Mathematics Department

Commutative algebra seminar

Date and time: 15 December 2022, 11 am

Venue: Ramanujan Hall

Title: Berger conjecture, valuations, and torsions
 
Abstract: Let R,m,k be a one-dimensional complete local reduced k-algebra over a field of characteristic zero. Berger conjectured that R is regular if and only if the universally finite module of differentials O is torsion-free. We discuss methods that have been used in the past to prove cases where the conjecture holds. When R is a domain, we prove the conjecture in several cases. Our techniques are primarily reliant on making use of the valuation semi-group of R. First, we establish a method of verifying the conjecture by analyzing the valuation semi-group of R and orders of units of the integral closure of R. We also prove the conjecture in the case when certain monomials are missing from the monomial support of the defining ideal of R. This also generalizes previously known results. This is joint work with Craig Huneke and Sarasij Maitra.

Speaker: Utsav Chowdhury, Indian Statical Institute, Kolkata, India
Date/Time: 16 December 2022, 5:30pm 

Gmeet link: meet.google.com/vxv-adfh-onj

Title: Characterisation of the affine plane using A^1 -homotopy theory

Abstract: Characterisation of the affine n-space is one of the major problems in affine algebraic geometry. Miyanishi showed an affine complex surface X is isomorphic to C^2 if O(X) is a U.F.D., O(X)^∗ = C^∗ and X has a non-trivial Ga-action [3, Theorem 1]. Since the orbits of a Ga-action are affine lines, the existence of a non-trivial Ga-action says that there is a non-constant A^1 in X. Ramanujam showed that a smooth complex surface is isomorphic to C^2 if it is topologically contractible and it is simply connected at infinity [5]. Topological contractibility, in particular, path connectedness says that there are non-constant intervals in X. On the other hand, A^1 -homotopy theory has been developed by F.Morel and V.Voevodsky [4] as a connection between algebra and topology. An algebrogeometric analog of topological connectedness is A^1 -connectedness. In this talk, using ghost homotopy techniques [2, Section 3] we will prove that if a surface X is A^1 -connected, then there is an open dense subset such that through every point there is a non-constant A^1 in X.

As a consequence using the algebraic characterization, we will prove that C^2 is the only A^1 -contractible smooth complex surface. This answers the conjecture that appeared in [1, Conjecture 5.2.3]. We will also see some other useful consequences of this result. This is joint work with Biman Roy.

References

[1] A. Asok, P. A. Østvær; A 1 -homotopy theory and contractible varieties: a survey, Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects. Lecture Notes in Mathematics, vol 2292. Springer, Cham. https://doi.org/10.1007/978-3-030-78977-05.

[2] C. Balwe, A. Hogadi and A. Sawant; A 1 -connected components of schemes. Adv Math, Volume 282, 2016.

[3] M. Miyanishi; An algebraic characterization of the affine plane. J. Math. Kyoto Univ. 15-1 (1975) 19-184.