commit 17f010b61447d7e12a02975d2d6c7040346ebce3 Author: Aise Johan de Jong Date: Thu Feb 9 10:38:18 2012 -0500 Crystals in QCoh <---> Modules with connections This is really an essentially trivial observation, but there is a lot of notation that obfuscates what is going on. crystalline.tex | 292 +++++++++++++++++++++++++++++++++++++++++++++++++------ 1 files changed, 261 insertions(+), 31 deletions(-) commit 07980d23b8b8585ead19249b8bcc2d4ef0869c05 Author: Aise Johan de Jong Date: Wed Feb 8 20:07:23 2012 -0500 Elementary properties module differential crystalline.tex | 67 +++++++++++++++++++++++++++++++++++++++++++++---------- 1 files changed, 55 insertions(+), 12 deletions(-) commit 11d659c18e0f3d7c40f043840f711ec3b1dcb19e Author: Aise Johan de Jong Date: Wed Feb 8 17:16:18 2012 -0500 Differentials of divided power envelopes Seems like a very general property but it seems hard to put your finger on exactly what is going on here... crystalline.tex | 167 ++++++++++++++++++++++++++++++++++++++++++++++++++----- 1 files changed, 152 insertions(+), 15 deletions(-) commit ea4677ffd9187ce07b6f6cc69cfa740e4c7b0057 Author: Aise Johan de Jong Date: Mon Feb 6 15:35:21 2012 -0500 Spell out proof of integrability Hopefully correct crystalline.tex | 102 +++++++++++++++++++++++++++++++++++++++--------------- 1 files changed, 73 insertions(+), 29 deletions(-) commit a51f3f8cc03e84ca03b3a7ae8ce52ba589763ba0 Author: Aise Johan de Jong Date: Sun Feb 5 22:41:17 2012 -0500 Fix some notation regarding divided power differentials Also some of the formulas regarding the explicit thickenings were wrong. crystalline.tex | 477 +++++++++++++++++++++++++++++++++++-------------------- 1 files changed, 307 insertions(+), 170 deletions(-) commit b633ca407b8fff0369a06710daf24ca8f5313792 Author: Aise Johan de Jong Date: Sat Feb 4 20:49:27 2012 -0500 Catch up with material in lectures crystalline.tex | 401 ++++++++++++++++++++++++++++++++++++++----------------- 1 files changed, 279 insertions(+), 122 deletions(-) commit 6f72eafae94e0c9a0c5469190d120f09e3a5e58e Author: Aise Johan de Jong Date: Sat Feb 4 14:53:16 2012 -0500 Bit of cleanup Went through the first few sections and cleaned up. crystalline.tex | 115 +++++++++++++++++++++++++++++++++--------------------- 1 files changed, 70 insertions(+), 45 deletions(-) commit 98ce5f1f99ecb2c948e0109204101580952709b4 Author: Aise Johan de Jong Date: Sat Feb 4 13:33:24 2012 -0500 Spell out existence colimits and divided power envelopes Formulate explicitly the fact that on the category of divided power rings a covariant functor to sets which commutes with limits and is bounded in a suitable sense is representable. This very general argument could be used elsewhere too, so may have to be moved somewhere else later. What is this result called? crystalline.tex | 165 ++++++++++++++++++++++++++++++++++++++++++------------- 1 files changed, 127 insertions(+), 38 deletions(-) commit 3f9b0f6443a7a04bd9a317ca29f3c93e0004fcf3 Merge: dd64a58 7d04816 Author: Aise Johan de Jong Date: Fri Feb 3 21:45:57 2012 -0500 Merge branch 'crystalline' commit 7d04816cc4050ca58f7fdd7f5910e39b961bcd1a Author: Aise Johan de Jong Date: Fri Feb 3 21:45:25 2012 -0500 More divided power algebra crystalline.tex | 487 +++++++++++++++++++++++++++++++++++++++++++------------ 1 files changed, 385 insertions(+), 102 deletions(-) commit