Lecture

Free Resolution of Modules

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In course

MATH-311: Algebra IV - rings and modules

Ring and module theory with a major emphasis on commutative algebra and a minor emphasis on homological algebra. ...

Description

This lecture covers the concept of free resolution of modules, where a chain complex of free modules is used to compute Ext-modules. The lecture also discusses projective modules, their definitions, and properties, emphasizing the importance of projective resolutions in module theory.

Instructors (2)

Domenico Valloni

Teaching & PhD Courses Algebraic geometry II - schemes and sheaves MATH-510 The aim of this course is to learn the basics of the modern scheme theoretic language of algebraic geometry.

Zsolt Patakfalvi

See CV here .

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

Related concepts (32)

Projective module

In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. Every free module is a projective module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains.

Module (mathematics)

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication.

Sheaf of modules

In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) → F(V) are compatible with the restriction maps O(U) → O(V): the restriction of fs is the restriction of f times that of s for any f in O(U) and s in F(U). The standard case is when X is a scheme and O its structure sheaf. If O is the constant sheaf , then a sheaf of O-modules is the same as a sheaf of abelian groups (i.

Flat module

In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion free modules. Formally, a module M over a ring R is flat if taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Flatness was introduced by in his paper Géometrie Algébrique et Géométrie Analytique.

Resolution (algebra)

In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of s of an ), which is used to define invariants characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions.

Related lectures (64)

Module Theory: Chain Conditions

Explores chain conditions in module theory, emphasizing Noetherian modules and stabilizing sequences of submodules.

Homomorphisms and Projective Resolutions

Covers homomorphisms, projective modules, and resolutions in chain complexes.

Ideals and Representations

Covers ideals, representations, modules, and maximal ideals in associative algebras.

Algebraic Kunneth Theorem

Covers the Algebraic Kunneth Theorem, explaining chain complexes and cohomology computations.

Long Exact Sequence of Ext-Modules