Finiteness problems in arithmetic deformation theory (08rit135)

Objectives

We plan to study a basic finiteness problem concerning deformations of complexes of modules for a a finitely generated profinite group. In a 2005 paper in the Annales de l\'Institut Fourier we showed how to generalize the deformation theory of Mazur and Schlessinger to the derived category of complexes of modules for the group. Such complexes arise naturally in arithmetic geometry, e.g. from the hypercohomology of complexes of \'etale sheaves.

Many important results in number theory amount to the computation of the universal deformation ring of a Galois module. A new problem arises in studying complexes of modules, however. This is to show that the universal deformation in question can be specified by a finite amount of linear algebra information with coefficients in the universal deformation ring.

This problem has an affirmative answer for modules, but for complexes we do not expect this to be so in general. We do conjecture, however, that it holds for complexes arising from arithmetic in a suitable sense.

Thus far we think we can show the finiteness property when the profinite group is the tame fundamental group of a regular local ring with finite residue field with respect to a divisor with strict normal crossings, but some of the details remain to be written. During the workshop we plan to work on finishing writing up this result and to investigate other arithmetic examples as well as potential counterexamples. If the conjecture survives close scrutiny, we believe it has the potential to become a central question in this area in the same way that other finiteness problems have motivated new developments in arithmetic geometry.

The Banff Research Center is an ideal environment in which to focus on this work together, away from the duties and obligations associated with our different home institutions. This proposal also responds to the objective of ensuring the representation of women stated as the last suggestion for workshop proposals, in that Frauke Bleher is a female faculty member at the University of Iowa.