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

## This Week

### 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.

## Spring 2018 meetings

### Upcoming Meetings

Date
Title
Speaker
Abstract
Apr 25 TBA Tomer Schlank (Hebrew U. (Jerusalem))
May 30 TBA Ilya Tyomkin (BGU)

### Past Meetings

Date
Title
Speaker
Abstract
Mar 7 Equations with singular moduli: effective aspects Yuri Bilu (University of Bordeaux)

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.

Mar 14 Infinite loop spaces in motivic homotopy theory Maria Yakerson (University of Duisburg-Essen (Essen))

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.

Mar 21 The structure of flat modules and the module structure of flat algebras over commutative rings Leonid Positselski (University of Haifa)

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.

Seminar run by Dr Ishai Dan-Cohen