NPTEL Course: Algebraic Geometry & Commutative Algebra | Prof. Dilip P. Patil
Course Details
| Exam Registration | 13 |
|---|---|
| Course Status | Ongoing |
| Course Type | Elective |
| Language | English |
| Duration | 12 weeks |
| Categories | Mathematics |
| Credit Points | 3 |
| Level | Undergraduate/Postgraduate |
| Start Date | 19 Jan 2026 |
| End Date | 10 Apr 2026 |
| Enrollment Ends | 02 Feb 2026 |
| Exam Registration Ends | 20 Feb 2026 |
| Exam Date | 18 Apr 2026 IST |
| NCrF Level | 4.5 — 8.0 |
Introduction to Algebraic Geometry and Commutative Algebra: A Foundational Course
Algebraic Geometry, once described by its 19th-century pioneers as a realm of profound but often inscrutable beauty, has undergone a radical transformation. The key to this modern, rigorous, and immensely powerful framework is Commutative Algebra. This 12-week NPTEL course, instructed by the eminent Prof. Dilip P. Patil from the Indian Institute of Science (IISc) Bangalore, offers a structured gateway into this fascinating intersection of two central mathematical disciplines.
About the Instructor: Prof. Dilip P. Patil
Prof. Dilip P. Patil brings decades of expertise and a distinguished academic lineage to this course. After earning his B.Sc. and M.Sc. from the University of Pune, he completed his Ph.D. from the University of Bombay through the prestigious Tata Institute of Fundamental Research (TIFR). As a Professor at IISc Bangalore and a frequent visiting professor at institutions like IIT Bombay, Ruhr-Universität Bochum, and Universität Leipzig, his research has significantly contributed to Commutative Algebra and Algebraic Geometry. His co-authored textbook is, in fact, one of the recommended readings for this very course.
Course Overview and Historical Significance
This course is designed to unravel the symbiotic relationship between geometry and algebra. Algebraic Geometry, central to the works of giants like Riemann and Poincaré, found its modern footing in the mid-20th century through the language of Commutative Algebra. This reconstruction provided the logical transparency needed to rigorously handle geometric intuition about curves, surfaces, and higher-dimensional spaces defined by polynomial equations.
Commutative Algebra provides the essential tools—rings, ideals, modules, localization—that allow us to define and study geometric objects algebraically. Concepts born in number theory, like Dedekind domains and integral extensions, became crucial in geometry. The simple yet profound idea of considering the set of prime ideals of a ring as a geometric space (equipped with the Zariski topology) is the very foundation of Alexander Grothendieck's revolutionary scheme theory.
Who Should Enroll?
INTENDED AUDIENCE: This course is tailored for advanced undergraduate (BS), postgraduate (MSc, ME), and doctoral (PhD) students in Mathematics.
PREREQUISITES: A solid foundation in:
- Linear Algebra
- Abstract Algebra (Groups, Rings, Fields)
INDUSTRY SUPPORT: The theoretical depth and problem-solving skills developed in this course are highly valued in advanced research and development sectors, including:
- IBM & Microsoft Research Labs
- SAP, TCS, Wipro, Infosys (R&D divisions)
Detailed 12-Week Course Layout
| Week | Topic |
|---|---|
| Week 01 | Algebraic Preliminaries 1 – Rings and Ideals |
| Week 02 | Algebraic Preliminaries 2 – Modules and Algebras |
| Week 03 | The K-Spectrum of a K-algebra and Affine algebraic sets |
| Week 04 | Noetherian and Artinian Modules |
| Week 05 | Hilbert's Basis Theorem and Consequences |
| Week 06 | Rings of Fractions |
| Week 07 | Modules of Fractions |
| Week 08 | Local Global Principle and Consequences |
| Week 09 | Hilbert’s Nullstellensatz and its equivalent formulations |
| Week 10 | Consequences of Hilbert's Nullstellensatz |
| Week 11 | Zariski Topology |
| Week 12 | Integral Extensions |
Recommended Textbooks and Resources
The course draws from a rich corpus of literature. Key texts include:
- Atiyah & Macdonald: The classic, concise introduction to Commutative Algebra.
- Eisenbud: A comprehensive modern text linking algebra directly to geometry.
- Patil & Storch: The instructor's own text, providing a tailored pathway for the course material.
- Shafarevich: A foundational text on Algebraic Geometry itself.
- Serre: For deeper insights into Local Algebra.
- Singh: An accessible text on Basic Commutative Algebra.
Conclusion: Embark on a Journey into Abstract Beauty
This course is more than a series of lectures; it's an initiation into a language that has reshaped modern mathematics. Under the expert guidance of Prof. Dilip P. Patil, you will build the algebraic toolkit necessary to explore geometric worlds defined by equations. From the fundamental Hilbert's Nullstellensatz—a cornerstone theorem bridging algebra and geometry—to the construction of the Zariski topology, you will gain the foundational knowledge that underpins contemporary research in pure mathematics, theoretical physics, and advanced cryptography. Enroll to begin your journey at the profound intersection of algebra and geometry.
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