symmetric monoidal (∞,1)-category of spectra
Given a (not necessarily commutative) unital ring , an -module is finitely presented (or of finite presentation) if there exists an exact sequence where are natural numbers.
For a module over a ring , the following are equivalent :
For a module over a ring , the following are equivalent :
is an isomophism ;
is an isomorphism.
. Using a finite presentation of one can write the commutative diagram
and deduce the isomorphism.
. Obvious
. The map is surjective, so there exists a finite number of elements and a finite number of functions such that the sum be the identity of . As a consequence must be finitely generated.
Consider a short exact sequence . Then using the commutative diagram
one gets that is surjective and thus that is finitely generated.
Last revised on July 25, 2023 at 16:38:55. See the history of this page for a list of all contributions to it.