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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 …