Stacks Project -- Overview

Tag 0239 points to Section 31.1: Introduction
Tag 023A points to Section 31.2: Descent data for quasi-coherent sheaves
- Tag 023B points to Definition 31.2.1
- Tag 023C points to Lemma 31.2.2
- Tag 023D points to Definition 31.2.3
- Tag 023E points to Lemma 31.2.4
Tag 023F points to Section 31.3: Descent for modules
- Tag 023G points to Definition 31.3.1
- Tag 023H points to Lemma 31.3.2
- Tag 023I points to Lemma 31.3.3
- Tag 023J points to Definition 31.3.4
  - Tag 023K points to Equation 31.3.4.1
- Tag 023L points to Lemma 31.3.5
- Tag 023M points to Lemma 31.3.6
- Tag 039W points to Lemma 31.3.7
- Tag 039X points to Lemma 31.3.8
- Tag 023N points to Proposition 31.3.9
- Tag 023O points to Remark 31.3.10
- Tag 039Y points to Remark 31.3.11
Tag 08WE points to Section 31.4: Descent for universally injective morphisms
- Tag 08WF points to Subsection 31.4.1: Category-theoretic preliminaries
  - Tag 08WG points to Equation 31.4.1.1
  - Tag 08WH points to Definition 31.4.2
    - Tag 08WI points to Equation 31.4.2.1
- Tag 08WJ points to Subsection 31.4.3: Universally injective morphisms
  - Tag 08WK points to Remark 31.4.4
  - Tag 08WL points to Definition 31.4.5
  - Tag 08WM points to Example 31.4.6
  - Tag 08WN points to Example 31.4.7
  - Tag 08WP points to Lemma 31.4.8
  - Tag 08WQ points to Definition 31.4.9
  - Tag 08WR points to Lemma 31.4.10
  - Tag 08WS points to Remark 31.4.11
    - Tag 08WT points to Equation 31.4.11.1
  - Tag 08WU points to Lemma 31.4.12
  - Tag 08WV points to Remark 31.4.13
- Tag 08WW points to Subsection 31.4.14: Descent for modules and their morphisms
  - Tag 08WX points to Equation 31.4.14.1
  - Tag 08WY points to Definition 31.4.15
  - Tag 08WZ points to Lemma 31.4.16
    - Tag 08X0 points to Equation 31.4.16.1
  - Tag 08X1 points to Lemma 31.4.17
    - Tag 08X2 points to Equation 31.4.17.1
  - Tag 08X3 points to Lemma 31.4.18
    - Tag 08X4 points to Equation 31.4.18.1
  - Tag 08X5 points to Definition 31.4.19
    - Tag 08X6 points to Equation 31.4.19.1
  - Tag 08X7 points to Lemma 31.4.20
    - Tag 08X8 points to Equation 31.4.20.1
  - Tag 08X9 points to Remark 31.4.21
  - Tag 08XA points to Theorem 31.4.22
- Tag 08XB points to Subsection 31.4.23: Descent for properties of modules
  - Tag 08XC points to Lemma 31.4.24
  - Tag 08XD points to Theorem 31.4.25
  - Tag 08XE points to Theorem 31.4.26
  - Tag 08XF points to Remark 31.4.27
Tag 023R points to Section 31.5: Fpqc descent of quasi-coherent sheaves
- Tag 023S points to Lemma 31.5.1
- Tag 023T points to Proposition 31.5.2
Tag 05AY points to Section 31.6: Descent of finiteness properties of modules
- Tag 05AZ points to Lemma 31.6.1
- Tag 05B0 points to Lemma 31.6.2
- Tag 082U points to Lemma 31.6.3
- Tag 05B1 points to Lemma 31.6.4
- Tag 05B2 points to Lemma 31.6.5
- Tag 05JZ points to Lemma 31.6.6
- Tag 05VF points to Remark 31.6.7
- Tag 05B3 points to Lemma 31.6.8
- Tag 05B4 points to Lemma 31.6.9
Tag 03DR points to Section 31.7: Quasi-coherent sheaves and topologies
- Tag 03DS points to Equation 31.7.0.1
- Tag 03DT points to Lemma 31.7.1
- Tag 03DU points to Definition 31.7.2
- Tag 03FG points to Remark 31.7.3
- Tag 070R points to Remark 31.7.4
- Tag 070S points to Lemma 31.7.5
- Tag 03FH points to Remark 31.7.6
- Tag 03DV points to Lemma 31.7.7
- Tag 03FI points to Lemma 31.7.8
- Tag 03FJ points to Lemma 31.7.9
- Tag 03DW points to Proposition 31.7.10
- Tag 03DX points to Proposition 31.7.11
- Tag 05VG points to Lemma 31.7.12
- Tag 06VE points to Lemma 31.7.13
- Tag 03LC points to Proposition 31.7.14
- Tag 071N points to Lemma 31.7.15
Tag 07AF points to Section 31.8: Parasitic modules
