Schedule, Abstracts and Slides

Saturday, April 21, 2012 | |

10:30 – 10:50 | Registration |

10:50 – 11:00 | Opening Remarks |

11:00 – 11:40 | Alexandra Seceleanu Algebras with good nilpotent actions and possible homological applications |

11:50 – 12:30 | Ben Anderson [Slides] NAK for Ext and the Blindness of $M$ |

12:30 – 2:00 | Lunch |

2:00 – 2:20 | Ela Celikbas [Slides] Fiber Products and Connected Sums of Local Rings |

2:30 – 3:10 | Alessandro De Stefani Artinian level algebras of low socle degree |

3:20 – 3:40 | Zheng Yang [Slides] Decomposing a Gorenstein Artin ring as a Connected Sum |

3:50 – 4:10 | 20 Minute Break |

4:20 – 5:00 | Sarang Sane [Slides] Projective Modules! Keep your Witts about! $C$ how its done! |

5:10 – 5:30 | Arindam Banerjee Bounds on Castelnuovo-Mumford Regularity |

Sunday, April 22, 2012 | |

9:30 – 10:10 | Saeed Nasseh [Slides] Factorizations of local homomorphisms |

10:20 – 10:40 | Jack Jeffries Finite $F$-Representation Type and $F$-Signature |

10:50 – 11:30 | Brian Johnson [Slides] Graded rings |

11:40 – 12:00 | Billy Sanders Semidualizing Modules |

**Speaker:** Alexandra Seceleanu

**Title:** Algebras with good nilpotent actions and possible homological applications

**Abstract:** Every Artinian algebra A comes naturally equipped with a natural class of nilpotent (vector space) homomorphisms. I will describe various combinatorial invariants arising from such nilpotent actions. Finally I will try to convince the audience that properties of these invariants and the decompositions that give rise to them can become useful tools in studying the homological properties of A.

**Speaker:** Ben Anderson

**Title:** NAK for Ext and the Blindness of $M$.

**Abstract:** Let $\varphi\colon (R,\mathfrak{m},k)\to (S,\mathfrak{m}S,k)$ be a flat local ring homomorphism,

and let $M$ be a finitely generated $R$-module. The following are equivalent:

- $M$ has an $S$-module structure

compatible with its $R$-module structure; - $\operatorname{Ext}^i_R(S,M)=0$ for $i\geq 1$;
- $\operatorname{Ext}^i_R(S,M)$ is finitely generated over $R$ for $i=1,\ldots,\dim_R(M)$;
- $\operatorname{Ext}^i_R(S,M)$ is finitely generated over $S$ for $i=1,\ldots,\dim_R(M)$;
- $\operatorname{Ext}^i_R(S,M)$ satisfies Nakayama’s Lemma over $R$ for $i=1,\ldots,\dim_R(M)$.

This improves upon recent results of Frankild, Sather-Wagstaff, and Wiegand and results of Christensen and Sather-Wagstaff. We will discuss this result, some generalizations, and equalities between some invariants over $R$ and $S$.

**Speaker:** Ela Celikbas

**Title:** Fiber Products and Connected Sums of Local Rings

**Abstract:** In this talk, we will introduce fiber products and connected sums of local rings.

We will give examples, describe their properties, and set up questions arising in this scenario.

**Speaker:** Alessandro De Stefani

**Title:** Artinian level algebras of low socle degree

**Abstract:** Macaulay’s inverse system gives a one-to-one correspondence between finitely generated modules over $S=k[[x_1,\ldots, x_n]]$, under a particular action, and ideals $I\subseteq S$ such that $S/I$ is Artinian. This correspondence can be restricted to suitable finitely generated $S$-modules and ideals $I$ such that $S/I$ is level, i.e. such that all socle elements in $S/I$ have maximal order. We use this tool to characterize Hilbert functions of level local algebras $(S/I,\mathfrak{m},k)$ such that $\mathfrak{m}^4= 0$, and to prove that level local algebras with maximal Hilbert function and $\mathfrak{m}^4=0$ are in fact graded.

**Speaker:** Zheng Yang

**Title:** Decomposing a Gorenstein Artin ring as a Connected Sum

**Abstract:** We will see more examples of connected sums and fiber products of rings. I will also discuss some new results in joint work with H.Ananthnarayan and E.Celikbas.

**Speaker:** Sarang Sane

**Title:** Projective Modules! Keep your Witts about! $C$ how its done!

**Abstract:** The aim of this talk will be to explain the title! While doing so, we might tour the world of projective modules and quadratic forms, and observe similarities between other worlds outside the realm of algebra.

**Speaker:** Arindam Banerjee

**Title:** Bounds on Castelnuovo-Mumford Regularity

**Abstract:** I’ll discuss bounds on Castelnuovo-Mumford regularity in some special cases. While working over polynomial ring under some condition on dimension, regularity of Tor modules have some upper bound which satisfies nice convexity properties. On the other hand, edge ideals of simple graphs whose complement graphs are chordal, have linear minimal free resolutions, that is regularities attain the minimal possible value in that case. However for many other types of simple graphs much less is known about regularities of edge ideals.

**Speaker:** Saeed Nasseh

**Title:** Factorizations of local homomorphisms

**Abstract:** Let $f\colon R \to S$ be a homomorphism of commutative rings. Many techniques for studying $R$-modules focus on finitely generated modules. As a consequence, these techniques are not well-suited for studying $S$ as an $R$-module. However, a technique of Avramov, Foxby, and Herzog sometimes allows one to replace the original homomorphism with a surjective one $R’\to S$ where $R$ and $R’$ are tightly connected. In this setting, $S$ is a cyclic $R’$-module, so one can study it using finitely generated techniques. I will give a general introduction to such factorizations, followed by a discussion of some new results on ``weakly functorial properties’’ of such factorizations and applications. The new results are joint with Sean Sather-Wagstaff.

**Speaker:** Jack Jeffries

**Title:** Finite $F$-Representation Type and $F$-Signature

**Abstract:** In 1999, an investigation of differential operators in positive characteristic led Smith and Van den Bergh to define the notion of rings of finite $F$-representation type. This property has many interesting connections with current research topics, but many open questions remain. We discuss some of these consequences of finite $F$-representation type, including a new result on the $F$-signature of such rings.

**Speaker:** Brian Johnson

**Title:** Graded rings

**Abstract:** We will briefly discuss (commutative) rings graded by $\mathbb Z^d$ and then consider rings graded by any abelian group. Looking at properties defined strictly in terms of homogeneous objects, we examine the relationships between them under gradings induced by a quotient of the grading group.

**Speaker:** Billy Sanders

**Title:** Semidualizing Modules

**Abstract:** An $R$ module $M$ is semidualizing if it is finitely generated, $\text{Hom}(M,M) = R$ and $\text{Ext}_R^i(M,M) = 0$ for all $i \geqslant 1$. These modules have similar properties to the canonical module of a ring. I will give examples of semidulizing modules and also talk about their properties and applications.