Properties of morphisms: Difference between revisions

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(Created page with "The following table is taken from page 179 of [http://www-math.mit.edu/~poonen/ Bjorn Poonen's] [http://math.mit.edu/~poonen/papers/Qpoints.pdf Rational points on varieties] {|c...")
 
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Line 14: Line 14:
|EGA II, 1.6.2(ii)
|EGA II, 1.6.2(ii)
|EGA II, 1.6.2(iii)
|EGA II, 1.6.2(iii)
|EGA IV.II, 2.7.1(xiii)
|EGA IV<sub>2</sub>, 2.7.1(xiii)
|EGA IV.III, 8.10.5(viii)
|EGA IV<sub>3</sub>, 8.10.5(viii)
|-
|-
|bijective
|bijective
Line 21: Line 21:
|YES
|YES
|NO
|NO
|EGA IV.II, 2.6.1(iv)
|EGA IV<sub>2</sub>, 2.6.1(iv)
|NO
|NO
|-
|-
Line 28: Line 28:
|EGA I, 2.2.7(i)
|EGA I, 2.2.7(i)
|NO
|NO
|EGA IV.II, 2.6.2(ii)
|EGA IV<sub>2</sub>, 2.6.2(ii)
|
|
|-
|-
Line 35: Line 35:
|EGA I, 4.2.5
|EGA I, 4.2.5
|EGA I, 4.3.2
|EGA I, 4.3.2
|EGA IV.II, 2.7.1(xii)
|EGA IV<sub>2</sub>, 2.7.1(xii)
|EGA IV.III, 8.10.5(iv)
|EGA IV<sub>3</sub>, 8.10.5(iv)
|-
|-
|dominant
|dominant
Line 46: Line 46:
|-
|-
|etale
|etale
|EGA IV.IV, 17.3.1
|EGA IV<sub>4</sub>, 17.3.1
|EGA IV.IV, 17.3.3(ii)
|EGA IV<sub>4</sub>, 17.3.3(ii)
|EGA IV.IV, 17.3.3(ii)
|EGA IV<sub>4</sub>, 17.3.3(ii)
|EGA IV.IV, 17.7.4(vi)
|EGA IV<sub>4</sub>, 17.7.4(vi)
|EGA IV.IV, 17.7.8(ii).
|EGA IV<sub>4</sub>, 17.7.8(ii).
|-
|-
|faithfully flat
|faithfully flat
Line 63: Line 63:
|EGA II, 6.1.5(ii)
|EGA II, 6.1.5(ii)
|EGA II, 6.1.5(iii)
|EGA II, 6.1.5(iii)
|EGA IV.II, 2.7.1(xv)
|EGA IV<sub>2</sub>, 2.7.1(xv)
|EGA IV.III, 8.10.5(x)
|EGA IV<sub>3</sub>, 8.10.5(x)
|-
|-
|finite presentation
|finite presentation
|EGA IV.I, 1.6.1
|EGA IV<sub>1</sub>, 1.6.1
|EGA IV.I, 1.6.2(ii)
|EGA IV<sub>1</sub>, 1.6.2(ii)
|EGA IV.I, 1.6.2(iii)
|EGA IV<sub>1</sub>, 1.6.2(iii)
|EGA IV.II, 2.7.1(vi)
|EGA IV<sub>2</sub>, 2.7.1(vi)
|
|
|-
|-
Line 77: Line 77:
|EGA I, 6.3.4(ii)
|EGA I, 6.3.4(ii)
|EGA I, 6.3.4(iv)
|EGA I, 6.3.4(iv)
|EGA IV.II, 2.7.1(v)
|EGA IV<sub>2</sub>, 2.7.1(v)
|
|
|-
|-
|flat
|flat
|EGA I, 0:6.7.1
|EGA I, 0:6.7.1
|EGA IV.II, 2.1.6
|EGA IV<sub>2</sub>, 2.1.6
|EGA IV.II, 2.1.4
|EGA IV<sub>2</sub>, 2.1.4
|EGA IV.II, 2.2.11(iv)
|EGA IV<sub>2</sub>, 2.2.11(iv)
|
|
|-
|-
|formally etale
|formally etale
|EGA IV.IV, 17.1.1
|EGA IV<sub>4</sub>, 17.1.1
|EGA IV.IV, 17.1.3(ii)
|EGA IV<sub>4</sub>, 17.1.3(ii)
|EGA IV.IV, 17.1.3(iii)
|EGA IV<sub>4</sub>, 17.1.3(iii)
|
|
|
|
|-
|-
|formally smooth
|formally smooth
|EGA IV.IV, 17.1.1
|EGA IV<sub>4</sub>, 17.1.1
|EGA IV.IV, 17.1.3(ii)
|EGA IV<sub>4</sub>, 17.1.3(ii)
|EGA IV.IV, 17.1.3(iii)
|EGA IV<sub>4</sub>, 17.1.3(iii)
|
|
|
|
|-
|-
|formally unram.