dd64a58d897d095361a7e6e212d50efb24d101a2 Author: Aise Johan de Jong Date: Thu Feb 2 19:48:46 2012 -0500 Typo Thanks to Amnon Yekutieli. stacks-sheaves.tex | 2 +- 1 files changed, 1 insertions(+), 1 deletions(-) commit c5681489971d1c26a6be93b8c121bf41c36f8b04 Author: Aise Johan de Jong Date: Wed Jan 25 16:37:54 2012 -0500 New proof cocontinuous functors give morphisms topoi This proof is uses more structure and is shorter too. sites.tex | 193 ++++++++++++++++--------------------------------------------- 1 files changed, 49 insertions(+), 144 deletions(-) commit 84bc01b03f4c779b060d182d4481ab53344217ed Author: Aise Johan de Jong Date: Sun Jan 22 15:02:01 2012 -0500 Some more lemmas on divided power structures crystalline.tex | 690 +++++++++++++++++++++++++++++++++++++++++++------------ 1 files changed, 543 insertions(+), 147 deletions(-) commit 262b6887097c2ce3494003ec0da6b9df0fd89fec Author: Aise Johan de Jong Date: Thu Jan 19 16:00:18 2012 -0500 A bit more on modules and connections crystalline.tex | 963 ++++++++++++++++++++++++++++++++++++++++++++++++++++++- 1 files changed, 949 insertions(+), 14 deletions(-) commit b3800fc541528f0643063ae8119c28b9f1419086 Author: Aise Johan de Jong Date: Tue Jan 17 21:01:20 2012 -0500 Small and big sites Compare them and explain how this lets you do functoriality. crystalline.tex | 184 ++++++++++++++++++++++++++++++++++++++++++++++--------- 1 files changed, 154 insertions(+), 30 deletions(-) commit 3338c28f5d2ff7cbfbc08a5758e0257d988547c7 Author: Aise Johan de Jong Date: Tue Jan 17 15:10:21 2012 -0500 Divided power schemes This is crappy notation perhaps... crystalline.tex | 241 ++++++++++++++++++++++++++++++++++++++++++++++++++---- 1 files changed, 223 insertions(+), 18 deletions(-) commit b82b004fa619d54b001b1a6fbe688e6fcd034ba2 Author: Aise Johan de Jong Date: Tue Jan 17 13:24:42 2012 -0500 Clarification surjectivity in proof Lemma Tag 01WN Hopefully this is a clarification and doesn't make the thing harder to read... morphisms.tex | 6 +++++- 1 files changed, 5 insertions(+), 1 deletions(-) commit a8546b6c101abca6cd34c5a2e71cbd20a688b95e Author: Alberto Gioia Date: Tue Jan 17 13:12:14 2012 -0500 Mistake in the proof of tag 01WN There is a mistake in the stacks project (in morphisms, proof of lemma 41.7, tag 01WM). The set denoted I' is said to be an ideal, but it is not since it does not contain any multiple of y. In the file I am sending you there is a corrected version I wrote with my advisor. The idea is the same, but the mistake I pointed out above is avoided and it does not mention the projective line. Best regards, Alberto Gioia. morphisms.tex | 56 +++++++++++++++++++++++++++++++++++--------------------- 1 files changed, 35 insertions(+), 21 deletions(-) commit e1d127e166ec729ca5843ee18eaaa2d26b3ba587 Author: Aise Johan de Jong Date: Tue Jan 17 11:09:17 2012 -0500 Affine crystalline site and compatibility with module of differentials crystalline.tex | 182 ++++++++++++++++++++++++++++++++++++++++++++++-------- 1 files changed, 155 insertions(+), 27 deletions(-) commit b23c3c13c892652f3d2be604b63d1b4c70f5b089 Author: Aise Johan de Jong Date: Tue Jan 17 09:37:27 2012 -0500 Correct version lemma on differentials. FUN! crystalline.tex | 71 +++++++++++++++++++++++++++++++++++++++++++++++++++++- 1 files changed, 69 insertions(+), 2 deletions(-) commit