commit 6fbd2254f87a913151019981d249c919a8f7795a Author: Aise Johan de Jong -- Hornet Date: Fri Nov 13 07:01:28 2009 -0500 Desirables: small changes This chapter is still lagging behind and maybe should be removed altogether. conventions.tex | 9 ++- desirables.tex | 228 ++++++++++++------------------------------------------- 2 files changed, 57 insertions(+), 180 deletions(-) commit 3517b720dd6f68ab7e054ddb5f6ac78f53863052 Author: Aise Johan de Jong -- Hornet Date: Fri Nov 13 06:56:15 2009 -0500 Algebraic stacks: Of course we want smooth not etale in definition Arghhhhhhh! algebraic.tex | 2 +- 1 files changed, 1 insertions(+), 1 deletions(-) commit 518837d7cf459d76fcec187415838c57538143cd Author: Aise Johan de Jong -- Hornet Date: Wed Nov 11 20:13:10 2009 -0500 Morphisms of spaces: More on relative conditions spaces-morphisms.tex | 135 ++++++++++++++++++++++++++++++++++++++++++++++--- spaces-properties.tex | 3 +- tags/tags | 8 +++ 3 files changed, 138 insertions(+), 8 deletions(-) commit 6849059e90399f1d93630e0bcc1f46a77baf5a09 Author: Aise Johan de Jong -- Hornet Date: Wed Nov 11 18:57:45 2009 -0500 Properties of Spaces: space etale over field is scheme spaces-properties.tex | 38 ++++++++++++++++++++++++++++++++++++++ 1 files changed, 38 insertions(+), 0 deletions(-) commit 22fbdba5cd0d3570e4a4047433eca46f31aeadda Author: Aise Johan de Jong -- Strider Date: Wed Nov 11 16:08:59 2009 -0500 Morphisms of Spaces: Relative conditions Trying to understand the relative versions of the local conditions found in Properties of Spaces Todo: Fix currently unfinished discussion of the above Add lemma about algebraic spaces etale over fields When does an algebraic space satisfy the sheaf condition for fpqc-coverings? This is missing in the discussion of algebraic space in the introductory chapter on algebraic spaces, but it doesn't have a high priority. Add remark discussing informally the relative conditions and what to do with them. spaces-morphisms.tex | 358 ++++++++++++++++++++++++++++++++++++++++--------- spaces-properties.tex | 17 ++- tags/tags | 27 ++++- 3 files changed, 330 insertions(+), 72 deletions(-) commit fddf37d7cbcfb28e42efd629afc585fa6d278b27 Author: Aise Johan de Jong -- Strider Date: Wed Nov 11 09:54:49 2009 -0500 Morphisms of Spaces: Valuative criterion separatedness schemes.tex | 4 +- spaces-morphisms.tex | 123 ++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 125 insertions(+), 2 deletions(-) commit 1a1b7573e25df9b16d30592e8ea86fb64afa98c5 Author: Aise Johan de Jong -- Cheetah Date: Wed Nov 11 09:00:03 2009 -0500 Morphisms of Spaces: more on separation axioms spaces-morphisms.tex | 177 +++++++++++++++++++++++++++++++++++++++++++++++-- spaces.tex | 48 +++++++++++++- 2 files changed, 216 insertions(+), 9 deletions(-) commit 1b54d5dbd9a261f935e242a17b07b6493295d186 Author: Aise Johan de Jong -- Strider Date: Tue Nov 10 17:00:34 2009 -0500 Morphisms of Spaces: Valuative criterion + finite separable extensions In the existence part of the valuative criterion for algebraic spaces it is enough to take a finite separable extension of the fraction field of the dvr. It is sometimes also necessary. It seems that if the morphism is separated, then it shouldn't be necessary, but we'll return to that later. spaces-morphisms.tex | 158 +++++++++++++++++++++++++++++++++++++++++++++---- spaces-properties.tex | 10 ++- 2 files changed, 151 insertions(+), 17 deletions(-) commit 15b591e3b46d022c4d3d8c39ef33f974a6c3fb2e Author: Aise Johan de Jong -- Strider Date: Mon Nov 9 17:11:30 2009 -0500 Morphisms of Spaces: Valuative criterion universal closednedd Finally, we have the other direction. We still have to reformulate this later for morphisms which are, say, quasi-separated. categories.tex | 12 +++ spaces-morphisms.tex | 190 +++++++++++++++++++++++++++++++++++++++++++++++++ spaces-properties.tex | 174 +++++++++++++++++++++++++++++++++----------- 3 files changed, 332 insertions(+), 44 deletions(-) commit 65620d4773a4aa7dea6180ab5553d0e53e848b19 Author: Aise Johan de Jong -- Hornet Date: Sun Nov 8 20:20:49 2009 -0500 Tags: New tags added tags/tags | 48 ++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 48 insertions(+), 0 deletions(-) commit e545e0124665a65c060753e47e89fc1265c4f985 Author: Aise Johan de Jong -- Hornet Date: Sun Nov 8 20:00:39 2009 -0500 Properties of Spaces: Split out arguments on points of spaces The purpose of this commit is to work out in more detail the arguments that lead to the result that a reasonable algebraic