- Tag 06ZL points to Definition 31.8.1
- Tag 0755 points to Lemma 31.8.2
- Tag 07AG points to Lemma 31.8.3
- Tag 0756 points to Lemma 31.8.4
Tag 023P points to Section 31.9: Fpqc coverings are universal effective epimorphisms
- Tag 02KI points to Lemma 31.9.1
- Tag 03N0 points to Lemma 31.9.2
- Tag 023Q points to Lemma 31.9.3
Tag 02KJ points to Section 31.10: Descent of finiteness properties of morphisms
- Tag 02KK points to Lemma 31.10.1
- Tag 0367 points to Lemma 31.10.2
- Tag 02KL points to Lemma 31.10.3
- Tag 02KM points to Lemma 31.10.4
- Tag 05B5 points to Lemma 31.10.5
- Tag 05B6 points to Remark 31.10.6
- Tag 05B7 points to Remark 31.10.7
- Tag 06NB points to Lemma 31.10.8
Tag 0347 points to Section 31.11: Local properties of schemes
- Tag 0348 points to Definition 31.11.1
- Tag 0349 points to Lemma 31.11.2
- Tag 034A points to Remark 31.11.3
Tag 034B points to Section 31.12: Properties of schemes local in the fppf topology
- Tag 034C points to Lemma 31.12.1
- Tag 0368 points to Lemma 31.12.2
Tag 0369 points to Section 31.13: Properties of schemes local in the syntomic topology
- Tag 036A points to Lemma 31.13.1
- Tag 036B points to Lemma 31.13.2
Tag 034D points to Section 31.14: Properties of schemes local in the smooth topology
- Tag 034E points to Lemma 31.14.1
- Tag 034F points to Lemma 31.14.2
- Tag 036C points to Lemma 31.14.3
- Tag 036D points to Lemma 31.14.4
- Tag 036E points to Lemma 31.14.5
Tag 06QL points to Section 31.15: Variants on descending properties
- Tag 06QM points to Lemma 31.15.1
- Tag 06QN points to Lemma 31.15.2
Tag 04QQ points to Section 31.16: Germs of schemes
- Tag 04QR points to Definition 31.16.1
- Tag 04QS points to Definition 31.16.2
Tag 04QT points to Section 31.17: Local properties of germs
- Tag 04N1 points to Definition 31.17.1
- Tag 04N4 points to Lemma 31.17.2
- Tag 04N8 points to Lemma 31.17.3
Tag 02KN points to Section 31.18: Properties of morphisms local on the target
- Tag 02KO points to Definition 31.18.1
- Tag 04QU points to Lemma 31.18.2
- Tag 06QP points to Lemma 31.18.3
- Tag 02KP points to Lemma 31.18.4
- Tag 034G points to Remark 31.18.5
Tag 02YJ points to Section 31.19: Properties of morphisms local in the fpqc topology on the target
- Tag 02KQ points to Lemma 31.19.1
- Tag 02KR points to Lemma 31.19.2
- Tag 02KS points to Lemma 31.19.3
- Tag 02KT points to Lemma 31.19.4
- Tag 02KU points to Lemma 31.19.5
- Tag 02KV points to Lemma 31.19.6
- Tag 02KW points to Lemma 31.19.7
- Tag 02KX points to Lemma 31.19.8
- Tag 02KY points to Lemma 31.19.9
- Tag 02KZ points to Lemma 31.19.10
- Tag 02L0 points to Lemma 31.19.11
- Tag 02L1 points to Lemma 31.19.12
- Tag 02L2 points to Lemma 31.19.13
- Tag 02L3 points to Lemma 31.19.14
- Tag 02L4 points to Lemma 31.19.15
- Tag 02L5 points to Lemma 31.19.16
- Tag 02L6 points to Lemma 31.19.17
- Tag 02L7 points to Lemma 31.19.18
- Tag 02L8 points to Lemma 31.19.19
- Tag 02L9 points to Lemma 31.19.20
- Tag 02LA points to Lemma 31.19.21
- Tag 02VI points to Lemma 31.19.22
- Tag 02VJ points to Lemma 31.19.23
- Tag 02VK points to Lemma 31.19.24
- Tag 02VL points to Lemma 31.19.25
- Tag 02VM points to Lemma 31.19.26
- Tag 02VN points to Lemma 31.19.27
- Tag 02VO points to Lemma 31.19.28
- Tag 02YK points to Lemma 31.19.29
- Tag 0694 points to Lemma 31.19.30
Tag 02YL points to Section 31.20: Properties of morphisms local in the fppf topology on the target
- Tag 02YM points to Lemma 31.20.1
Tag 02LB points to Section 31.21: Application of fpqc descent of properties of morphisms
- Tag 06NC points to Lemma 31.21.1
- Tag 02LC points to Lemma 31.21.2
Tag 036F points to Section 31.22: Properties of morphisms local on the source
