In the first part of this paper, we study Koszul property of directed graded categories. In the second part of this paper, we prove a general criterion for an infinite directed category to be Koszul. We show that infinite directed categories in the theory of representation stability are Koszul over a field of characteristic zero

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.

We extend our family rigidity and vanishing theorems in [{\bf LiuMaZ}] to the Spin^ c case. In particular, we prove a K-theory version of the main results of [{\bf H}],[{\bf Liu1}, Theorem B] for a family of almost complex manifolds.

In [W], Witten considered the indices of elliptic operators on the free loop space CM of a spin manifold M. In particular the index of the formal signature operator on the loop space is exactly the elliptic genus of Landweber-Stong-Ochanine [LS],[O]. Motivated by physics, Witten made the conjectures about the rigidity of these elliptic operators which say that their 51-equivariant indices on M are independent of g G Sl. See [L] for the early history of the subject.

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.