Invariance and Localization for Cyclic Homology of Dg Algebras

We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [19] [3] preserves cyclic homology. This completes results of Rickard’s [48] and Happel’s [18]. It also extends well known results on preservation of cyclic homology under Morita equivalence [10], [39], [25], [26], [41], [42]. We then show that under suitable flatness hypotheses, an exact sequence of derived categories of DG algebras yields a long exact sequence in cyclic homology. This may be viewed as an analogue of Thomason-Trobaugh’s [51] and Yao’s [58] localization theorems in K-theory (cf. also [55]). I am grateful to the referee for his careful reading of the manuscript. Summary This paper is concerned with cyclic homology of (unbounded, non-commutative) differential Z-graded algebras. The case of positively graded DG algebras was first considered by ViguéBurghelea [53] and T. Goodwillie [15]. We need the slightly more general setting to allow for the algebras appearing in Morita theory for derived categories. For simplicity, in this summary, we only state the results for the special case of ‘ordinary’ algebras. We point out however, that the range of possible applications is greatly enlarged if one admits general differential graded algebras. Let k be a commutative ring. In this summary, all k-algebras are assumed to be projective over k. Let A and B be k-algebras. Consider the full subcategory rep (A,B) of the derived category of A-B-bimodules formed by the bimodule complexes X which when restricted to B become quasiisomorphic to perfect complexes (i.e. finite complexes of finitely generated projective B-modules). Generalizing results of C. Kassel [25] [26] we show in (2.4) that each such complex X gives rise to a morphism in cyclic homology HC∗ (X) : HC∗(A) → HC∗(B). This morphism is functorial in the sense that if we view A as an A-A-bimodule complex, then HC∗(A) = 1 and if Y ∈ rep (B,C) then HC∗(X ⊗LB Y ) = HC∗(Y ) ◦ HC∗(X). This implies in particular that HC∗ is an invariant for Morita equivalence of derived categories [48] [49], that is, if the derived functor ?⊗LA X : DA→ DB is an equivalence, then HC∗(X) is invertible. Moreover, we show that HC∗(X) only depends on the class of X in the Grothendieck group of the triangulated category rep (A,B). These Grothendieck groups are naturally viewed as the morphism spaces of a category whose objects are all algebras. A K-theoretic equivalence is an isomorphism of this category. Thus, cyclic homology is invariant under K-theoretic equivalence. For example, a finite-dimensional algebra of finite global dimension over an algebraically closed field is K-theoretically equivalent to its largest semi-simple quotient (2.5). Thus, if k is an algebraically closed field, the cyclic homology of a finite-dimensional algebra A of finite global dimension only depends on the number of isomorphism classes of simple A-modules. This yields the ‘no loops To appear in Journal of Pure and Applied Algebra