A note on minimal Galois embeddings of abelian surfaces
We extend the definition of Hochschild and cyclic homologies of a scheme over a commutative ring k to define the Hochschild homologies HH⁎(X/S) and cyclic homologies HC⁎(X/S) of a scheme X with respect to an arbitrary base scheme S. Our main purpose is to study product structures on the Hochschild homology groups HH⁎(X/S). In particular, we show that carries the structure of a graded algebra.
A concept of a slice of a semisimple derivation is introduced. Moreover, it is shown that a semisimple derivation d of a finitely generated commutative algebra A over an algebraically closed field of characteristic 0 is nothing other than an algebraic action of a torus on Max(A), and, using this, that in some cases the derivation d is linearizable or admits a maximal invariant ideal.
We give a shorter proof to a recent result by Neuberger [Rocky Mountain J. Math. 36 (2006)], in the real case. Our result is essentially an application of the global asymptotic stability Jacobian Conjecture. We also extend some of the results of Neuberger's paper.