- Tag 036G points to Definition 31.22.1
- Tag 04QV points to Lemma 31.22.2
- Tag 036H points to Lemma 31.22.3
- Tag 036I points to Remark 31.22.4
Tag 036J points to Section 31.23: Properties of morphisms local in the fpqc topology on the source
- Tag 036K points to Lemma 31.23.1
- Tag 036L points to Lemma 31.23.2
Tag 036M points to Section 31.24: Properties of morphisms local in the fppf topology on the source
- Tag 036N points to Lemma 31.24.1
- Tag 036O points to Lemma 31.24.2
- Tag 036P points to Lemma 31.24.3
- Tag 036Q points to Lemma 31.24.4
Tag 036R points to Section 31.25: Properties of morphisms local in the syntomic toplogy on the source
- Tag 036S points to Lemma 31.25.1
Tag 036T points to Section 31.26: Properties of morphisms local in the smooth topology on the source
- Tag 036U points to Lemma 31.26.1
Tag 036V points to Section 31.27: Properties of morphisms local in the étale topology on the source
- Tag 036W points to Lemma 31.27.1
- Tag 03X4 points to Lemma 31.27.2
- Tag 03YV points to Lemma 31.27.3
Tag 04QW points to Section 31.28: Properties of morphisms étale local on source-and-target
- Tag 04QX points to Example 31.28.1
- Tag 04QY points to Example 31.28.2
- Tag 04QZ points to Definition 31.28.3
- Tag 04R0 points to Lemma 31.28.4
- Tag 04R1 points to Lemma 31.28.5
- Tag 04R2 points to Lemma 31.28.6
- Tag 04R3 points to Remark 31.28.7
- Tag 04R4 points to Remark 31.28.8
Tag 04R5 points to Section 31.29: Properties of morphisms of germs local on source-and-target
- Tag 04NB points to Definition 31.29.1
- Tag 04R6 points to Lemma 31.29.2
- Tag 04R7 points to Lemma 31.29.3
- Tag 04ND points to Lemma 31.29.4
- Tag 04NI points to Lemma 31.29.5
- Tag 04NJ points to Lemma 31.29.6
- Tag 04NK points to Lemma 31.29.7
- Tag 04NL points to Lemma 31.29.8
Tag 023U points to Section 31.30: Descent data for schemes over schemes
- Tag 023V points to Definition 31.30.1
- Tag 02VP points to Remark 31.30.2
- Tag 023W points to Definition 31.30.3
- Tag 02VQ points to Remark 31.30.4
- Tag 023X points to Lemma 31.30.5
- Tag 023Y points to Lemma 31.30.6
- Tag 02VR points to Definition 31.30.7
- Tag 02VS points to Lemma 31.30.8: Pullback of descent data for schemes over families
- Tag 02VT points to Definition 31.30.9
- Tag 023Z points to Definition 31.30.10
- Tag 02VU points to Definition 31.30.11
Tag 02VV points to Section 31.31: Fully faithfulness of the pullback functors
- Tag 02VW points to Lemma 31.31.1
- Tag 02VX points to Lemma 31.31.2
- Tag 0240 points to Lemma 31.31.3
- Tag 0241 points to Lemma 31.31.4
- Tag 0242 points to Lemma 31.31.5
- Tag 0243 points to Lemma 31.31.6
- Tag 040J points to Lemma 31.31.7
- Tag 040K points to Lemma 31.31.8
- Tag 02VZ points to Lemma 31.31.9
- Tag 02VY points to Lemma 31.31.10
- Tag 02W0 points to Lemma 31.31.11
- Tag 040L points to Remark 31.31.12
Tag 02W1 points to Section 31.32: Descending types of morphisms
- Tag 02W2 points to Definition 31.32.1
- Tag 02W3 points to Lemma 31.32.2
Tag 0244 points to Section 31.33: Descending affine morphisms
- Tag 0245 points to Lemma 31.33.1
- Tag 03I0 points to Lemma 31.33.2
Tag 0246 points to Section 31.34: Descending quasi-affine morphisms
- Tag 0247 points to Lemma 31.34.1
Tag 02W4 points to Section 31.35: Descent data in terms of sheaves
- Tag 02W5 points to Lemma 31.35.1
- Tag 02W6 points to Remark 31.35.2
Tag 0248 points to Section 31.36: Descent in terms of simplicial schemes
- Tag 0249 points to Definition 31.36.1
- Tag 07TC points to Lemma 31.36.2
- Tag 024A points to Definition 31.36.3
- Tag 024B points to Lemma 31.36.4
- Tag 024C points to Lemma 31.36.5

The Stacks Project

Tree view for Chapter 31: Descent