algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...

, a free presentation of a

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...

''M'' over a commutative ring ''R'' is an

exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the conte ...

of ''R''-modules: :

\bigoplus_ R \ \overset \to\ \bigoplus_ R \ \overset\to\ M \to 0.

Note the image under ''g'' of the standard basis generates ''M''. In particular, if ''J'' is finite, then ''M'' is a

finitely generated module In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts i ...

. If ''I'' and ''J'' are finite sets, then the presentation is called a finite presentation; a module is called finitely presented if it admits a finite presentation. Since ''f'' is a

module homomorphism In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a ring ''R'', then a function f: M \to N is called an ''R''-''module homomorphism'' or an ' ...

between free modules, it can be visualized as an (infinite) matrix with entries in ''R'' and ''M'' as its cokernel. A free presentation always exists: any module is a quotient of a free module:

F \ \overset\to\ M \to 0

, but then the kernel of ''g'' is again a quotient of a free module:

F' \ \overset \to\ \ker g \to 0

. The combination of ''f'' and ''g'' is a free presentation of ''M''. Now, one can obviously keep "resolving" the kernels in this fashion; the result is called a

free resolution In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to d ...

. Thus, a free presentation is the early part of the free resolution. A presentation is useful for computation. For example, since tensoring is right-exact, tensoring the above presentation with a module, say ''N'', gives: :

\bigoplus_ N \ \overset \to\ \bigoplus_ N \to M \otimes_R N \to 0.

This says that

M \otimes_R N

is the cokernel of

f \otimes 1

. If ''N'' is also a ring (and hence an ''R''-algebra), then this is the presentation of the ''N''-module

M \otimes_R N

; that is, the presentation extends under base extension. For left-exact

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

s, there is for example Proof: Applying ''F'' to a finite presentation

R^ \to R^ \to M \to 0

results in :

0 \to F(M) \to F(R^) \to F(R^).

This can be trivially extended to :

0 \to 0 \to F(M) \to F(R^) \to F(R^).

The same thing holds for

G

. Now apply the

five lemma In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is not only valid for abelian categories but also wor ...

\square

References

* Eisenbud, David, ''Commutative Algebra with a View Toward Algebraic Geometry'', Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, . {{algebra-stub Algebra

See also

References