aa9825f3370e8c354ee0078940a9f0a3636be110 Author: Aise Johan de Jong Date: Mon Jan 16 21:48:15 2012 -0500 Two errors, one fixed crystalline.tex | 7 +++---- 1 files changed, 3 insertions(+), 4 deletions(-) commit a3e18450cf673a33a5032af10d97d2a862784a16 Author: Aise Johan de Jong Date: Mon Jan 16 21:17:19 2012 -0500 Divided power algebras... crystalline.tex | 187 +++++++++++++++++++++++++++++++++++++++++++++++++++---- 1 files changed, 175 insertions(+), 12 deletions(-) commit acaa8cf9a42ca5191107eebbf2755a327649faae Author: Aise Johan de Jong Date: Mon Jan 16 19:02:47 2012 -0500 Definition of divided power structure crystalline.tex | 115 ++++++++++++++++++++++++++++++++++++++++++++++++++++--- 1 files changed, 109 insertions(+), 6 deletions(-) commit f9d466aa8cbed5073e6b6bfd755323e81a1a80ca Author: Aise Johan de Jong Date: Mon Jan 16 16:01:07 2012 -0500 Start notes crystalline cohomology Makefile | 4 +- browse.html | 81 ++++++++++++---------- chapters.tex | 1 + crystalline.tex | 161 ++++++++++++++++++++++++++++++++++++++++++++ my.bib | 28 ++++++++ preamble.tex | 1 + scripts/fix_titles_refs.py | 1 + tags/Makefile | 4 +- 8 files changed, 240 insertions(+), 41 deletions(-) commit 035f82b8828f74cb8fffd6a6d08ee81d521ce6b8 Author: Aise Johan de Jong Date: Mon Jan 16 13:38:48 2012 -0500 Definition G-rings smoothing.tex | 31 +++++++++++++++++++++++++++++++ 1 files changed, 31 insertions(+), 0 deletions(-) commit 93b3e1d3b4861b0d8876349fc48084782bd697f7 Author: Aise Johan de Jong Date: Sat Jan 14 15:12:15 2012 -0500 Tags: Added new tags tags/tags | 98 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++- 1 files changed, 96 insertions(+), 2 deletions(-) commit 7995a9933afd7ed3043060a4d2cb6fe63714278a Author: Aise Johan de Jong Date: Sat Jan 14 15:11:19 2012 -0500 Final fixups on chapter Smoothing Ring Maps In particular move it to the correct spot and remove "under construction" from the title. Makefile | 5 +- browse.html | 140 +++++++++++++++++++++++++++--------------------------- chapters.tex | 2 +- more-algebra.tex | 2 +- preamble.tex | 2 +- smoothing.tex | 2 +- tags/Makefile | 5 +- 7 files changed, 78 insertions(+), 80 deletions(-) commit 773743e134e05a7c16f9562c4143a480ee6bffe4 Author: Aise Johan de Jong Date: Sat Jan 14 14:58:57 2012 -0500 Popescu's proof of GND done It's like chess! algebra.tex | 1 + smoothing.tex | 295 +++++++++++++++++++++++++++++++++++++++++++++++++++----- 2 files changed, 269 insertions(+), 27 deletions(-) commit 4a6b2ddaf99412e8a8eafc6affe399849caf6193 Author: Aise Johan de Jong Date: Fri Jan 13 16:40:47 2012 -0500 More details in smoothing.tex smoothing.tex | 347 ++++++++++++++++++++++++++++++++++++++++++++++---------- 1 files changed, 285 insertions(+), 62 deletions(-) commit aff72b69535a59507ea6dfcfadb49294ed139a8d Author: Aise Johan de Jong Date: Fri Jan 13 16:39:37 2012 -0500 Constructing flat maps If A is local and geometrically regular over a field k then it is easy to make flat maps from polynomial rings into A. more-algebra.tex | 44 ++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 44 insertions(+), 0 deletions(-) commit 710d36c95dd53d56bbb44f2430d1f5462217e99b Author: Aise Johan de Jong Date: Fri Jan 13 16:38:21 2012 -0500 Flat over regular If R ---> S is a local map from a regular local ring to a Noetherian local ring and a regular system of parameters maps to