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.
(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
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.
See at morphism of finite type for the notion in algebraic geometry.
An object in an AB5-category is of finite type if one of the following equivalent conditions hold:
(i) any complete directed set of subobjects of is stationary
(ii) for any complete directed set of subobjects of an object the natural morphism is an isomorphism.
An object is finitely presented if it is of finite type and if for any epimorphism where is of finite type, it follows that is also of finite type. An object in an AB5 category is coherent if it is of finite type and for any morphism of finite type is of finite type.
For an exact sequence in an AB5 category the following hold:
(a) if and are finitely presented, then is finitely presented;
(b) if is finitely presented and of finite type, then is finitely presented;
(c) if is coherent and of finite type then is also coherent.
For a module over a ring this is equivalent to being finitely generated -module. It is finitely presented if it is finitely presented in the usual sense of existence of short exact sequence of the form where and are finite.
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.
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.
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
Last revised on July 9, 2021 at 07:22:10. See the history of this page for a list of all contributions to it.