A proper morphism of schemes is a separated morphism that is a morphism of finite type
and a universally closed morphism.
Universal closedness means that, for every morphism 
, the base change 
is a closed
map (Stacks Project 2026).
Properness is the algebraic-geometric analogue of the compact space condition. For example, projective space
over a field is proper over that field. A resolution
of singularities is normally required to be a proper birational
morphism.
See also
Base Change,
Birational Morphism,
Fiber Product,
Morphism,
Morphism of Finite Type,
Projective
Space,
Resolution of Singularities,
Separated Morphism,
Universally
Closed Morphism
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References
Hartshorne, R. Algebraic Geometry. New York: Springer-Verlag, 1977.The Stacks Project
Authors. "Proper Morphisms." §29.42 in The Stacks Project,
Tag 01W0, 2026. https://stacks.math.columbia.edu/tag/01W0.
Cite this as:
Weisstein, Eric W. "Proper Morphism."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ProperMorphism.html
Subject classifications