a regular sequence then R ---> S is flat. algebra.tex | 26 +++++++++++++++++++++++++- 1 files changed, 25 insertions(+), 1 deletions(-) commit 16d52f4c3be245b46aac602a8aa43c22318b4995 Author: Aise Johan de Jong Date: Fri Jan 13 10:47:15 2012 -0500 Shorten proof in morphisms.tex by using algebra result morphisms.tex | 27 +++++---------------------- 1 files changed, 5 insertions(+), 22 deletions(-) commit 35d8431a9e6b07b4ee2eec42136444f114f73548 Author: Aise Johan de Jong Date: Fri Jan 13 10:46:40 2012 -0500 Finiteness conditions for maps to Artinian local rings algebra.tex | 92 +++++++++++++++++++++++++++++++++++++++++++++++++++------- 1 files changed, 81 insertions(+), 11 deletions(-) commit 1a1d29c1996a234da9fc34fe4daca9d18b20db57 Author: Aise Johan de Jong Date: Thu Jan 12 20:08:43 2012 -0500 Fixup of separable residue field case Add proofs and small corrections. smoothing.tex | 118 +++++++++++++++++++++++++++++++++++--------------------- 1 files changed, 74 insertions(+), 44 deletions(-) commit 0142d4dc7de96baa28d958317a52fa0a038ddd46 Author: Aise Johan de Jong Date: Thu Jan 12 17:12:00 2012 -0500 Layout of proof Popescu's theorem Very sketchy... will work this out soon. smoothing.tex | 374 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 374 insertions(+), 0 deletions(-) commit a5294c485cdbc63e04bcf44001da4acb1f698949 Author: Aise Johan de Jong Date: Thu Jan 12 17:10:35 2012 -0500 A local complete intersection homomorphism is a perfect ring map This result we've already dealt with in the setting of morphisms of schemes, but it seems like a good idea to also prove it directly in commutative algebra. more-algebra.tex | 20 ++++++++++++++++++-- 1 files changed, 18 insertions(+), 2 deletions(-) commit 553aaa287392dc47a0d6e498a6edba46118fee9c Author: Aise Johan de Jong Date: Thu Jan 12 17:08:04 2012 -0500 Define (regular) system of parameters Should have done so earlier. Also: Try to be correct and define a local ring to be a "regular local ring" if there exists a regular system of parameters, i.e., don't say a regular local ring is "regular" until after we have shown that a regular local ring has localizations which are also regular local rings... algebra.tex | 44 +++++++++++++++++++++++++------------------- 1 files changed, 25 insertions(+), 19 deletions(-) commit 07858979aa26410048aa6368979083b35842708b Author: Aise Johan de Jong Date: Thu Jan 12 17:07:22 2012 -0500 Regular sequences and permutations algebra.tex | 75 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++-- 1 files changed, 72 insertions(+), 3 deletions(-) commit 5535509ad5a11afb180dca6f46d97435b6c60e0f Author: Aise Johan de Jong Date: Thu Jan 12 17:05:30 2012 -0500 Formally define ideal of definition in local ring algebra.tex | 31 +++++++++++++++++++------------ 1 files changed, 19 insertions(+), 12 deletions(-) commit ba4716f7c2670cea4d27b036aa7973dbdd7b079d Author: Aise Johan de Jong Date: Tue Jan 10 14:33:25 2012 -0500 Various notions of standard elements This is needed as the approach using just what we now call "elementary standard" elements wasn't flexible enough for the last part of the proof. Or at least we don't know... algebra.tex | 57 ++++++++--- smoothing.tex | 288 +++++++++++++++++++++++++++++++++++++++++++++----------- 2 files changed, 272 insertions(+), 73 deletions(-) commit 1562af70fb6153eb40a4be3d416667f6672bfccf