A generalization of Lichtenbaum's theorem on the cohomological dimension of algebraic varieties (Q2277038)

Let X be a separated n-dimensional scheme of finite type over a field. The cohomological dimension of X is the largest integer i such that the i-th cohomology group of X with coefficients in some quasicoherent sheaf is non-zero. Lichtenbaum's theorem says that the cohomological dimension of X is less than n if and only if every n-dimensional irreducible component of X is non-proper. Let Y be a locally closed subscheme of X. We define the cohomological dimension of X with respect to Y to be the largest integer i such that the i-th local cohomology group of X with support in Y and with coefficients in some quasicoherent sheaf on X is non-zero. The goal of this paper is to prove that the cohomological dimension of X with respect to Y is less than n if and only if every connected component of the preimage of Y in every n-dimensional irreducible component of the normalization of \(X_{red}\) is non-proper.