In mathematics, there are several theorems basic to
algebraic ''K''-theory.
Throughout, for simplicity, we assume when an
exact category is a subcategory of another exact category, we mean it is strictly full subcategory (i.e., isomorphism-closed.)
Theorems
The localization theorem generalizes the
localization theorem for abelian categories.
Let
be exact categories. Then ''C'' is said to be
cofinal in ''D'' if (i) it is closed under extension in ''D'' and if (ii) for each object ''M'' in ''D'' there is an ''N'' in ''D'' such that
is in ''C''. The prototypical example is when ''C'' is the category of
free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
s and ''D'' is the category of
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 characterizati ...
s.
See also
*
Fundamental theorem of algebraic K-theory
References
*
*Ross E. Staffeldt
On Fundamental Theorems of Algebraic K-Theory*GABE ANGELINI-KNOLL
FUNDAMENTAL THEOREMS OF ALGEBRAIC K-THEORY*Tom Harris
Algebraic proofs of some fundamental theorems in algebraic K-theory
{{algebra-stub
Algebraic K-theory
Theorems in algebraic topology