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Summer Graduate School

Combinatorial and DG-Algebra Techniques for Free Resolutions (Tianjin, China) June 08, 2020 - June 19, 2020
Parent Program: --
Location: Chern Institute of Mathematics, Nankai University, Tianjin, China
Organizers Chengming Bai (Chern Institute of Mathematics), LEAD Dave Bayer (Barnard College), Claudia Miller (Syracuse University)
Description
2020 sgs combinatorial and dg algebra techniques nankai proposal bayer.2019.01.06 %285%29 page 1
A cellular resolution of the real projective plan

The two topics, combinatorial theory of free resolutions and differential graded algebra techniques in homological algebra, each have a long and rich history in commutative algebra and its applications to algebraic geometry. Free resolutions are at the center of much of the study in the field and these two approaches give powerful tools for their study and their application to other problems. Neither of these topics is generally covered in graduate courses. Furthermore, recent developments have exhibited exciting interplay between the two subjects. The purpose of the school is to introduce the graduate students to these subjects and these new developments. The school will consist of two lectures each day and carefully planned problem sessions designed to reinforce the foundational material and to give them a chance to experiment with problems involving the interplay between the two subjects.

 

Suggested Prerequisites

I. Some commutative algebra, including prime ideals, graded rings, regular sequences, Hilbert functions, and the Koszul complex.

 

This material can be found in any of the following books:

1) Matsumura "Commutative Ring Theory" (Cambridge University Press)

Chapter 2, Section 4 (localization, Spec)

Chapter 5, beginning of Section 13 (graded rings)

Chapter 6, Section 16 (regular sequences)

 

2) Atiyah, MacDonald "Introduction to Commutative Algebra" (Addison-Wesley)

Chapter 10 (graded rings)

Chapter 11 (Hilbert functions)

 

3) Bruns, Herzog "Cohen-Macaulay Rings" (Cambridge University Press)

Section 1.1 (regular sequences)

Section 1.6 (Koszul complex)

 

4) Eisenbud “Commutative Algebra with a View toward Algebraic Geometry” (Springer)

Section 1.5 (graded rings)

Section 1.9 (Hilbert functions)

Sections 17.1 and 17.2 (Koszul complex)

 

II. Basic homological algebra: exact sequences, free and projective resolutions, Tor and Ext, comparison theorem, quasi-isomorphism, homotopy, long exact sequences of homology.

 

This material can be found in the following book:

1) Weibel “An Introduction to Homological Algebra" (Cambridge University Press)

Section 1.1 (complexes, chain maps, quasi-isomorphisms) Theorem 1.3.1 (long exact sequences)

Section 1.4 (homotopies)

Section 2.2 (projective resolutions)

Section 2.3, just the beginning (injective resolutions)

Section 2.5, just definition of derived functors and Tor and Ext (Tor and Ext).

 

III. In addition, students who wish to get a head start understanding combinatorial free resolutions are encouraged to look at the following book:

 

1) Miller, Sturmfels “Combinatorial Commutative Algebra” (Springer)

 

Your University library may have a SpringerLink subscription providing online access to PDF editions of these Springer books.

 

For eligibility and how to apply, see the Summer Graduate Schools homepage

Due to the small number of students supported by MSRI, only one student per institution will be funded by MSRI.