Author: Aise Johan de Jong Date: Tue Jan 10 09:38:19 2012 -0500 Small changes smoothing.tex | 36 ++++++++++++++++++------------------ 1 files changed, 18 insertions(+), 18 deletions(-) commit 3059621e1f310399fa0909a3d9e37fd3f5f57f09 Author: Aise Johan de Jong Date: Sun Jan 8 17:58:20 2012 -0500 Clarify proof Elkik's lemma Now we use a result of the chapter "More on Algebra" so we have to move the chapter "Smoothing Ring Maps" later. Since we expect to use restriction of scalars (which really should be explained in the context of schemes) we'll have to move it after some of the chapters on schemes. We'll do this later (so temporarily there is a forward reference in the stacks project). more-algebra.tex | 33 +++++++--- smoothing.tex | 175 +++++++++++++++++++++++++++++++++++++---------------- 2 files changed, 145 insertions(+), 63 deletions(-) commit 4084aab51a27af1b71ba4f119ffdd680e569b161 Author: Aise Johan de Jong Date: Sun Jan 8 12:40:09 2012 -0500 Proof delocalization lemma (height zero case) smoothing.tex | 35 ++++++++++++++++++++++++++++++++--- 1 files changed, 32 insertions(+), 3 deletions(-) commit 7756e095ac56f7a0cb9799005bbc22d11decfcc1 Author: Aise Johan de Jong Date: Sun Jan 8 09:40:41 2012 -0500 Fix typo my.bib my.bib | 36 ++++++++++++++++++------------------ 1 files changed, 18 insertions(+), 18 deletions(-) commit e692985f2a944f4c5a1dce355713b6aa9b37a0a8 Author: Aise Johan de Jong Date: Sun Jan 8 08:56:04 2012 -0500 Formulation of local problem in smoothing.tex smoothing.tex | 331 ++++++++++++++++++++++++++++++++++++++++++++++++++------- 1 files changed, 290 insertions(+), 41 deletions(-) commit 6bf75caaee3ad6d0818dc780c414a815711d22a7 Author: Aise Johan de Jong Date: Sun Jan 8 08:53:54 2012 -0500 Introduction smoothing.tex changed a bit Tried to get some of the history straight but it seems to be confused (even by the main players themselves). my.bib | 40 ++++++++++++++++++++++++++++++++++++++++ smoothing.tex | 48 +++++++++++++++++++++++++++++++++++------------- 2 files changed, 75 insertions(+), 13 deletions(-) commit 6ad95f5c9aa971fdbe4104e51c3f03ec5f4d69a0 Author: Aise Johan de Jong Date: Fri Jan 6 20:44:22 2012 -0500 Fill in details Left over from commit 2ee09eb smoothing.tex | 112 ++++++++++++++++++++++++++++++++++++++++++++++++-------- 1 files changed, 96 insertions(+), 16 deletions(-) commit 2ee09ebc2e6e0717f92ca39012e8f7a6180b9c97 Author: Aise Johan de Jong Date: Fri Jan 6 17:05:25 2012 -0500 Reduce to base field case This is really very straightforward using the lifting and desingularization lemmas at hand. smoothing.tex | 101 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 101 insertions(+), 0 deletions(-) commit 323dece3a1466324e9e21fb9df7f2e641b1102ba Author: Aise Johan de Jong Date: Fri Jan 6 16:36:34 2012 -0500 Added ``under construction'' to smoothing chapter Seems like a good idea as things are very much still in flux. Also some improvements on the lifting and desingularization lemmas. smoothing.tex | 118 +++++++++++++++++++++++++++++++++----------------------- 1 files changed, 69 insertions(+), 49 deletions(-) commit 05ddbd8eeee4e9e36b3d3d1ff1e297513a05aac6 Author: Aise Johan de Jong Date: Thu Jan 5 13:50:09 2012 -0500 Two examples formally smooth algebras more-algebra.tex | 27 ++++++++++++++++++++++++++- 1 files changed, 26 insertions(+), 1 deletions(-)