Extending higher derivations to rings and modules of quotients

A torsion theory is called differential (higher differential) if a derivation (higher derivation) can be extended from any module to the module of quotients corresponding to the torsion theory. We study conditions equivalent to higher differentiability of a torsion theory. It is known that the Lambek, Goldie and any perfect torsion theories are differential. We show that these classes of torsion theories are higher differential as well. Then, we study conditions under which a higher derivation extended to a right module of quotients extends also to a right module of quotients with respect to a larger torsion theory. Lastly, we define and study the symmetric version of higher differential torsion theories. We prove that the symmetric versions of the results on higher differential (one-sided) torsion theories hold for higher derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie and any perfect torsion theories are higher differential

Absolute torsion and eta-invariant

In a recent joint work with V. Turaev (cf. math.DG/9810114) we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion it has the advantage of having a well-defined sign. Also, the absolute torsion is defined for arbitrary orientable flat vector bundles, and not only for unimodular ones, as is classical Reidemeister torsion. In this paper I show that the sign behavior of the absolute torsion, under a continuous deformation of the flat bundle, is determined by the eta-invariant and the Pontrjagin classes.Comment: 10 page