Properties of morphisms: Difference between revisions
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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] | =Bjorn Poonen's table= | ||
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]. I copied it because it's slightly more accessible to me as a web page. | |||
{|class="wikitable" | {|class="wikitable" | ||
| Line 50: | Line 52: | ||
|EGA IV<sub>4</sub>, 17.3.3(ii) | |EGA IV<sub>4</sub>, 17.3.3(ii) | ||
|EGA IV<sub>4</sub>, 17.7.4(vi) | |EGA IV<sub>4</sub>, 17.7.4(vi) | ||
|EGA IV<sub>4</sub>, 17.7.8(ii) | |EGA IV<sub>4</sub>, 17.7.8(ii) | ||
|- | |- | ||
|faithfully flat | |faithfully flat | ||
| Line 202: | Line 204: | ||
|EGA II, 5.5.5(iii) | |EGA II, 5.5.5(iii) | ||
|NO. | |NO. | ||
|EGA IV<sub>3</sub>, 8.10.5(xiii) | |EGA IV<sub>3</sub>, 8.10.5(xiii) | ||
|- | |- | ||
|proper | |proper | ||
| Line 234: | Line 236: | ||
|quasi-projective | |quasi-projective | ||
|EGA II, 5.3.1 | |EGA II, 5.3.1 | ||
|EGA II, 5.3.4(ii) | |EGA II, 5.3.4(ii) | ||
|EGA II, 5.3.4(iii) | |EGA II, 5.3.4(iii) | ||
|NO | |NO | ||
|EGA IV<sub>3</sub>, 8.10.5(xiv) | |EGA IV<sub>3</sub>, 8.10.5(xiv) | ||
|- | |- | ||
|quasi-separated | |quasi-separated | ||
| Line 272: | Line 274: | ||
|EGA IV<sub>4</sub>, 17.3.3(iii) | |EGA IV<sub>4</sub>, 17.3.3(iii) | ||
|EGA IV<sub>4</sub>, 17.7.4(v) | |EGA IV<sub>4</sub>, 17.7.4(v) | ||
|EGA IV<sub>4</sub>, 17.7.8(ii) | |EGA IV<sub>4</sub>, 17.7.8(ii) | ||
|- | |- | ||
|surjective | |surjective | ||
| Line 314: | Line 316: | ||
|EGA IV<sub>4</sub>, 17.3.3(iii) | |EGA IV<sub>4</sub>, 17.3.3(iii) | ||
|EGA IV<sub>4</sub>, 17.7.4(iv) | |EGA IV<sub>4</sub>, 17.7.4(iv) | ||
|EGA IV<sub>4</sub>, 17.7.8(ii) | |EGA IV<sub>4</sub>, 17.7.8(ii) | ||
|} | |} | ||
Revision as of 14:16, 7 October 2011
Bjorn Poonen's table
The following table is taken from page 179 of Bjorn Poonen's Rational points on varieties. I copied it because it's slightly more accessible to me as a web page.
| 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) |