A non-commutative differential module approach to Alexander modules

Abstract

The theory of R. Crowell on derived modules is approached within the theory of non-commutative differential modules. We also seek analogies to the theory of cotangent complex from differentials in the commutative ring setting. Finally, we give examples motivated from the theory of Galois coverings of curves.

Galois action on homology of generalized Fermat Curves

“…The situation is similar with the pro-ℓ Burau representation, defined in [12]. We also in section 5.1.5 how we can pass from the Z s−1 ℓ -covers corresponding to generalized Fermat curves, to the Z ℓ -case corresponding to the pro-ℓ Burau representation, using the ideas of [13]. Section 3 is an introduction to Ihara's ideas on the study of the absolute Galois groups as a profinite braid [9], [10] following [11].…”

“…For a description of the Alexander module in terms of differentials in non-commutative algebras we refer to [13]. Notice that when the group…”

“…Let us now combine the two Crowell sequences together. For an explanation of these two combined sequences in terms of the "cotangent sequence" we refer to [13].…”

“…Let Fk,s−1 = F k,s−1 ⊗ SpecQ Spec Q. Consider also the inverse limit We now compare the Crowell sequences for the cyclic covers and the Fermat covers, following [13]…”

“…In section 4.1 we compute the Alexander module for the generalized Fermat curves, while section 5 is devoted to the Z s−1 ℓ cover of the projective line, seen as a limit of F k,s−1 -curves and the relation to the Tate modules of them. Finally we consider the passage to the Burau representation by comparing the corresponding Crowell sequences, in terms of the viewpoint developed in [13].…”