finite type


Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories


Rational homotopy theory


See also compact object.

In algebraic geoemtry

See at morphism of finite type for the notion in algebraic geometry.

In Abelian categories

An object XX in an AB5-category? CC is of finite type if one of the following equivalent conditions hold:

(i) any complete directed set {X i} iI\{X_i\}_{i\in I} of subobjects of XX is stationary

(ii) for any complete directed set {Y i} iI\{Y_i\}_{i\in I} of subobjects of an object YY the natural morphism colim iC(X,Y i)C(X,Y)colim_i C(X,Y_i) \to C(X,Y) is an isomorphism.

An object XX is finitely presented if it is of finite type and if for any epimorphism p:YXp:Y\to X where YY is of finite type, it follows that kerpker\,p is also of finite type. An object XX in an AB5 category is coherent if it is of finite type and for any morphism f:YXf: Y\to X of finite type kerfker\,f is of finite type.

For an exact sequence 0XXX00\to X'\to X\to X''\to 0 in an AB5 category the following hold:

(a) if XX' and XX'' are finitely presented, then XX is finitely presented;

(b) if XX is finitely presented and XX' of finite type, then XX'' is finitely presented;

(c) if XX is coherent and XX' of finite type then XX'' is also coherent.

For a module MM over a ring RR this is equivalent to MM being finitely generated RR-module. It is finitely presented if it is finitely presented in the usual sense of existence of short exact sequence of the form R IR JM0R^I\to R^J\to M\to 0 where II and JJ are finite.

In homotopical algebra

A graded object is often said to be of finite type if it is degreewise of finite dimension/rank, in some sense.

The terminology is used specifically in rational homotopy theory.

Notably a rational space is said to be of finite type if all its rational homotopy groups are finite dimensional vector spaces over the rational numbers.

Accordingly, chain complex of vector spaces, possibly that generating a semifree dga is said to be of finite type if it is degreewise finite dimensional.

Beware however that the terminology clashes with the use in homotopy theory, there the concept of finite homotopy type is crucially different from homotopy type with finite homotopy groups.

In stable homotopy theory

A spectrum (in the sense of stable homotopy theory) is one whose cohomology is finitely generated in each degree, but could exist in infinitely many degrees.


Revised on January 17, 2015 09:20:53 by Urs Schreiber (