Flat module
Overview
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
, and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a flat module over a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
R is an R-module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
M such that taking the tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
over R with M preserves exact sequence
Exact sequence
An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...
s. 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.
Vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
are flat modules. Free module
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
s, or more generally projective module
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
s, are also flat, over any R.
Unanswered Questions
