Cofiniteness

Abstract: Let X be a scheme of finite type over a field k. The cohomological dimension of X is the smallest integer n > 0 such that H'(X, F) = 0 for all i > n, and for all quasi-coherent sheaves F on X. There are two well-known theorems about the cohomological dimension of X. Serre's theorem states that X is affine if and only if its cohomological dimension is zero. Lichtenbaum's theorem states that if X is irreducible of dimension d, then its cohomological dimension is equal to d (the largest possible) if and only if X is proper over k. The purpose of this paper is to study situations which lie in between these two extremes. In particular, we would like to find the cohomogical dimension of projective n-space minus a closed subvariety. Our main results are the following. 1. A local vanishing theorem (3.1). This says that if A is a local ring of dimension n, and J is an ideal, such that the variety of J, V(J), meets every formal branch of Spec A in a subset of dimension > 1, then Hj3(M) = 0 for all A-modules M. This is a local analogue of Lichtenbaum's theorem, and as a corollary, it gives a new proof of Lichtenbaum's theorem. It also gives a new proof of a theorem of Nagata, which says that a normal affine surface, minus a closed subset of pure codimension one, is again affine. 2. A theorem on meromorphic functions (6.8). Let X be a closed subset of a non-singular proper scheme Z over k. Assume that X is a local complete

Abstract: We investigate trace functions of modules for vertex operator algebras (VOA) satisfying C2 -cofiniteness. For the modular invariance property, Zhu assumed two conditions in [Z]: (1) A(V) is semisimple and (2) C2 -cofiniteness. We show that C2 -cofiniteness is enough to prove a modular invariance property. For example, if a VOA V= ⊕ m=0 ∞ V m is C2 -cofinite, then the space spanned by generalized characters (pseudotrace functions of the vacuum element) of V -modules is a finite-dimensional $\SL_2(\mathbb{Z})$ SL 2 ( Z) -invariant space and the central charge and conformal weights are all rational numbers. Namely, we show that C2 -cofiniteness implies "rational conformal field theory" in a sense as expected in Gaberdiel and Neitzke [GN]. Viewing a trace map as a symmetric linear map and using a result of symmetric algebras, we introduce "pseudotraces" and pseudotrace functions and then show that the space spanned by such pseudotrace functions has a modular invariance property. We also show that C2 -cofiniteness is equivalent to the condition that every weak module is an N -graded weak module that is a direct sum of generalized eigenspaces of L(0) .

Abstract: We demonstrate that, for vertex operator algebras of CFT type, C 2 -cofiniteness and rationality is equivalent to regularity. For C 2 -cofinite vertex operator algebras, we show that irreducible weak modules are ordinary modules and C 2 -cofinite, V + L is C 2 -cofinite, and the fusion rules are finite.