nLab finitely presented module

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Definition

Given a (not necessarily commutative) unital ring RR, an RR-module is finitely presented (or of finite presentation) if there exists an exact sequence R qR pM0R^q\to R^p\to M\to 0 where p,qp,q are natural numbers.

Equivalent characterisations

Proposition

For a module MM over a ring RR, the following are equivalent :

  1. MM is finitely presented ;
  2. MM is a compact object of the category of RR-modules.

Proposition

For a module MM over a ring RR, the following are equivalent :

  1. MM is finitely presented ;
  2. for every family {Q α} αA\{Q_\alpha\}_{\alpha \in A} of RR-modules, the canonical map
    M R αQ α αM RQ α M \otimes_R \prod_\alpha Q_\alpha \rightarrow \prod_\alpha M \otimes_R Q_\alpha

    is an isomophism ;

  3. for every RR-module QQ and every set AA, the canonical map
    M RR AM A M \otimes_R R^A \rightarrow M^A

    is an isomorphism.

Proof

121 \Rightarrow 2. Using a finite presentation R qR pMR^q \to R^p \to M of MM one can write the commutative diagram

and deduce the isomorphism.

232 \Rightarrow 3. Obvious

313 \Rightarrow 1. The map MR MM MM \otimes R^M \to M^M is surjective, so there exists a finite number of elements {m i} iI\{m_i\}_{i \in I} and a finite number of functions {f i} iI:MR\{f_i\}_{i \in I} : M \to R such that the sum im if i\sum_i m_i f^i be the identity of MM. As a consequence MM must be finitely generated.

Consider a short exact sequence 0KR pM00 \to K \to R^p \to M \to 0. Then using the commutative diagram

one gets that KR KK KK \otimes R^K \to K^K is surjective and thus that KK 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.