Canonical modules of complexes

Abstract

We define the notion of the canonical module of a complex. We then consider Serre's conditions for a complex and study their relationship to the local cohomology of the canonical module and its ring of endomorphisms. M : . . .The derived category is triangulated, the suspension functor Σ being defined by the formulas (ΣM) n = M n−1 and d ΣM n = −d n−1 . The symbol "≃" is reserved for isomorphisms in D(R). We use the subscripts "b", "+" and "−" to denote the homological boundness, the homological boundness from below and the homological boundness from above, respectively. The superscript "f " denotes the homological finiteness. So the full subcategory of D(R) consisting of complexes with finitely generated homology modules is denoted by D f (R). As usual, we identify the category of R-modules as the full subcategory of D(R) of complexes M satisfying H i (M) = 0 for i = 0. For a complex M ∈ D(R), by sup M and inf M, we mean its homological supremum and infimum. Let M