|formally unram.
|EGA IV.IV, 17.1.1
|EGA IV<sub>4</sub>, 17.1.1
|EGA IV.IV, 17.1.3(ii)
|EGA IV<sub>4</sub>, 17.1.3(ii)
|EGA IV.IV, 17.1.3(iii)
|EGA IV<sub>4</sub>, 17.1.3(iii)
|
|
|
|
|-
|-
|fppf  
|fppf
|Definition 3.4.1
|Definition 3.4.1
|YES
|YES
Line 126: Line 126:
|YES
|YES
|NO
|NO
|EGA IV.II, 2.6.2(iv)
|EGA IV<sub>2</sub>, 2.6.2(iv)
|-
|-
|immersion
|immersion
Line 133: Line 133:
|EGA I, 4.3.2
|EGA I, 4.3.2
|
|
|EGA IV.III, 8.10.5(iii)
|EGA IV<sub>3</sub>, 8.10.5(iii)
|-
|-
|injective
|injective
Line 139: Line 139:
|YES
|YES
|NO
|NO
|EGA IV.II, 2.6.1(ii)
|EGA IV<sub>2</sub>, 2.6.1(ii)
|NO
|NO
|-
|-
Line 146: Line 146:
|YES
|YES
|YES
|YES
|EGA IV.II, 2.7.1(viii)
|EGA IV<sub>2</sub>, 2.7.1(viii)
|EGA IV.III, 8.10.5(i)
|EGA IV<sub>3</sub>, 8.10.5(i)
|-
|-
|loc. immersion
|loc. immersion
Line 164: Line 164:
|-
|-
|loc. of finite pres.
|loc. of finite pres.
|EGA IV.I, 1.4.2
|EGA IV<sub>1</sub>, 1.4.2
|EGA IV.I, 1.4.2(ii)
|EGA IV<sub>1</sub>, 1.4.2(ii)
|EGA IV.I, 1.4.2(iii)
|EGA IV<sub>1</sub>, 1.4.2(iii)
|EGA IV.II, 2.7.1(iv)
|EGA IV<sub>2</sub>, 2.7.1(iv)
|
|
|-
|-
Line 174: Line 174:
|EGA I, 6.6.6(ii)
|EGA I, 6.6.6(ii)
|EGA I, 6.6.6(iii)
|EGA I, 6.6.6(iii)
|EGA IV.II, 2.7.1(iii)
|EGA IV<sub>2</sub>, 2.7.1(iii)
|-
|-
|monomorphism
|monomorphism
Line 180: Line 180:
|YES
|YES
|YES
|YES
|EGA IV.II, 2.7.1(ix)
|EGA IV<sub>2</sub>, 2.7.1(ix)
|EGA IV.III, 8.10.5(ii)
|EGA IV<sub>3</sub>, 8.10.5(ii)
|-
|-
|open
|open
Line 187: Line 187:
|EGA I, 2.2.7(i)
|EGA I, 2.2.7(i)
|NO
|NO
|EGA IV.II, 2.6.2(i)
|EGA IV<sub>2</sub>, 2.6.2(i)
|
|
|-
|-
Line 194: Line 194:
|EGA I, 4.2.5
|EGA I, 4.2.5
|EGA I, 4.3.2
|EGA I, 4.3.2
|EGA IV.II, 2.7.1(x)
|EGA IV<sub>2</sub>, 2.7.1(x)
|EGA IV.III, 8.10.5(iii)
|EGA IV<sub>3</sub>, 8.10.5(iii)
|-
|-
|projective
|projective
Line 202: Line 202:
|EGA II, 5.5.5(iii)
|EGA II, 5.5.5(iii)
|NO.
