finite type

(also nonabelian homological algebra)

and

See also *compact object*.

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

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.

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*.

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.

- 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

Revised on January 17, 2015 09:20:53
by Urs Schreiber
(195.113.30.252)