Abstract. For every finite-to-one map λ : Γ → Γ and for every abelian group K, the generalized shift σ λ of the direct sum Γ K is the endomorphism defined by (x i ) i∈Γ → (x λ(i) ) i∈Γ [3]. In this paper we analyze and compute the algebraic entropy of a generalized shift, which turns out to depend on the cardinality of K, but mainly on the function λ. We give many examples showing that the generalized shifts provide a very useful universal tool for producing counter-examples.We denote by Z, P, and N respectively the set of integers, the set of primes, and the set of natural numbers; moreover N 0 = N ∪ {0}. For a set Γ, P fin (Γ) denotes the family of all finite subsets of Γ. For a set Λ and an abelian group G we denote by G Λ the direct product i∈Λ G i , and by G (Λ) the direct sum i∈Λ G i , where all G i = G. For a set X, n ∈ N, and a function f : X → X let Per(f ) be the set of all periodic points and Per n (f ) the set of all periodic points of order at most n of f in X.
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
We present in the context of Gorenstein homological algebra the notion of a "G-Gorenstein complex" as the counterpart of the classical notion of a Gorenstein complex. In particular, we investigate equivalences between the category of G-Gorenstein complexes of fixed dimension and the G-class of modules.
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