INdAM-COFUND Incoming fellow between 2012-11-19 and 2014-11-18

Project information

Title

KASCMRM: Koszul Algebras, their syzygies and Castelnuovo-Mumford Regularity of Modules over them

Abstract

The Koszul complex is perhaps the most important freecomplex in commutative algebra. It has three main features: it has a DG-algebra structure, it resolves the residue field over regular local rings, it is a linear complex. Algebras whose residue field is resolved by a free complex satisfying the third condition are called Koszul. Koszul algebras can be characterized in various ways, one of  the most elegant is due to Avramov, Eisenbud and Peeva and it asserts that the Koszul property is equivalent to the finiteness of the relative Castelnuovo-Mumford regularity of every module or, equivalently, of the residue field. To prove that a ring is Koszul is usually a difficult task but several techniques and methods are available and they involve deformations, filtrations and combinatorics.

In this proposal, our main goal to study the
following three basic questions:

1. Can the regularity of finitely generated modules
   over a Koszul ring be computed algorithmically?
2. What are the numerical constrains that syzygies
   of Koszul rings satisfy?
3. Is it true that any Koszul algebra can be
   deformed via Grobner bases and lifting to and algebra defined by
   monomials of degree 2?

Fellow information

Research interests

• Commutative Algebra
• Schubert Varieties

Previous positions, awards

• June 2011-November 2012: Post Doctoral Fellow, The Institute of Mathematical Sciences, Chennai, India
• January 2006-May 2011: Research Scholar, Indian Institute of Technology Madras, Chennai, India

Publications, preprints, other works

1. A. V. Jayanthan, Ramakrishna Nanduri, Castelnuovo-Mumford regularity and Gorensteinness of fiber cone, Comm. Algebra, 40 (2012), no.4, 1338-1351.
2. A. V. Jayanthan, Ramakrishna Nanduri, Depths and Hilbert coefficients of fiber cones of stretched m-primary ideals, appeared in the Proceedings of the 7th
   Japan-Vietnam Joint Seminar on Commutative Algebra, held at Quy Nhon
   University, VIETNAM, December 2011.
3. A. V. Jayanthan, Ramakrishna Nanduri, On the depth of fiber cones of stretched m-primary ideals, submitted (2012)
4. Ramakrishna Nanduri, Hilbert coefficients of Schubert varieties in Grassmannians, submitted (2012)
5. A. V. Jayanthan and Ramakrishna Nanduri, On the relation type of fiber cones, preprint (2012)

Conferences, schools, other events

• The 20th National School on Algebra on “Discrete Invariants in Commutative Algebra and in
  Algebraic Geometry
  Mangalia, Romania, September 2012
• The 7th Japan-Vietnam Joint Conference/Seminar on Commutative Algebra
  Quy Nhon University, Vietnam, December 2011
• International Conference on Commutative Algebra and Algebraic Geometry (CAAG 2010)
  Indian Institute of Science (IISc), Bangalore, India, December 2010
• International congress of Mathematicians (ICM), 2010
  Hyderabad, India, August 2010
• School and Workshop on local rings and local study of algebraic varieties
  International Centre for Theoretical Physics (ICTP), Trieste, Italy, May 2010
• Golden Jubilee Workshop on Multiplicity Conjectures, Resolution of Singularieties and Toric
  Geometry
  Indian Institute of Technology Madras, Chennai, India, December 2008