Properties of morphisms
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 IV.II, 2.7.1(xiii) | EGA IV.III, 8.10.5(viii) |
| bijective | YES | NO | EGA IV.II, 2.6.1(iv) | NO | |
| closed | EGA I, 2.2.6 | EGA I, 2.2.7(i) | NO | EGA IV.II, 2.6.2(ii) | |
| closed immersion | EGA I, 4.2.1 | EGA I, 4.2.5 | EGA I, 4.3.2 | EGA IV.II, 2.7.1(xii) | EGA IV.III, 8.10.5(iv) |
| dominant | EGA I, 2.2.6 | EGA I, 2.2.7(i) | NO | ||
| etale | EGA IV.IV, 17.3.1 | EGA IV.IV, 17.3.3(ii) | EGA IV.IV, 17.3.3(ii) | EGA IV.IV, 17.7.4(vi) | EGA IV.IV, 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 IV.II, 2.7.1(xv) | EGA IV.III, 8.10.5(x) |
| finite presentation | EGA IV.I, 1.6.1 | EGA IV.I, 1.6.2(ii) | EGA IV.I, 1.6.2(iii) | EGA IV.II, 2.7.1(vi) | |
| finite type | EGA I, 6.3.1 | EGA I, 6.3.4(ii) | EGA I, 6.3.4(iv) | EGA IV.II, 2.7.1(v) | |
| flat | EGA I, 0:6.7.1 | EGA IV.II, 2.1.6 | EGA IV.II, 2.1.4 | EGA IV.II, 2.2.11(iv) | |
| formally etale | EGA IV.IV, 17.1.1 | EGA IV.IV, 17.1.3(ii) | EGA IV.IV, 17.1.3(iii) | ||
| formally smooth | EGA IV.IV, 17.1.1 | EGA IV.IV, 17.1.3(ii) | EGA IV.IV, 17.1.3(iii) | ||
| formally unram. | EGA IV.IV, 17.1.1 | EGA IV.IV, 17.1.3(ii) | EGA IV.IV, 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 IV.II, 2.6.2(iv) | ||
| immersion | EGA I, 4.2.1 | EGA I, 4.2.5 | EGA I, 4.3.2 | EGA IV.III, 8.10.5(iii) | |
| injective | EGA I, 3.5.11 | YES | NO | EGA IV.II, 2.6.1(ii) | NO |
| isomorphism | EGA I, 2.2.2 | YES | YES | EGA IV.II, 2.7.1(viii) | EGA IV.III, 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 IV.I, 1.4.2 | EGA IV.I, 1.4.2(ii) | EGA IV.I, 1.4.2(iii) | EGA IV.II, 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 IV.II, 2.7.1(iii) | |
| monomorphism | EGA I, 0:4.1.1 | YES | YES | EGA IV.II, 2.7.1(ix) | EGA IV.III, 8.10.5(ii) |
| open | EGA I, 2.2.6 | EGA I, 2.2.7(i) | NO | EGA IV.II, 2.6.2(i) | |
| open immersion | EGA I, 4.2.1 | EGA I, 4.2.5 | EGA I, 4.3.2 | EGA IV.II, 2.7.1(x) | EGA IV.III, 8.10.5(iii) |
| projective | EGA II, 5.5.2 | EGA II, 5.5.5(ii). | EGA II, 5.5.5(iii) | NO. | EGA IV.III, 8.10.5(xiii). |
| proper | EGA II, 5.4.1 | EGA II, 5.4.2(ii) | EGA II, 5.4.2(iii) | EGA IV.II, 2.7.1(vii) | EGA IV.III, 8.10.5(xii) |
| quasi-affine | EGA II, 5.1.1 | EGA II, 5.1.10(ii) | EGA II, 5.1.10(iii) | EGA IV.II, 2.7.1(xiv) | EGA IV.III, 8.10.5(ix) |
| quasi-compact | EGA I, 6.6.1 | EGA I, 6.6.4(ii) | EGA I, 6.6.4(iii) | EGA IV.II, 2.6.4(v) | |
| quasi-finite | EGA II, 6.2.3 | EGA II, 6.2.4(ii) | EGA II, 6.2.4(iii) | EGA IV.II, 2.7.1(xvi) | EGA IV.III, 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 IV.III, 8.10.5(xiv). |
| quasi-separated | EGA IV.I, 1.2.1 | EGA IV.I, 1.2.2(ii) | EGA IV.I, 1.2.2(iii) | EGA IV.II, 2.7.1(ii) | |
| radicial | EGA I, 3.5.4 | EGA I, 3.5.6(i) | EGA I, 3.5.7(ii) | EGA IV.II, 2.6.1(v) | EGA IV.III, 8.10.5(vii) |
| sch.-th. dominant | EGA IV.III, 11.10.2 | YES | NO | EGA IV.III, 11.10.5(i) | |
| separated | EGA I, 5.4.1 | EGA I, 5.5.1(ii) | EGA I, 5.5.1(iv) | EGA IV.II, 2.7.1(i) | EGA IV.III, 8.10.5(v) |
| smooth | EGA IV.IV, 17.3.1 | EGA IV.IV, 17.3.3(ii) | EGA IV.IV, 17.3.3(iii) | EGA IV.IV, 17.7.4(v) | EGA IV.IV, 17.7.8(ii). |
| surjective | EGA I, 2.2.6 | EGA I, 2.2.7(i) | EGA I, 3.5.2(ii) | EGA IV.II, 2.6.1(i) | EGA IV.III, 8.10.5(vi) |
| univ. bicontinuous | EGA IV.II, 2.4.2 | EGA IV.II, 2.4.3(ii) | EGA IV.II, 2.4.3(iii) | ||
| univ. closed | EGA II, 5.4.9 | EGA IV.II, 2.4.3(ii) | EGA IV.II, 2.4.3(iii) | EGA IV.II, 2.6.4(ii) | |
| univ. homeom. | EGA IV.II, 2.4.2 | EGA IV.II, 2.4.3(ii) | EGA IV.II, 2.4.3(iii) | EGA IV.II, 2.6.4(iv) | |
| univ. open | EGA IV.II, 2.4.2 | EGA IV.II, 2.4.3(ii) | EGA IV.II, 2.4.3(iii) | EGA IV.II, 2.6.4(i) | |
| unramified | EGA IV.IV, 17.3.1 | EGA IV.IV, 17.3.3(ii) | EGA IV.IV, 17.3.3(iii) | EGA IV.IV, 17.7.4(iv) | EGA IV.IV, 17.7.8(ii). |