Consider a graded ideal in the polynomial ring in several variables. We shall discuss criterion for the graded ideal and its power to have linear resolution. Then we focus our attention to study linear resolution of monomial ideals. Monomial ideals are the bridge between commutative algebra and the combinatorics. Monomial ideals are also significant because they appear as initial ideals of arbitrary ideals. Since many properties of an initial ideal are inherited by its original ideal, one often adopt this strategy to decipher properties of general ideals. The first talk is meant for covering the preliminary results on resolution and regularity of monomial ideal.The aim of this series of talk is to present the result in arXiv:1709.05055 https://arxiv.org/abs/1709.05055>.

We'll study free resolutions of monomial ideals via the notion of a labelled simplicial complex. We derive a criterion due to Bayer, Peeva and Sturmels for a labelled simplicial complex to define a free resolution. As consequences, we show that the Koszul complex is exact and prove the Hilbert syzygy theorem.

We will continue with the material in Chapter 1 in Cassels and Frohlich.

Let S be a Noetherian ring, and R = S/I. It is always possible to construct a differential graded algebra (DG-algebra) resolution of R over S due to a result of Tate. If R is the residue field of S, then Gulliksen proved that such a DG-algebra resolution is minimal. We shall discuss the construction of the Tate resolution in our talk.

Huneke and Itoh independently proved a celebrated result on integral closure of powers of an ideal generated by a regular sequence. As a consequence of this theorem, one can find the Hilbert-Samuel polynomial of the integral closure filtration of I if the normal reduction number is at most 2. We prove Hong and Ulrich's version of the intersection theorem.

We will define special singularities of algebraic or analytic varieties called rational singularities introduced my M. Artin. After discussing some equivalent criterion for rationality we will give many naturally occuring examples. Next, we will describe the results of Artin in dimension 2. Important results due to Brieskorn, Lipman, Mumford, Tjurina will be mentioned. If time permits some results by Spivakovsky, Le dung Trang-M. Tosun, Gurjar-Wagh will be mentioned. No proofs wil be given.

We will continue with the material in Chapter 1 in Cassels and Frohlich.

This talk will be introduction to syzygies: basic theorems, examples and some early applications. This is the first seminar on this topic.

Let S be a Noetherian ring, and R = S/I. It is always possible to construct a differential graded algebra (DG-algebra) resolution of R over S due to a result of Tate. If R is the residue field of S, then Gulliksen proved that such a DG-algebra resolution is minimal. We shall discuss the construction of the Tate resolution in our talk.

In these talks, we will see relations between Hilbert series of a module and its graded Betti numbers. This gives relations between the graded Betti numbers of a modules which are known as Herzog-Kuhl equations. As an application, we show that the property of R being Cohen-Macaulay is characterized by the existence of a pure Cohen-Macaulay R-module of finite projective dimension.