|NO.
|EGA IV.III, 8.10.5(xiii).
|EGA IV<sub>3</sub>, 8.10.5(xiii).
|-
|-
|proper
|proper
Line 208: Line 208:
|EGA II, 5.4.2(ii)
|EGA II, 5.4.2(ii)
|EGA II, 5.4.2(iii)
|EGA II, 5.4.2(iii)
|EGA IV.II, 2.7.1(vii)
|EGA IV<sub>2</sub>, 2.7.1(vii)
|EGA IV.III, 8.10.5(xii)
|EGA IV<sub>3</sub>, 8.10.5(xii)
|-
|-
|quasi-affine
|quasi-affine
Line 215: Line 215:
|EGA II, 5.1.10(ii)
|EGA II, 5.1.10(ii)
|EGA II, 5.1.10(iii)
|EGA II, 5.1.10(iii)
|EGA IV.II, 2.7.1(xiv)
|EGA IV<sub>2</sub>, 2.7.1(xiv)
|EGA IV.III, 8.10.5(ix)
|EGA IV<sub>3</sub>, 8.10.5(ix)
|-
|-
|quasi-compact
|quasi-compact
Line 222: Line 222:
|EGA I, 6.6.4(ii)
|EGA I, 6.6.4(ii)
|EGA I, 6.6.4(iii)
|EGA I, 6.6.4(iii)
|EGA IV.II, 2.6.4(v)
|EGA IV<sub>2</sub>, 2.6.4(v)
|
|
|-
|-
Line 229: Line 229:
|EGA II, 6.2.4(ii)
|EGA II, 6.2.4(ii)
|EGA II, 6.2.4(iii)
|EGA II, 6.2.4(iii)
|EGA IV.II, 2.7.1(xvi)
|EGA IV<sub>2</sub>, 2.7.1(xvi)
|EGA IV.III, 8.10.5(xi)
|EGA IV<sub>3</sub>, 8.10.5(xi)
|-
|-
|quasi-projective
|quasi-projective
Line 237: Line 237:
|EGA II, 5.3.4(iii)
|EGA II, 5.3.4(iii)
|NO.
|NO.
|EGA IV.III, 8.10.5(xiv).
|EGA IV<sub>3</sub>, 8.10.5(xiv).
