and VI_q, and answer positively an open question of Nagpal on representation stability theory.

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.

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.

We show that if X is a cocompact G-CW-complex such that each isotropy subgroup Gσ is L(2)-good over an arbitrary commutative ring k, then X satisfies some fixed-point formula which is an L(2)-analogue of Brown’s formula in 1982. Using this result we present a fixed point formula for a cocompact proper G-CW-complex which relates the equivariant L(2)-Euler characteristic of a fixed point CW-complex Xs and the Euler characteristic of X/G. As corollaries, we prove Atiyah’s theorem in 1976, Akita’s formula in 1999 and a result of Chatterji-Mislin in 2009. We also show that if X is a free G-CW-complex such that C∗(X) is chain homotopy equivalent to a chain complex of finitely generated projective Zπ1(X)-modules of finite length and X satisfies some fixed-point formula over Q or C which is an L(2)-analogue of Brown’s formula, then χ(X/G)=χ(2)(X). As an application, we prove that the weak Bass conjecture holds for any finitely presented group G satisfying the following condition: for any finitely dominated CW-complex Y with π1(Y )=G, Y satisfies some fixed-point formula over Q or C which is an L(2)-analogue of Brown’s formula.