Finiteness conditions of algebraic geometry

Many statements in the mathematical subfield of commutative algebra and algebraic geometry depend on certain finiteness conditions.

If a definition is formulated only for algebras , the corresponding statement for geometric objects is defined by local maps.

Let it be a ring. ${\ displaystyle A}$

term	Explanation
at last	an -algebra is called finite if the -module is finitely generated, i.e. H. if there is a surjection of modules. ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle B}$ ${\ displaystyle A}$ ${\ displaystyle A ^ {n} \ to B}$ ${\ displaystyle A}$
finite type (algebra)	an -algebra is of finite type if it is generated finitely as an -algebra, i.e. H. if there is a surjection of algebras. ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle A}$ ${\ displaystyle A [X_ {1}, \ ldots, X_ {n}] \ to B}$ ${\ displaystyle A}$
finite type (module)	a module is of finite type if it is finitely generated, i.e. H. if there is a surjection of modules. ${\ displaystyle A}$ ${\ displaystyle M}$ ${\ displaystyle A ^ {n} \ to M}$ ${\ displaystyle A}$
finite type (scheme)	a schema morphism is of finite type if the archetype of an open affine subset of is a finite union of affine subsets such that for each an -algebra is of finite type. ${\ displaystyle X \ to Y}$ ${\ displaystyle U}$ ${\ displaystyle Y}$ ${\ displaystyle V_ {i}}$ ${\ displaystyle \ Gamma (V_ {i}, {\ mathcal {O}} _ {X})}$ ${\ displaystyle i}$ ${\ displaystyle \ Gamma (U, {\ mathcal {O}} _ {Y})}$
finally presented (module)	a -module is finite if it is a coke of a homomorphism between free modules of finite type. ${\ displaystyle A}$ ${\ displaystyle M}$
locally finite type	a schema morphism is locally finite type if there is a neighborhood as well as a neighborhood for each point , so that the morphism is finite type . ${\ displaystyle f \ colon X \ to Y}$ ${\ displaystyle x \ in X}$ ${\ displaystyle U}$ ${\ displaystyle V \ supseteq f (U)}$ ${\ displaystyle f}$ ${\ displaystyle U \ to V}$
quasi-endless	A schema morphism is quasi- endless if it is of finite type and all fibers are discrete; equivalent to: if it is of finite type and the fibers are finite (as morphisms). A schema morphism is quasi- endless at a point if there are affine open environments or from or with , so that is quasi-endless. ${\ displaystyle X \ to Y}$ ${\ displaystyle X \ to Y}$ ${\ displaystyle x \ in X}$ ${\ displaystyle U}$ ${\ displaystyle V}$ ${\ displaystyle x}$ ${\ displaystyle f (x)}$ ${\ displaystyle f (U) \ subseteq V}$ ${\ displaystyle U \ to V}$
locally quasi-indulgent	A schema morphism is locally quasi-infinite if it is quasi-infinite at every point. ${\ displaystyle X \ to Y}$
quasi-compact	a schema morphism is quasi-compact if the archetype of every open, quasi-compact subset of is again quasi-compact. ${\ displaystyle X \ to Y}$ ${\ displaystyle Y}$