|-
|-
|quasi-separated
|quasi-separated
|EGA IV.I, 1.2.1
|EGA IV<sub>1</sub>, 1.2.1
|EGA IV.I, 1.2.2(ii)
|EGA IV<sub>1</sub>, 1.2.2(ii)
|EGA IV.I, 1.2.2(iii)
|EGA IV<sub>1</sub>, 1.2.2(iii)
|EGA IV.II, 2.7.1(ii)
|EGA IV<sub>2</sub>, 2.7.1(ii)
|
|
|-
|-
Line 250: Line 250:
|EGA I, 3.5.6(i)
|EGA I, 3.5.6(i)
|EGA I, 3.5.7(ii)
|EGA I, 3.5.7(ii)
|EGA IV.II, 2.6.1(v)
|EGA IV<sub>2</sub>, 2.6.1(v)
|EGA IV.III, 8.10.5(vii)
|EGA IV<sub>3</sub>, 8.10.5(vii)
|-
|-
|sch.-th. dominant
|sch.-th. dominant
|EGA IV.III, 11.10.2
|EGA IV<sub>3</sub>, 11.10.2
|YES
|YES
|NO
|NO
|EGA IV.III, 11.10.5(i)
|EGA IV<sub>3</sub>, 11.10.5(i)
|
|
|-
|-
Line 264: Line 264:
|EGA I, 5.5.1(ii)
|EGA I, 5.5.1(ii)
|EGA I, 5.5.1(iv)
|EGA I, 5.5.1(iv)
|EGA IV.II, 2.7.1(i)
|EGA IV<sub>2</sub>, 2.7.1(i)
|EGA IV.III, 8.10.5(v)
|EGA IV<sub>3</sub>, 8.10.5(v)
|-
|-
|smooth
|smooth
|EGA IV.IV, 17.3.1
|EGA IV<sub>4</sub>, 17.3.1
|EGA IV.IV, 17.3.3(ii)
|EGA IV<sub>4</sub>, 17.3.3(ii)
|EGA IV.IV, 17.3.3(iii)
|EGA IV<sub>4</sub>, 17.3.3(iii)
|EGA IV.IV, 17.7.4(v)
|EGA IV<sub>4</sub>, 17.7.4(v)
|EGA IV.IV, 17.7.8(ii).
|EGA IV<sub>4</sub>, 17.7.8(ii).
|-
|-
|surjective
|surjective
Line 278: Line 278:
|EGA I, 2.2.7(i)
|EGA I, 2.2.7(i)
|EGA I, 3.5.2(ii)
|EGA I, 3.5.2(ii)
|EGA IV.II, 2.6.1(i)
|EGA IV<sub>2</sub>, 2.6.1(i)
|EGA IV.III, 8.10.5(vi)
|EGA IV<sub>3</sub>, 8.10.5(vi)
|-
|-
|univ. bicontinuous
|univ. bicontinuous
|EGA IV.II, 2.4.2
|EGA IV<sub>2</sub>, 2.4.2
|EGA IV.II, 2.4.3(ii)
|EGA IV<sub>2</sub>, 2.4.3(ii)
|EGA IV.II, 2.4.3(iii)
|EGA IV<sub>2</sub>, 2.4.3(iii)
|
|
|
|
Line 290: Line 290:
|univ. closed
|univ. closed
|EGA II, 5.4.9
|EGA II, 5.4.9
|EGA IV.II, 2.4.3(ii)
|EGA IV<sub>2</sub>, 2.4.3(ii)
|EGA IV.II, 2.4.3(iii)
|EGA IV<sub>2</sub>, 2.4.3(iii)
|EGA IV.II, 2.6.4(ii)
|EGA IV<sub>2</sub>, 2.6.4(ii)
|
|
|-
|-
|univ. homeom.
|univ. homeom.
|EGA IV.II, 2.4.2
|EGA IV<sub>2</sub>, 2.4.2
|EGA IV.II, 2.4.3(ii)
|EGA IV<sub>2</sub>, 2.4.3(ii)
|EGA IV.II, 2.4.3(iii)
|EGA IV<sub>2</sub>, 2.4.3(iii)
|EGA IV.II, 2.6.4(iv)
|EGA IV<sub>2</sub>, 2.6.4(iv)
|
|
|-
|-
|univ. open
|univ. open
|EGA IV.II, 2.4.2
|EGA IV<sub>2</sub>, 2.4.2
|EGA IV.II, 2.4.3(ii)
|EGA IV<sub>2</sub>, 2.4.3(ii)
|EGA IV.II, 2.4.3(iii)
|EGA IV<sub>2</sub>, 2.4.3(iii)
|EGA IV.II, 2.6.4(i)
|EGA IV<sub>2</sub>, 2.6.4(i)
|
|
|-
|-
|unramified
|unramified
|EGA IV.IV, 17.3.1
|EGA IV<sub>4</sub>, 17.3.1
|EGA IV.IV, 17.3.3(ii)
|EGA IV<sub>4</sub>, 17.3.3(ii)
|EGA IV.IV, 17.3.3(iii)
|EGA IV<sub>4</sub>, 17.3.3(iii)
|EGA IV.IV, 17.7.4(iv)
|EGA IV<sub>4</sub>, 17.7.4(iv)
|EGA IV.IV, 17.7.8(ii).
