nLab object of finite type

This entry is about “objects of finite type” in algebra, homological algebra and rational homotopy theory. For finite homotopy types and π-finite homotopy types in homotopy theory see there. For related notions in category theory see at compact object. For finite types in type theory and in homotopy type theory see at inductive family. For more disambigation see at finite type.

Contents

Idea

The terminology of “objects of finite type” or of objects that “have finite type” is common in various contexts and usually means something like finitely generated object; but beware that conventional usage across contexts is not fully systematic.

Definitions

In algebraic geometry

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

In Abelian categories

An object $X$ in an AB5-category $C$ is of finite type if one of the following equivalent conditions hold:

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

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

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

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

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

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

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

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

In homotopical algebra and rational homotopy theory

A graded object is often said to be of finite type if it is degreewise of finite dimension/rank, in some sense. This terminology is used specifically in rational homotopy theory.

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

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

Beware however that the terminology clashes somewhat 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 of finite type (in the sense of stable homotopy theory) is one whose cohomology is finitely generated in each degree, but could exist in infinitely many degrees.

check

See also/instead at finite spectrum.

References

• Nicolae Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375

• MathOverflow m-oplus-n-is-of-finite-type-if-m-n-are-of-finite-type