The Stacks project tables
Stability of properties of morphisms
base change
composition
local on the source
local on the target
cancellation
IND
fpqc local
descent
base change
composition
local on the source
local on the target
cancellation
IND
fpqc local
descent
closed immersion
arbitrary
✓
✗
✓
separated
✓
✓
✓
open immersion
arbitrary
✓
✗
✓
unramified
✓
✓
✓
locally closed immersion
arbitrary
✓
quasi-compact
arbitrary
✓
✗
✓
quasi-separated
✓
✓
✓
quasi-compact immersion
✓
✓
✗
✓
quasi-separated
✓
✓
universally closed
arbitrary
✓
✗
✓
separated
✗
✓
✓
separated
arbitrary
✓
✗
✓
✓
✓
✓
quasi-separated
arbitrary
✓
✗
✓
✓
✓
✓
✓
monomorphism
arbitrary
✓
✗
✓
✓
✓
✓
✓
surjective
arbitrary
✓
✗
✓
injective
✗
✓
✓
universally injective
arbitrary
✓
✗
✓
✓
✓
✓
✓
affine
arbitrary
✓
✗
✓
separated
✓
✓
✓
quasi-affine
arbitrary
✓
✗
✓
quasi-separated
✓
✓
✓
of finite type
arbitrary
✓
✗
✓
quasi-separated
✓
✓
✓
locally of finite type
arbitrary
✓
✓
✓
✓
✓
✓
✓
quasi-finite
arbitrary
✓
✗
✓
quasi-separated
✓
✓
✓
locally quasi-finite
arbitrary
✓
✓
of finite presentation
arbitrary
✓
✗
✓
quasi-separated and locally of finite type
✓
✓
locally of finite presentation
arbitrary
✓
✓
✓
locally of finite type
✓
✓
✓
locally of finite type of relative dimension $d$
arbitrary
✓
local complete intersection
flat
✓
✓
universally open
arbitrary
✓
✗
✓
unramified
✗
✓
✓
flat
arbitrary
✓
✓
✓
unramified
✓
✓
✓
faithfully flat
✓
✓
✗
✓
open immersion
✗
✓
syntomic
arbitrary
✓
✓
smooth
arbitrary
✓
✓
✓
unramified
✓
✓
✓
unramified
arbitrary
✓
✓
G-unramified
arbitrary
✓
✓
etale
arbitrary
✓
✓
✓
unramified
✓
✓
proper
arbitrary
✓
✓
separated
✓
✓
✓
H-projective
arbitrary
✓
projective
arbitrary
✗
✗
separated
✓
✗
✗
quasi-projective
✓
quasi-compact
✗
✗
quasi-separated
✓
✗
locally projective
arbitrary
finite
arbitrary
✓
✗
✓
separated
✓
✓
integral
arbitrary
✓
✗
✓
separated
✓
✓
✓
finite locally free
arbitrary
✓
✗
✓
✗
✓
✓
bijective
✗
✓
✗
✓
✗
✗
✓
closed
✗
✓
✗
✓
✗
✗
✓
dominant
✗
quasi-compact and dominant
flat
✓
✗
✗
✗
✗
✓
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