|EGA IV<sub>4</sub>, 17.7.8(ii).
|}
|}

Revision as of 21:23, 6 October 2011

The following table is taken from page 179 of Bjorn Poonen's Rational points on varieties

Definition Composition Base Change fpqc Descent Spreading Out
affine EGA II, 1.6.1 EGA II, 1.6.2(ii) EGA II, 1.6.2(iii) EGA IV2, 2.7.1(xiii) EGA IV3, 8.10.5(viii)
bijective YES NO EGA IV2, 2.6.1(iv) NO
closed EGA I, 2.2.6 EGA I, 2.2.7(i) NO EGA IV2, 2.6.2(ii)
closed immersion EGA I, 4.2.1 EGA I, 4.2.5 EGA I, 4.3.2 EGA IV2, 2.7.1(xii) EGA IV3, 8.10.5(iv)
dominant EGA I, 2.2.6 EGA I, 2.2.7(i) NO
etale EGA IV4, 17.3.1 EGA IV4, 17.3.3(ii) EGA IV4, 17.3.3(ii) EGA IV4, 17.7.4(vi) EGA IV4, 17.7.8(ii).
faithfully flat EGA I, 0:6.7.8 YES YES YES
finite EGA II, 6.1.1 EGA II, 6.1.5(ii) EGA II, 6.1.5(iii) EGA IV2, 2.7.1(xv) EGA IV3, 8.10.5(x)
finite presentation EGA IV1, 1.6.1 EGA IV1, 1.6.2(ii) EGA IV1, 1.6.2(iii) EGA IV2, 2.7.1(vi)
finite type EGA I, 6.3.1 EGA I, 6.3.4(ii) EGA I, 6.3.4(iv) EGA IV2, 2.7.1(v)
flat EGA I, 0:6.7.1 EGA IV2, 2.1.6 EGA IV2, 2.1.4 EGA IV2, 2.2.11(iv)
formally etale EGA IV4, 17.1.1 EGA IV4, 17.1.3(ii) EGA IV4, 17.1.3(iii)
formally smooth EGA IV4, 17.1.1 EGA IV4, 17.1.3(ii) EGA IV4, 17.1.3(iii)
formally unram. EGA IV4, 17.1.1 EGA IV4, 17.1.3(ii) EGA IV4, 17.1.3(iii)
fppf Definition 3.4.1 YES YES YES
fpqc Vis05, 2.34 Vis05, 2.35(i) Vis05, 2.35(v)
homeomorphism YES NO EGA IV2, 2.6.2(iv)
immersion EGA I, 4.2.1 EGA I, 4.2.5 EGA I, 4.3.2 EGA IV3, 8.10.5(iii)
injective EGA I, 3.5.11 YES NO EGA IV2, 2.6.1(ii) NO
isomorphism EGA I, 2.2.2 YES YES EGA IV2, 2.7.1(viii) EGA IV3, 8.10.5(i)
loc. immersion EGA I, 4.5.1 EGA I, 4.5.5(i) EGA I, 4.5.5(iii)
loc. isomorphism EGA I, 4.5.2 EGA I, 4.5.5(i) EGA I, 4.5.5(iii)
loc. of finite pres. EGA IV1, 1.4.2 EGA IV1, 1.4.2(ii) EGA IV1, 1.4.2(iii) EGA IV2, 2.7.1(iv)
loc. of finite type EGA I, 6.6.2 EGA I, 6.6.6(ii) EGA I, 6.6.6(iii) EGA IV2, 2.7.1(iii)
monomorphism EGA I, 0:4.1.1 YES YES EGA IV2, 2.7.1(ix) EGA IV3, 8.10.5(ii)
open EGA I, 2.2.6 EGA I, 2.2.7(i) NO EGA IV2, 2.6.2(i)
open immersion EGA I, 4.2.1 EGA I, 4.2.5 EGA I, 4.3.2 EGA IV2, 2.7.1(x) EGA IV3, 8.10.5(iii)
projective EGA II, 5.5.2 EGA II, 5.5.5(ii). EGA II, 5.5.5(iii) NO. EGA IV3, 8.10.5(xiii).
