Finite presentability (module)

The finite presentability is a concept from the mathematical theory of modules . A module is finitely presentable if it has a finite generating system for which the relations that may exist between its elements are subject to a finiteness condition.

Presentation of a module

Let it be a link module over a ring . If a generating system of and denotes the -fold direct sum of with the base -elements , then there is exactly one homomorphism with . Since the module generates is surjective and you get a short exact sequence ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle (m_{i})_{i\in I}}$ ${\displaystyle M}$ ${\displaystyle R^{(I)}}$ ${\displaystyle I}$ ${\displaystyle R}$ ${\displaystyle e_{i},\,i\in I}$ ${\displaystyle f:R^{(I)}\rightarrow M}$ ${\displaystyle f(e_{i})=m_{i}}$ ${\displaystyle (m_{i})_{i\in I}}$ ${\displaystyle M}$ ${\displaystyle f}$

${\displaystyle 0\rightarrow \mathrm {ker} (f)\rightarrow R^{(I)}{\xrightarrow {f}}M\rightarrow 0}$,

which is called the presentation of which belongs to the generating system . ${\displaystyle M}$

In the above definition , the so-called relation module contains information about the relations that exist between the generating elements. Is in the extreme case , then is an isomorphism that maps the canonical basis to, that is, the latter is a basis of , in particular in this case it is a free module . The term to be defined here demands the existence of a finite generating system, the elements of which are not subject to too many relations: ${\displaystyle \mathrm {ker} (f)}$ ${\displaystyle \mathrm {ker} (f)=\{0\}}$ ${\displaystyle f:R^{(I)}\rightarrow M}$ ${\displaystyle (e_{i})_{i\in I}}$ ${\displaystyle (m_{i})_{i\in I}}$ ${\displaystyle M}$ ${\displaystyle M}$

A module is called finitely presentable if there is a finitely generated free module and a surjective homomorphism such that it is finitely generated. ${\displaystyle M}$ ${\displaystyle F}$ ${\displaystyle f:F\rightarrow M}$ ${\displaystyle \mathrm {ker} (f)}$

Since all the free finitely generated -modules to with isomorphic so you do a short exact sequence ${\displaystyle R}$ ${\displaystyle R^{n}}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle 0\rightarrow \mathrm {ker} (f)\rightarrow R^{n}{\xrightarrow {f}}M\rightarrow 0}$

with finitely generated . ${\displaystyle \mathrm {ker} (f)}$

Examples

• Finally generated modules over a Noetherian ring are finite presentable, because in the above definition is finitely generated as a sub-module of the Noetherian module . ${\displaystyle \mathrm {ker} (f)}$ ${\displaystyle R^{n}}$
• Every finitely generated projective module is finitely presentable.

properties

Relation modules

If finitely presentable, then, by definition, the core of a certain surjection of a finitely generated free module is finitely generated. It turns out that every relational module for a finite generating system is finitely generated, it even applies: ${\displaystyle M}$ ${\displaystyle M}$

• If it is finite presentable and surjective with a finitely generated module , then finitely generated. ${\displaystyle M}$ ${\displaystyle f:N\rightarrow M}$ ${\displaystyle N}$ ${\displaystyle \mathrm {ker} (f)}$

To prove this, consider next to the short exact sequence

${\displaystyle 0\rightarrow \mathrm {ker} (f)\rightarrow N{\xrightarrow {f}}M\rightarrow 0}$

also the short exact sequence

${\displaystyle 0\rightarrow K\rightarrow R^{n}\rightarrow M\rightarrow 0}$

from the definition of finite presentability with a finitely generated module . If one also assumes that is projective, it follows from Schanuel's lemma that , that is, it is the direct summand of a finitely generated module and therefore finitely generated itself. The general case can be traced back to it. ${\displaystyle K}$ ${\displaystyle N}$ ${\displaystyle \mathrm {ker} (f)\oplus R^{n}\cong N\oplus K}$ ${\displaystyle \mathrm {ker} (f)}$

Localization of homomorphisms

If a multiplicative subset of the commutative ring is , then one can localize -modules by . If a -linear mapping is, then ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle f:M\rightarrow N}$ ${\displaystyle R}$

${\displaystyle f_{S}:M_{S}\rightarrow N_{S},\quad f_{S}\left({\frac {m}{s}}\right):={\frac {f(x)}{s}}}$

a -linear mapping, and the assignment ${\displaystyle R_{S}}$

${\displaystyle \mathrm {Hom} _{R}(M,N)\rightarrow \mathrm {Hom} _{R_{S}}(M_{S},N_{S}),\,f\mapsto f_{S}}$

induces a -linear mapping ${\displaystyle R_{S}}$

${\displaystyle \mathrm {Hom} _{R}(M,N)_{S}\rightarrow \mathrm {Hom} _{R_{S}}(M_{S},N_{S}).}$

The question now arises when this mapping is an isomorphism. It applies

• Let it be a commutative ring, multiplicative and and modules. If it is finite presentable, then is the figure above ${\displaystyle R}$ ${\displaystyle S\subset R}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle R}$ ${\displaystyle M}$

${\displaystyle \mathrm {Hom} _{R}(M,N)_{S}\rightarrow \mathrm {Hom} _{R_{S}}(M_{S},N_{S}).}$

an isomorphism.

literature

• Ernst Kunz : Introduction to Commutative Algebra and Algebraic Geometry (Vieweg Studies; Vol. 46). Vieweg, Braunschweig 1980, ISBN 3-528-07246-6 .
• Louis H. Rowen: Ring Theory, Vol. 1 (Pure and applied mathematics; Vol. 127). Academic Press, Boston, Mass. 1988, ISBN 0-12-599841-4 .

Individual evidence

1. ^ Ernst Kunz: Introduction to commutative algebra and algebraic geometry , definition IV.1.8.
2. ^ Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Definition IV.1.9.
3. ↑ Louis H. Rowen: Ring Theory, Vol. 1 , Examples 2.8.28.
4. ↑ Louis H. Rowen: Ring Theory, Vol. 1 , Proposition 2.8.29.
5. ^ Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Theorem IV.1.10.