Saul Lubkin

Professor of Mathematics

Office: Hylan 705
Office Hours: M 10:30-12:15am, W 10:30-11:15am or by appointment
Phone: (585) 733-3537
Fax: (585) 244-6631
E-mail: lubkin@math.rochester.edu

Courses

Mth162, Calculus IIA, MW 12:30-1:45pm, Hylan 201

Course home page for Mth162, Fall, 2020

Mth142, Calculu I, MW 9:00-10:15, Hylan 201

Course home page for Mth142, Fall, 2020

Research Interests

Algebraic Geometry

Homological Algebra

Commutative Algebra

Algebraic Topology

Links to some useful papers

On Several Points of Homological Algebra, by Alexandre Grothendieck

Algebraic Coherent Sheaves, by Jean Pierre Serre

A well-written paper about sheaves

An addendum to Kelley's "General Topology" written by myself that constructs the completion of a uniform space directly without using metrization theorems.

Imbedding of Abelian Categories, by Saul Lubkin: A paper that reduces proving many theorems in abelian categories to the case of the category of abelian groups.

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.

Generalization of p-adic cohomology; bounded Witt vectors. A canonical lifting of a variety in charecteristic p back to characteristic zero, by Saul Lubkin.