Local cohomology for non-commutative graded algebras

Abstract

We generalize the theory of local cohomology and local duality to a large class of non-commutative N‐graded noetherian algebras; specifically, to any algebra, B, that can be obtained as graded quotient of some noetherian AS‐Gorenstein algebra, A.

As an application, we generalize three “classical” commutative results. For any graded module M over B we have the Bass-numbers u_i (M) = dim_k Extⁱ _b(k, M), and we can then prove that for M finitely generated, we have

• id(M) =sup{i|uⁱ(M)≠0};

• the Bass-theorem: if id(M) < ∞, then id(M) = depth(B);

• the “No Holes”-theorem: if depth(M) ≤i≤(M),

then μⁱ (M) ≠ 0,

where id(M) is M's injective dimension as an object in the category of graded modules, while depth(M) is the smallest i such that Extⁱ _B(k, M) ≠ 0. As a further application, we also generalize a non‐vanishing result for local cohomology. It states that if M is a finitely generated graded B‐ module, then

Here is the i'th local cohomology‐module of M. To prove this result, we need the AS‐Gorenstein algebra, A, of which B is a quotient, to satisfy the so-called Similar Submodule Condition, SSC, defined in [11] (for instance, A could be PI).

Keywords:

• Non-Commutative graded algebras
• Local cohomology
• local duality
• Bass -numbers