NCMW  Commutative algebra and algebraic geometry in positive characteristics (2018)
Speakers and Syllabus
Speaker 
Affiliation 
Title 
Krishna Hanumanthu (KH) 
Chennai Mathematical Instituite 
Introduction to schemes and sheaf cohomology 
H. Ananthnarayan (HA) 
Indian Institute of Technology Bombay 
Introduction to Local Cohomology modules 
Manoj Kummini (MK) 
Chennai Mathematical Instituite 
Frational rings 
Vijaylaxmi Trivedi (VT) 
Tata Institute of Fundamental Research 
HilbertKunz multiplicity 
Srikanth Iyengar (SI) 
University of Utah 
Homological conjectures 
Anurag Singh (AS) 
University of Utah 
Frational, Fregular, Fpure and Finjective rings 
V. Srinivas (VS) 
Tata Institute of Fundamental Research 
Rational singularities are of Frational type 
K.i. Watanabe (KW) 
Nihon University 
Ideal Theory of 2dimensional normal local rings via resolution of singularities 
Tutorial Instructors 

Dipankar Ghosh (DG) 
CMI, Chennai 

Mitra Koley (MK) 
TIFR, Mumbai 

Shreedevi Masuti (SM) 
CMI, Chennai 

Mandira Mondal (MM) 
CMI, Chennai 
Abstracts of lectures
 Krishna Hanumanthu. Introduction to schemes and sheaf cohomology
 Basics of scheme theory. A very brief overview of schemes and their imporant properties. We will not prove any results here but only give an outline asneeded for the following topics.
 Cohomology of coherent sheaves on schemes. (a) We will introduce sheaf cohomology for coherent sheaves. All basic properties of sheaf cohomology will be developed. (b) Grothendieck vanishing theorem. (c) Let $(A,\mathfrak m)$ be a local ring and let $X$ be a projective scheme over $A$. Let $F$ be a coherent sheaf on $X$. Then we will explain the result which says that the completion of $H^i(X,F)$ is isomorphic to the inverse limit of $H^i(X_n,F_n)$, where $X_n,F_n$ are obtained via the base change $A \to A/\mathfrak m^n.$
 Blow ups and introduction to the problem of resolution of singularities.
 Introduction to intersection of curves on surfaces
 Hariharan Ananthnarayan. Introduction to local cohomology modules
 Definition of local cohomology. Computation of local cohomology via various ways in particular using the Cech complex.
 Behavior of local cohomology with respect to flat maps. Independence theorem. Grothendieck vanishing theorem
 Grothendieck nonvanishing theorem. Frobenius action on local cohomology.
 Some applications of local cohomology. In particular Hoa's theorem on asymptotic reduction number of ideals. Proof of Monomial conjecture in char $p > 0.$
 Manoj Kummini. Frational rings
 Preliminary lectures. Review of local cohomology from Hartshorne, Springer LNM. Quick introduction to doublecomplex spectral sequences. Review of blowups, Rees algebras and birational maps.
 Main Lectures. These lectures will primarily be based on the paper of K. E. Smith, Frational rings have rational singularities, Amer. J. Math. 1997. Pseudorationality, rational singularities Introduction to Frational rings. Frationality and local cohomology Frationality implies pseudorationality.
 K.i. Watanabe.Ideal Theory of 2dimensional normal local rings via resolution of singularities.
 Intersection theory on the resolution, Fundamental cycle, rational singularities.
 RiemannRoch theorem, log resolution of integrally closed ideals and expres sion of $\ell(A/I)$ and multiplicity via cycle on the resolution.
 $p_g$ideals. Definition, existence and fundamental properties.
 Core of integrally closed ideals, calculation of core for $p_g$ideals and com parison of core($I$) and $I$.
 Anurag Singh. $F$rational, $F$regular, $F$pure and $F$injective rings.
 Preliminary lectures. Demazure's description of normal graded rings in terms of $Q$divisors (2) Local cohomology and canonical modules in terms of $Q$divisors 3) $F$injective, $F$rational, $F$pure, and $F$regular rings. various constructions.
 Main lectures. (1) Classification of Fregular graded rings of dimension two (2) Cyclic covers; behaviour of singularities under cyclic covers (3) Frobenius representation type and Fsignature (4) Fthresholds following TakagiWatanabe, and calculations of these.
 Srikanth Iyengar. Homological conjectures.
 Preliminary lectures. (1) Homological invariants of modules over local rings (2) Perfect complexes (3) Local duality.
 Main lectures. (1) Homological properties of modules over Frobenius endomorphism (2) Perfect extensions of Kunz’s theorem. (3) The Frobenius endomorphism and perfect complexes. (4) The homological conjectures.
 V. Srinivas.Rational singularities are of Frational type.
 This lecture series will go over the proof that rational singularities are of Frational type, based on my paper with V. B. Mehta. This result is one of the first in the emerging area of Fsingularities. The series will sketch also the background material used in this proof, including Cartier operators and the results of Deligne and Illusie on splitting of the de Rham complex. The proof also yields the GrauertRiemenschneider theorem as a byproduct.
 V. B. Mehta and V. Srinivas, {\em A characterisation of rational singularities, } Asian J. Math. 1 (1997) 249271.
 Vijaylaxmi Trivedi. HilbertKunz multiplicity.
 Preliminary lectures. These lectures will be a selfcontained introduction to the theory of HilbertKunz multiplicity $e_{HK}$. In particular, following Huneke's notes, we give a proof that the limit exists. We also discuss a bit about the second coefficient of the HilbertKunz function.
 Main lectures. Here we work in the graded setup. In case of two dimensional rings, we discuss relation of $e_{HK}$ and the associated syzygy vector bundle. We give an application of such a characterization (i) to the notion of $e_{HK}$ in characteristic $0$, (ii) to the Frobenius semistability behaviour of the syzygy bundles over trinomial curves, using Monsky's notion of taxicab distance.
 HilbertKunz density function. We give an introduction to the notion of {\it HilbertKunz density function} and its properties, also its applications to various question about HilbertKunz multiplicity such as Asymptotic growth of $e_{HK},$ an approach towards HilbertKunz multiplicity in characteristic $0$ and Its relation with $F$pure threshold of the maximal ideal.
Time Table
Date  Day  10.00 To 11.15 
11.45 To 1.00 
2.00 To 3.15 
Tea  3.30 To 5.00 

10/12/2018  Monday  KH 
T E A 
HA 
L U N C H 
MK  KH MM+MK 
S N A C K S 

11/12/2018  Tuesday  KH  HA  MK  HA DG+SM 

12/12/2018  Wednesday  KH  HA  VT  MK MM+MK 

13/12/2018  Thursday  KH  HA  VT  KH MM+MK 

14/12/2018  Friday  KW  VT  MK  HA DG+SM 

15/12/2018  Saturday  KW  VT  MK  VT MM+MK 

17/12/2018  Monday  KW 
T E A 
SI 
L U N C H 
AS  MK MM+MK 
S N A C K S 

18/12/2018  Tuesday  KW  SI  AS  VT MM+MK 

19/12/2018  Wednesday  VS  SI  AS  AS DG+SM 

20/12/2018  Thursday  VS  SI  AS  SI DG+SM 

21/12/2018  Friday  VS  SI  AS  AS DG+SM 

22/12/2018  Saturday  VS  SI  AS  SI DG+SM 
Val 
3.305.00 Problem session