The seminar meets on Wednesdays, 15:10-16:30, in Math -101

This Week

Leonid Positselski (University of Haifa)

The structure of flat modules and the module structure of flat algebras over commutative rings

The Govorov-Lazard theorem describes flat modules over any
associative ring as filtered direct limits of finitely generated free modules,
but a more informative description may be desirable. I will explain how
to obtain the flat modules over a Noetherian commutative ring which has
either Krull dimension 1, or an arbitrary Krull dimension but countable
spectrum, as the direct summands of transfinitely iterated extensions of
localizations of the ring with respect to countable multiplicative subsets.
An even more precise description, with localizations of the ring at its
elements (= multiplicative subsets generated by one element), is obtained
for the underlying modules of flat finitely presented commutative algebras
over arbitrary commutative rings. I will start with a discussion of
complete cotorsion pairs in module categories and proceed to formulate
the above mentioned results and explain some of the ideas behind their
proofs. This talk is based on the speaker’s joint work with Alexander Slavik.

A singular modulus is the j-invariant of an elliptic curve with complex multiplication. André (1998) proved that a polynomial equation F(x,y)=0 can have only finitely many solutions in singular moduli (x,y), unless the polynomial F(x,y) is “special” in a certain precisely defined sense. Pila (2011) extended this to equations in many variables, proving the André-Oort conjecture on C^n. The arguments of André and Pila were non-effective (used Siegel-Brauer).

I will report on a recent work by Allombert, Faye, Kühne, Luca, Masser, Pizarro, Riffaut, Zannier and myself about partial effectivization of these results.

Classically in homotopy theory, infinite loop spaces are recognized as spaces with an additional structure: grouplike $E_{\infty}$-spaces. The category of such spaces is equivalent to the category of connective spectra. Replacing topological spaces with smooth schemes, we end up in the realm of motivic homotopy theory, where an analogous statement was sought for since the theory has appeared. In this talk we will discuss the motivic recognition principle, which provides an equivalence between the category of motivic connective spectra and the category of grouplike motivic spaces with so called framed transfers.

This is joint work with Elden Elmanto, Marc Hoyois, Adeel Khan and Vladimir Sosnilo.

The Govorov-Lazard theorem describes flat modules over any
associative ring as filtered direct limits of finitely generated free modules,
but a more informative description may be desirable. I will explain how
to obtain the flat modules over a Noetherian commutative ring which has
either Krull dimension 1, or an arbitrary Krull dimension but countable
spectrum, as the direct summands of transfinitely iterated extensions of
localizations of the ring with respect to countable multiplicative subsets.
An even more precise description, with localizations of the ring at its
elements (= multiplicative subsets generated by one element), is obtained
for the underlying modules of flat finitely presented commutative algebras
over arbitrary commutative rings. I will start with a discussion of
complete cotorsion pairs in module categories and proceed to formulate
the above mentioned results and explain some of the ideas behind their
proofs. This talk is based on the speaker’s joint work with Alexander Slavik.