Dissertation Defense Seminar

In this thesis, we prove that algebraic de Rham cohomology as a functor defined on smooth F_p-algebras is formally \'etale in a precise sense. This result shows that given de Rham cohomology, one automatically obtains the theory of crystalline cohomology as its \textit{unique} functorial deformation. To prove this, we define and study the notion of a pointed {G}_a^{perf}-module and its refinement which we call a quasi-ideal in {G}_a^{perf} -- following Drinfeld's terminology. Our main constructions show that there is a way to ``unwind" any pointed {G}_a^{perf}-module and define a notion of a cohomology theory for algebraic varieties. We use this machine to redefine de Rham cohomology theory and deduce its formal \'etalness and a few other properties.

Shubhodip's advisor is Bhargav Bhatt. Speaker(s): Shubhodip Mondal (UM)