space X has a sober space of points |X|. In this reworking we discover the notion of an ``almost reasonable space''. An algebraic space X is almost reasonable if for every affine scheme U and etale morphism U --> X the fibres of U --> X are universally bounded. Later we will encouter the following question: Suppose given a fibre square diagram X' --> X | | v V V' --> V with V' --> V a surjective etale morphism of affine schemes, such that X' is reasonable. Is X reasonable? If you know how to (dis)prove this then please email stacks.project@gmail.com Anyway, the corresponding result for ``almost reasonable'' spaces is easy. Moreover, an almost reasonable space is a colimit of quasi-separated algebraic spaces. But on the other hand, we do not know how to prove that an almost reasonable space X has an open dense subspace which is a scheme, nor do we know how to prove that |X| is sober. morphisms.tex | 68 +++- spaces-properties.tex | 1103 ++++++++++++++++++++++++++++++++++++++----------- spaces.tex | 21 +- tags/tags | 4 +- 4 files changed, 946 insertions(+), 250 deletions(-) commit ee1bc94849b2353e72c0b14d9202f2f468c5b363 Author: Aise Johan de Jong -- Hornet Date: Fri Nov 6 17:41:33 2009 -0500 Morphisms: Morphisms with universally bounded fibres Is there a more suitable name for this property? If so, please email stacks.project@gmail.com. exercises.tex | 6 ++ morphisms.tex | 191 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 197 insertions(+), 0 deletions(-) commit 0ce90d6bb8037d602e012860ec74b99d16523a34 Author: Aise Johan de Jong -- Strider Date: Fri Nov 6 16:05:12 2009 -0500 Morphisms: nowhere dense sets and quasi-finite morphisms Analogue of the lemma on images of nowhere desne closed sets under finite morphisms. The only difference is that the image may not be closed. morphisms.tex | 35 +++++++++++++++++++++++++++++++++++ topology.tex | 18 ++++++++++++++++++ 2 files changed, 53 insertions(+), 0 deletions(-) commit 029407658b388e4dd36330e7d7e95cb0e3288f4d Merge: f2f2181 c62b8c3 Author: Aise Johan de Jong -- Hornet Date: Fri Nov 6 09:09:30 2009 -0500 Merge branch 'master' of git://paard.math.columbia.edu/var/git/stacks commit f2f218148010d9b90ad5282de737b09c8dbea852 Author: Aise Johan de Jong -- Hornet Date: Fri Nov 6 09:06:29 2009 -0500 Properties of Spaces: Finite fibres A lemma on finite fibres of affines mapping in an etale manner into an algebraic space. Before adding this to the online project we should discuss bounds for fibres of quasi-finite morphisms, perhaps by introducing a class of morphisms characterized by having universally bounded fibres. more-morphisms.tex | 2 +- spaces-properties.tex | 29 +++++++++++++++++++++++++++++ 2 files changed, 30 insertions(+), 1 deletions(-) commit c62b8c3300b1be64cea3ee9cbfb5481f8098baf0 Author: Aise Johan de Jong -- Strider Date: Wed Nov 4 11:22:12 2009 -0500 Properties of spaces: Special covering for reasonable spaces Not sure this is really all that useful... spaces-properties.tex | 37 +++++++++++++++++++++++++++++++++++-- 1 files changed, 35 insertions(+), 2 deletions(-) commit 6975741388b99069fb5c01980c2d100f738380e7 Author: Aise Johan de Jong -- Strider Date: Wed Nov 4 11:20:38 2009 -0500 Morphisms of Spaces: Valuative criterion universal closedness Only in one direction, the other is more confusing spaces-morphisms.tex | 71 ++++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 71 insertions(+), 0 deletions(-) commit 4a4442a96b632d25b45b6160388af0baad7c140e Author: Aise Johan de Jong -- Strider Date: Tue Oct 27 13:34:41 2009 -0400 Homology: small change homology.tex | 13 +++++++++---- 1 files changed, 9 insertions(+), 4 deletions(-) commit f1dd890e715863c7c73b5cc64835db39d8ec6320 Author: Aise Johan de Jong -- Hornet Date: Tue Oct 27 08:58:07 2009 -0400 Morphisms of Spaces: Valutaive criteria Allow for a field extension in the definition of the existence part of the valuative criterion for morphisms of algebraic spaces. Design decision: Any field extension is allowed. If one can get away with less (e.g. finite separable extensions or no extension at all) then this should be formulated in a separate lemma. We do this in the case of representable morphisms in this commit also. spaces-morphisms.tex | 110 +++++++++++++++++++++++++++++++++++++++++++++----- 1 files changed, 100 insertions(+), 10 deletions(-) commit 5f7871b827fe79e96d3151abf62eab0a7a98510c Author: Aise Johan de Jong -- Strider Date: Mon Oct 26 15:43:03 2009 -0400 Properties of Spaces: sobriety attained