An exact functor is a functor between categories of modules or abelian groups that preserves the exactness of sequences, meaning it maps exact sequences to exact sequences. This property is crucial in homological algebra as it ensures that the functor respects the algebraic structure and relationships between objects in a category.

Exact Functor

exact functor

categories of modules

An Abelian group is a set equipped with an operation that combines any two elements to form a third element, satisfying the properties of closure, associativity, identity, invertibility, and commutativity. Named after mathematician Niels Henrik Abel, these groups are foundational in abstract algebra and are characterized by the fact that the order of operation does not affect the result.

abelian groups

exactness of sequences

An exact sequence is a sequence of algebraic objects and morphisms between them, where the image of one morphism equals the kernel of the next. This concept is fundamental in homological algebra and provides a framework for understanding the structure and relationships between various algebraic entities, such as groups, modules, or vector spaces.

exact sequences

Homological algebra is a branch of mathematics that studies homology in a general algebraic setting, providing a framework for analyzing and solving problems in algebraic topology, algebraic geometry, and beyond. It uses chain complexes and exact sequences to explore the relationships between different algebraic structures, revealing deep insights into their properties and interactions.

homological algebra

An algebraic structure is a set equipped with one or more operations that follow specific axioms, providing a framework to study algebraic systems like groups, rings, and fields. These structures allow mathematicians to abstract and generalize patterns and properties across different mathematical systems, facilitating deeper understanding and applications across various domains.

algebraic structure

Understanding the relationships between objects is essential to deciphering the underlying structure and function within various scientific, mathematical, and technological fields. These relationships define how objects interact, influence each other, and form complex systems that can be analyzed for insights and applications.

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