北京雁栖湖应用数学研究院

Local commutative algebra studies the algebraic structures that arise when one focuses on the behavior of rings and modules near a prime ideal or a point of an algebraic variety. It provides the foundational language for understanding singularities, intersection theory, deformation theory, and the local geometry of algebraic spaces. At the same time, local algebra has become increasingly computational: many fundamental questions—such as computing dimensions, multiplicities, tangent spaces, resolutions, and invariants of singularities—can now be addressed algorithmically using Groebner bases, standard bases, and computer algebra systems.

This course introduces the fundamental concepts and techniques of local commutative algebra, with a strong emphasis on explicit computation. Alongside theoretical developments, students will learn how to perform concrete calculations in polynomial rings and local rings using Groebner basis methods and computational tools such as Macaulay2, Singular, or SageMath. Topics include localization, primary decomposition, dimension theory, Hilbert functions, multiplicity, integral dependence, regular sequences, Cohen–Macaulay rings, and homological invariants.

The goal of the course is twofold:

Conceptual understanding: to develop the structural viewpoint of local algebra and understand its role in modern algebraic geometry.
Computational ability: to solve explicit problems involving ideals, modules, and singularities through algorithmic methods.

The course is suitable for graduate students in algebra, algebraic geometry, number theory, topology, and computational mathematics, as well as researchers interested in symbolic computation and applications of algebraic methods.