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
We prove that the cover ideals of all unimodular hypergraphs have the nonincreasing
depth function property. Furthermore, we show that the index of depth stability of
these ideals is bounded by the number of variables.
We develop a local cohomology theory for FI^m-modules, and show that it in many ways mimics the classical theory for multi-graded modules over a polynomial ring. In particular, we define an invariant of FI^m-modules using this local cohomology theory which closely resembles an invariant of multi-graded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.
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.