Title: Periodicity in Cyclic Cohomology and Monodromy at Archimedean Infinity
Speaker: Abhishek Banerjee
Speaker Info: Johns Hopkins University
The cohomology of the "fibre at infinity" of an arithmetic variety can be computed by means of a complex first introduced by Consani. At archimedean infinity, this complex replaces Steenbrink’s complex for the cohomology of the universal fibre of a degeneration over a disc. The nearby cycles complex associated to this degeneration carries a monodromy operator $N$ and we can show that the graded pieces of the filtration on the cohomology of the nearby cycles complex by $Ker(N^j)$, $j\geq 0$, are isomorphic to the cyclic homology of a sheaf of differential operators (using some results of Wodzicki). Further, we can show that, under this isomorphism, the periodicity operator in cyclic homology coincides with the (logarithm of) the monodromy on the nearby cycles complex.Date: Tuesday, January 27, 2009
In this talk, we will do the same at archimedean infinity, where we have to work with "global sections" rather than with sheaves, and therefore show that there is a natural map from the cyclic cohomology of the ring of differential operators to the graded pieces of a filtration on the cohomology of the fibre at infinity, and that in this framework, the periodicity operator in cyclic cohomology is again the counterpart of the monodromy operator on Consani’s complex. The switch between cyclic homology and cohomology is a consequence of the fact that the monodromy operators on the nearby cycles complex and on Steenbrink’s complex are equal only upto homotopy. This is followed up by defining a complex with monodromy that plays the role of a nearby cycles complex for the fibre at infinity. Again, the monodromy operator on the latter is equivalent to the monodromy on Consani’s complex upto homotopy.
Finally, we consider the long exact sequence of Connes and Karoubi involving the algebraic, topological and relative $K$-theories of a Frechet Algebra. This long exact sequence lies above the periodicity sequence in cyclic homology. In this talk, we will construct the same sequence for the $K$ theories of the sheaf of differential operators, using the cohomologies of simplicial sheaves as defined by Brown and Gersten. This long exact sequence then lies above the periodicity sequence of cyclic (hyper)cohomologies.