# BCAM-Severo Ochoa Course: Introduction to multiplicity theory

Date: Tue, Apr 4 - Fri, May 12 2023

Hour: 15:30-17:30

Location: Seminar room at Mathematics department of UPV/EHU and Online

Speakers: Ilya Smirnov (BCAM-Ikerbasque)

**LINK FOR THE SESSION** HERE**DATES and TIME:**

Preliminaries: April 4 and 5, from 15:00 to 17:15

Main lectures: April 18, 19, 20, 21, 25, 26, 27 and 28, from 15:30 to 17:30

Extras: May 11 and 12 from 15:30 to 17:30

(total of 24 hours)**LOCATION:** Seminar room at Mathematics department of UPV/EHU and Online

Sessions on 4 and 5 April are preliminary and can be skipped by some.

The topics that will be studied:

1) the Hilbert polynomial of a graded ring

2) associated graded rings, the Hilbert polynomial of an ideal

3) consequence: the Krull height theorem and the main theorem of the dimension theory**ABSTRACT:**

The course concerns the notion of the multiplicity of a local ring which is a generalization of the multiplicity of a root of a polynomial in one variable to systems of polynomial equations in several variables. Geometrically, it is the appropriate notion of degree for a point on a variety: a geometric shape described by one or several polynomial equations. For instance, it gives the multiplicity of the intersection of two varieties at a point.

The multiplicity is a measure of singularity of a point on a variety: Nagata proved that points of multiplicity one are non-singular. It is perhaps the most widely-used measure of singularity and was extensively developed after its appearance in Hironaka's seminal work on the resolution of singularities. Despite its age the multiplicity theory is still has deep open prob-

lems, such as the Lech's conjecture was raised in 1960s and still resists (although see https://arxiv.org/abs/2005.02338 for a recent major advance) and Huneke's extension of Nagata's theorem is open in mixed characteristic (https://www.ams.org/journals/proc/1982-085-03/S0002-9939-1982-0656095-2/S0002-9939-1982-0656095-2.pdf). Another recent direction in

multiplicity theory is the study of Lech's inequality, the various means of improving it (for example, https://arxiv.org/abs/1711.06951), which connects to algebraic geometry in at least two ways: by providing a uniform convergence estimate in a recent work of Blum{Liu on the normalized volume of a singularity and by Mumford's restriction on the singularities appearing on limits of smooth varieties.**TOPICS:**

The goal of this course is to prove the equivalence of several different-looking definitions and discuss basic properties of multiplicity.

1. Hilbert polynomials of graded rings.

2. Samuel's definition of multiplicity and its basic properties.

3. Superficial elements. Existence of parameter reductions.

4. Koszul complex. Serre's definition of multiplicity as the Euler characteristic.

5. Further properties of the multiplicity of parameter ideals: associativity formula, Lech's for-

mula, etc.

6. Integral closure and reductions.

Literature: Many of these results can be found in the book of Huneke and Swanson Integral closure of ideals rings and modules" which is freely available online. For example, it contains the basics of multiplicity theory, the connection between multiplicity and the integral closure of ideals, and the theory of superificial elements. Serre Local algebra" contains his definition and the background leading to it, this is also covered in the book Cohen{Macaulay rings" by Bruns and Herzog. Many aspects of multiplicity theory (especially for parameter ideals) are covered in Nagata's book Local rings", but it is unfortunately hard to read and find.

Background: The background for this class is covered by a solid knowledge of the Atiyah MacDonald book (Chapters 1-10 with some omissions). Most important topics are the following:

- Rings, ideals, modules. Artinian and Noetherian properties. Composition series and length

of an Artinian and Noetherian module.

- Associated primes and prime filtrations.

- Krull's dimension. Height of ideals and Krull's height theorem.

- The Artin{Rees lemma.

It might be also helpful to know some idea about integral ring extensions and of the completion of the local ring. These two topics are covered in the Atiyah{MacDonald book as well. The two preliminary lectures on April 4 and 5 will include a discussion of Krull's height theorem and the Artin-Rees lemma.***Registration is free, but mandatory before 12 April 2023. **To sign-up go to form link and fill the registration form

## Organizers:

BCAM

## Confirmed speakers:

Ilya Smirnov (BCAM-Ikerbasque)

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