The Stacks project tables

Stability of properties of morphisms

base changecompositionlocal on the sourcelocal on the targetcancellationINDfpqc localdescent
closed immersionarbitraryseparated
open immersionarbitraryunramified
locally closed immersionarbitrary
quasi-compactarbitraryquasi-separated
quasi-compact immersionquasi-separated
universally closedarbitraryseparated
separatedarbitrary
quasi-separatedarbitrary
monomorphismarbitrary
surjectivearbitraryinjective
universally injectivearbitrary
affinearbitraryseparated
quasi-affinearbitraryquasi-separated
of finite typearbitraryquasi-separated
locally of finite typearbitrary
quasi-finitearbitraryquasi-separated
locally quasi-finitearbitrary
of finite presentationarbitraryquasi-separated and locally of finite type
locally of finite presentationarbitrarylocally of finite type
locally of finite type of relative dimension $d$arbitrary
local complete intersectionflat
universally openarbitraryunramified
flatarbitraryunramified
faithfully flatopen immersion
syntomicarbitrary
smootharbitraryunramified
unramifiedarbitrary
G-unramifiedarbitrary
etalearbitraryunramified
properarbitraryseparated
H-projectivearbitrary
projectivearbitraryseparated
quasi-projectivequasi-compactquasi-separated
locally projectivearbitrary
finitearbitraryseparated
integralarbitraryseparated
finite locally freearbitrary
bijective
closed
dominant
quasi-compact and dominantflat
formally etale
formally smooth
formally unramified
geometrically connected
geometrically integral
geometrically reduced
homeomorphism
immersion?quasi-compact
injective
isomorphism
local immersion
local isomorphism
open
purely inseparable
radicial
scheme-theoretically dominant
universally bijective
universally bicontinuous
universal homeomorphism

Stability of properties of objects in derived categories