proper EGA II, 5.4.1 EGA II, 5.4.2(ii) EGA II, 5.4.2(iii) EGA IV2, 2.7.1(vii) EGA IV3, 8.10.5(xii)
quasi-affine EGA II, 5.1.1 EGA II, 5.1.10(ii) EGA II, 5.1.10(iii) EGA IV2, 2.7.1(xiv) EGA IV3, 8.10.5(ix)
quasi-compact EGA I, 6.6.1 EGA I, 6.6.4(ii) EGA I, 6.6.4(iii) EGA IV2, 2.6.4(v)
quasi-finite EGA II, 6.2.3 EGA II, 6.2.4(ii) EGA II, 6.2.4(iii) EGA IV2, 2.7.1(xvi) EGA IV3, 8.10.5(xi)
quasi-projective EGA II, 5.3.1 EGA II, 5.3.4(ii). EGA II, 5.3.4(iii) NO. EGA IV3, 8.10.5(xiv).
quasi-separated EGA IV1, 1.2.1 EGA IV1, 1.2.2(ii) EGA IV1, 1.2.2(iii) EGA IV2, 2.7.1(ii)
radicial EGA I, 3.5.4 EGA I, 3.5.6(i) EGA I, 3.5.7(ii) EGA IV2, 2.6.1(v) EGA IV3, 8.10.5(vii)
sch.-th. dominant EGA IV3, 11.10.2 YES NO EGA IV3, 11.10.5(i)
separated EGA I, 5.4.1 EGA I, 5.5.1(ii) EGA I, 5.5.1(iv) EGA IV2, 2.7.1(i) EGA IV3, 8.10.5(v)
smooth EGA IV4, 17.3.1 EGA IV4, 17.3.3(ii) EGA IV4, 17.3.3(iii) EGA IV4, 17.7.4(v) EGA IV4, 17.7.8(ii).
surjective EGA I, 2.2.6 EGA I, 2.2.7(i) EGA I, 3.5.2(ii) EGA IV2, 2.6.1(i) EGA IV3, 8.10.5(vi)
univ. bicontinuous EGA IV2, 2.4.2 EGA IV2, 2.4.3(ii) EGA IV2, 2.4.3(iii)
univ. closed EGA II, 5.4.9 EGA IV2, 2.4.3(ii) EGA IV2, 2.4.3(iii) EGA IV2, 2.6.4(ii)
univ. homeom. EGA IV2, 2.4.2 EGA IV2, 2.4.3(ii) EGA IV2, 2.4.3(iii) EGA IV2, 2.6.4(iv)
univ. open EGA IV2, 2.4.2 EGA IV2, 2.4.3(ii) EGA IV2, 2.4.3(iii) EGA IV2, 2.6.4(i)
unramified EGA IV4, 17.3.1 EGA IV4, 17.3.3(ii) EGA IV4, 17.3.3(iii) EGA IV4, 17.7.4(iv) EGA IV4, 17.7.8(ii).
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