Algebraic geometry played a central role in 19th century math. The deepest results of Abel, Riemann, Weierstrass, and the most important works of Klein and Poincar/’e were part of this subject. In the middle of the 20th century algebraic geometry had been through a large reconstruction. The domain of application of its ideas had grown tremendously, in the direction of algebraic varieties over arbitrary fields and more general complex manifolds. Many of the best achievements of algebraic geometry could be cleared of the accusation of incomprehensibility or lack of rigour. The foundation for this reconstruction was (commutative) algebra. In the 1950s and 60s have brought substantial simplifications to the foundation of algebraic geometry, which significantly came closer to the ideal combination of logical transparency and geometric intuition. Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers in number fields constitute an important class of commutative rings — the Dedekind domains. This has led to the notions of integral extensions and integrally closed domains. The notion of localization of a ring (in particular the localization with respect to a prime ideal leads to an important class of commutative rings — the local rings. The set of the prime ideals of a commutative ring is naturally equipped with a topology — the Zariski topology. All these notions are

widely used in algebraic geometry and are the basic technical tools for the definition of scheme theory — a generalization of algebraic geometry introduced by Grothendieck.

INTENDED AUDIENCE : BS / ME / MSc / PhD

PREREQUISITES : Linear Algebra ;Algebra – First Course

INDUSTRY SUPPORT : R & D Departments of IBM / Microsoft Research Labs SAP /TCS / Wipro / Infosys

Week 01 : Algebraic Preliminaries 1 --- Rings and Ideals

Week 02 : Algebraic Preliminaries 2 --- Modules and Algebras

Week 02 : Algebraic Preliminaries 2 --- Modules and Algebras

Week 03 : The K -Spectrum of a K -algebra and Affine algebraic sets

Week 05 : Hilbert's Basis Theorem and Consequences

Week 06 : Rings of Fractions

Week 07 : Modules of Fractions

Week 09 : Hilbert’s Nullstellensatz and its equivalent formulations

Week 11 : Zariski Topology

Week 12 : Integral Extensions

[1] Atiyah, M. F. ; Macdonald I. G. : Introduction to Commutative Algebra, Addison-Wesley, London, 1969.

[2] Eisenbud, D. : Commutative Algebra With a View Towards Algebraic Geometry, GTM 150, Springer, New York/Berlin/Heidelberg, 1995.

[3] Patil, D. P. ; Storch, U. : Introduction to Algebraic Geometry and Commutative Algebra, IISc Lecture Notes Series, No. 1,

IISc Press/World Scientific Publications Singapore/Chennai, 2010. — Indian Edition Published by Cambridge University Press India Pvt. Ltd. 2012.

IISc Press/World Scientific Publications Singapore/Chennai, 2010. — Indian Edition Published by Cambridge University Press India Pvt. Ltd. 2012.

[4] Shafarevich, I. R. : Basic Algebraic Geometry 1, Springer-Verlag, Berlin Heidelberg 2013.

[5] Serre, J. -P. : Local Algebra, Springer Monographs in Mathematics, Springer, Berlin/Heidelberg, 2000.

[6] Singh, B. : Basic Commutative Algebra, World Scientific Publications Singapore, 2011.

Dilip P. Patil received B. Sc. and M. Sc. in Mathematics from the University of Pune in 1976 and 1978,
respectively. From 1979 till 1992 he studied Mathematics at School of Mathematics, Tata Institute of
Fundamental Research, Bombay and received Ph. D. through University of Bombay in 1989. Currently he is
a Professor of Mathematics at the Departments of Mathematics, Indian Institute of Science, Bangalore. At
present he is a Visiting Professor at the Department of Mathematics, IIT Bombay. He has been a Visiting
Professor at Ruhr-Universitt Bochum, Universitt Leipzig, Germany and several universities in Europe and
Canada. His research interests are mainly in Commutative Algebra and Algebraic Geometry.

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams:**24 April 2022** Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.

Registration url: Announcements will be made when the registration form is open for registrations.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

**CRITERIA TO GET A CERTIFICATE**

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.

Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

**YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.**

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IISc Bangalore.It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams:

Registration url: Announcements will be made when the registration form is open for registrations.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.

Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IISc Bangalore.It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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