# 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 F-rational rings Vijaylaxmi Trivedi (VT) Tata Institute of Fundamental Research Hilbert-Kunz multiplicity Srikanth Iyengar (SI) University of Utah Homological conjectures Anurag Singh (AS) University of Utah F-rational, F-regular, F-pure and F-injective rings V. Srinivas (VS) Tata Institute of Fundamental Research Rational singularities are of F-rational type K.-i. Watanabe (KW) Nihon University Ideal Theory of 2-dimensional 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
1. 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.
2. 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 $A \to A/\mathfrak m^n.$
3. Blow ups and introduction to the problem of resolution of singularities.
4. Introduction to intersection of curves on surfaces
• Hariharan Ananthnarayan. Introduction to local cohomology modules
1. Definition of local cohomology. Computation of local cohomology via various ways in particular using the Cech complex.
2. Behavior of local cohomology with respect to flat maps. Independence theorem. Grothendieck vanishing theorem
3. Grothendieck non-vanishing theorem. Frobenius action on local cohomology.
4. 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. F-rational rings
1. Preliminary lectures. Review of local cohomology from Hartshorne, Springer LNM. Quick introduction to double-complex spectral sequences. Review of blow-ups, Rees algebras and birational maps.
2. Main Lectures. These lectures will primarily be based on the paper of K. E. Smith, F-rational rings have rational singularities, Amer. J. Math. 1997. Pseudo-rationality, rational singularities Introduction to F-rational rings.  F-rationality and local cohomology F-rationality implies pseudo-rationality.
• K.-i. Watanabe.Ideal Theory of 2-dimensional normal local rings via resolution of singularities.
1. Intersection theory on the resolution, Fundamental cycle, rational singularities.
2. Riemann-Roch theorem, log resolution of integrally closed ideals and expres- sion of $\ell(A/I)$ and multiplicity via cycle on the resolution.
3. $p_g$-ideals. Definition, existence and fundamental properties.
4. 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.
1. 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.
2. Main lectures. (1) Classification of F-regular graded rings of dimension two (2) Cyclic covers; behaviour of singularities under cyclic covers (3) Frobenius representation type and F-signature (4) F-thresholds following Takagi-Watanabe, and calculations of these.
• Srikanth Iyengar. Homological conjectures.
1. Preliminary lectures. (1) Homological invariants of modules over local rings (2) Perfect complexes (3) Local duality.
2. 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 F-rational type.
1. This lecture series will go over the proof that rational singularities are of F-rational type, based on my paper with V. B. Mehta. This result is one of the first in the emerging area of F-singularities. 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 Grauert-Riemenschneider theorem as a byproduct.
2. V. B. Mehta and V. Srinivas, {\em A characterisation of rational singularities, } Asian J. Math. 1 (1997) 249-271.
• Vijaylaxmi Trivedi. Hilbert-Kunz multiplicity.
1. Preliminary lectures. These lectures will be a self-contained  introduction to the theory of Hilbert-Kunz 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 Hilbert-Kunz function.
2. 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.
3. Hilbert-Kunz density function. We give an introduction to  the  notion of {\it Hilbert-Kunz density function}  and its properties, also  its applications to various question about Hilbert-Kunz multiplicity such as Asymptotic growth of $e_{HK},$ an approach towards Hilbert-Kunz multiplicity in characteristic  $0$ and Its relation with  $F$-pure threshold of the maximal ideal.

## Time Table

 Date Day 10.00To11.15 11.45To1.00 2.00To3.15 Tea 3.30To5.00 10/12/2018 Monday KH T E A HA L U N C H MK KHMM+MK S N A C K S 11/12/2018 Tuesday KH HA MK HADG+SM 12/12/2018 Wednesday KH HA VT MKMM+MK 13/12/2018 Thursday KH HA VT KHMM+MK 14/12/2018 Friday KW VT MK HADG+SM 15/12/2018 Saturday KW VT MK VTMM+MK 17/12/2018 Monday KW T E A SI L U N C H AS MKMM+MK S N A C K S 18/12/2018 Tuesday KW SI AS VTMM+MK 19/12/2018 Wednesday VS SI AS ASDG+SM 20/12/2018 Thursday VS SI AS SIDG+SM 21/12/2018 Friday VS SI AS ASDG+SM 22/12/2018 Saturday VS SI AS SIDG+SM Val

3.30-5.00 Problem session

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