On the derived invariance of cohomology theories for coalgebras

Abstract

We study the derived invariance of the cohomology theories $Hoc{h}^{\ast }$, $H^{\ast }$ and $H{C}^{\ast }$ associated to coalgebras over a field. We prove a theorem characterizing derived equivalences. As particular cases, it describes the two following situations: 1) $f:C\to D$ a quasi-isomorphism of differential graded coalgebras, 2) the existence of a "cotilting" bicomodule ${}_C{T}_D$. In these two cases we construct a derived-Morita equivalence context, and consequently we obtain isomorphisms $Hoch^*(C)\cong\Hoch^*(D)$ and $H^{\ast }(C)\cong H^{\ast }(D)$. Moreover, when we have a coassociative map inducing an isomorphism $H^{\ast }(C)\cong H^{\ast }(D)$ (for example when there is a quasi-isomorphism $f:C\to D$), we prove that $H{C}^{\ast }(C)\cong H{C}^{\ast }(D)$.

21 pages