# Saul Lubkin

## Professor of Mathematics

Office: Hylan 705 MW, 11:45am-1:00PM, via my personal zoom: rochester.zoom.us/my/slubkin, passcode 2486, and also in person in my office, Hylan 705. (585) 733-3537 (585) 244-6631 lubkin@math.rochester.edu

## Courses

Mth165, Differential Equations with Linear Algebra, in person, MW 9:00-10:15, Meliora 221.

Mth143, Calculus III, entirely on-line, MW 10:30-11:45, via zoom, at rochester.zoom.us/my/slubkin, passcode 2486.

## Research Interests

Algebraic Geometry

Homological Algebra

Commutative Algebra

Algebraic Topology

## Current Research

The paper below constructs a $p$-adic cohomology theory on algebraic varieties in characteristic $p$, that appears to be superior to all other $p$-adic cohomology theories so far constructed (For example, for Euclidean space of dimension $n$ over a ring $A$ in characteristic $p$, it is isomorphic to EXACTLY the de Rham cohomology of Euclidean space of dimension $n$ over $W_(A)$: there is no extra torsion, and, a foriori, no topological torsion. $W_(A)$ is a subring of $W(A)$ that I call the bounded Witt vectors on $A$.). I am now approaching a different method of perfecting p-adic cohology -- to have exactly the right torsion. The above method works for Euclidean space, but is very difficult to compute on other finite algebraic varieties of finite type over a field of characteristic p. My new approach: Given a finitely generated algebra A over the ring of p-adic integers $B=\bf Z_p^ (or more generally over any p-adic algebra B), let A! be the A-algebra formed by dividing powers at the ideal$pA$. Then we have the usual A-module D_B(A) of differntials of A over B. We define the module D!_B(A) of divided power differntials of A over B similarly, to be the quotient module of this A-module by dividing by the submodule generated by elements of the form d(x^{} - x^{} (where, for every element x in pA, x^{} = "x^n/n!", if the ring A is free of p-torsion; defn slightly more complicated is A has non-zeo torsion). Taking the usual exterior powers of this module D!_B(A) gives a skew symmetric differential graded algebra, call it \Gamma^*_B(A). Take the dagger completion of this differential graded algebra, which is \Gamma^*_B(A)\dagger. The cohomology of this cochain complex is the "divided power dagger cohomology" of$A$over$B\$. This construction can also be made by universal mapping properties -- it is quite natural. For Euclidean space, A=B[T_1,...,T_n], this cohomology theory gives exactly the same results as the usual cohomology of the usual exterior algebra of the module of differentials of A over B -- the de Rham cohology. And this construction is much easier to compute than the Bounded Witt cohomology that I have discussed above.