This article studies the algebraic structure of homology theories defined by
a left Hopf algebroid U over a possibly noncommutative base algebra A, such as for example
Hochschild, Lie algebroid (in particular Lie algebra and Poisson), or group and étale
groupoid (co)homology. Explicit formulae for the canonical Gerstenhaber algebra structure on
Ext_U(A, A) are given. The main technical result constructs a Lie derivative satisfying a generalised
Cartan–Rinehart homotopy formula whose essence is that Tor^U(M, A) becomes for
suitable right U-modules M a Batalin–Vilkovisky module over Ext_U(A, A), or in the words
of Nest, Tamarkin, Tsygan and others, that Ext_U(A, A) and Tor^U(M, A) form a differential
calculus. As an illustration, we show how the well-known operators from differential geometry
in the classical Cartan homotopy formula can be obtained. Another application consists in
generalising Ginzburg’s result that the cohomology ring of a Calabi–Yau algebra is a Batalin–
Vilkovisky algebra to twisted Calabi–Yau algebras
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