In mathematics, specifically algebraic geometry, the valuative criteria are a collection of results that make it possible to decide whether a morphism of algebraic varieties, or more generally schemes, is universally closed, separated, or proper.
Recall that a valuation ring A is a domain, so if K is the field of fractions of A, then Spec K is the generic point of Spec A.
Then the following are equivalent:[1][2] The lifting condition is equivalent to specifying that the natural morphism is injective (resp.
Furthermore, in the special case when Y is (locally) Noetherian, it suffices to check the case that A is a discrete valuation ring.
This algebraic geometry–related article is a stub.