Let X be a reasonable algebraic space. To prove that every irreducible closed subset of |X| has a generic point we first write it as the underlying set of points of a closed subspace, which is then also reasonable, and then we use that it has an open dense subspace which is a scheme, and finally we use the generic point of that scheme! It is a little convoluted but it works. schemes.tex | 2 +- spaces-properties.tex | 421 ++++++++++++++++++++++++++++++------------------- 2 files changed, 258 insertions(+), 165 deletions(-) commit 843b51d8ec5858d22eb04cb79ae50501af375abe Author: Aise Johan de Jong -- Strider Date: Mon Oct 26 14:58:32 2009 -0400 Properties of Spaces: Reasonable spaces have dense schematic locus This is kind of tricky... spaces-properties.tex | 159 +++++++++++++++++++++++++++++++++++++++++++++++-- 1 files changed, 154 insertions(+), 5 deletions(-) commit 14c1d74efc4932e8d7f1828610c4cb919e9b0a39 Author: Aise Johan de Jong -- Strider Date: Mon Oct 26 14:55:37 2009 -0400 Properties: Quasi-compact schemes have open dense separated subschemes This is obvious. Surprisingly it is also sometimes useful. This just all points to the fact that it is very easy to show things hold on dense open sets, i.e., in codimension 0. properties.tex | 25 ++++++++++++++++++++++++- 1 files changed, 24 insertions(+), 1 deletions(-) commit 12ef586def804b11c14af2e1874374d8252abad9 Author: Aise Johan de Jong -- Strider Date: Mon Oct 26 14:52:55 2009 -0400 Groupoids: Lemma on affine invariant opens If (U, R, s, t, c) is a groupoid scheme and s, t are finite locally free, then any point whose orbit is contained in an affine open, is actually contained in an R-invariant affine open. groupoids.tex | 56 +++++++++++++++++++++++++++++++++++++++++++++++++++++++- 1 files changed, 55 insertions(+), 1 deletions(-) commit 97b4c1bb2f2d144da9193354148764fbbf08a307 Author: Aise Johan de Jong -- Strider Date: Mon Oct 26 09:57:27 2009 -0400 Groupoids: Locus where s, t have a property is invariant Given a groupoid (U, R, s, t, c) the open subset of U over which s, t have a given property P of morphisms of schemes is R-invariant provided this property is tau-local on the base and s, t define tau-coverings. There is a potential for misuse here, and in some other places where the statement of a lemma depends on knowing what a covering is. Namely, some authors have a different definition of what constitutes a covering. For example in some places in the literature the ring map k[x] ---> k[x] \times k[x]/(x) may be said to define an fppf covering, whereas we do not allow this... groupoids.tex | 58 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++- 1 files changed, 57 insertions(+), 1 deletions(-) commit 2ad48003176074b003af9cd34bccc9946a6a0ed8 Author: Aise Johan de Jong -- Hornet Date: Sun Oct 25 22:41:35 2009 -0400 Tags: New tags added and two fixed tags/tags | 55 +++++++++++++++++++++++++++++++++++++++++++++++++++++-- 1 files changed, 53 insertions(+), 2 deletions(-) commit 75109f8b91f555bd08ca0a3213e251398493122b Author: Aise Johan de Jong -- Hornet Date: Sun Oct 25 22:26:48 2009 -0400 More on morphisms: Neat lemma on quasi-finite morphisms If Y --> X is quasi-finite, then there exists a dense open U of X such that Y --> X is finite over U. It would be cool if we could, by a simple argument, avoid appealing to Zariski's main theorem (lemma-characterize-finite) at the end of the proof. Namely, then we could move it to the section on finite morphisms in Morphisms. more-morphisms.tex | 84 +++++++++++++++++++++++++++++++++++++++++++++++++++- 1 files changed, 83 insertions(+), 1 deletions(-) commit b36cf1c7412f6b7a43d4c6b1bd2d64b73dc9f8e7 Author: Aise Johan de Jong -- Hornet Date: Sun Oct 25 22:18:21 2009 -0400 Morphisms: Structure of finite maps A finite map of affine schemes becomes, after a finite locally free base change, the composition of a closed immersion by a morphism Y --> X where Y is a union of closed subschemes mapping isomorphically to X. We use this to show that a nowhere dense set is mapped to a nowhere dense set by a finite morphism. We also add an example that shows that a non-quasi-separated quasi-finite morphism does not become finite after localizing around a generic point of the target. morphisms.tex | 175 ++++++++++++++++++++++++++++++++++++++++++++++++++++++--- 1 files changed, 166 insertions(+), 9 deletions(-) commit 4f3c094b09bb9a8797fe65da71866a1e3bf56388 Author: Aise Johan de Jong -- Hornet Date: Sun Oct 25 22:17:13 2009 -0400 Topology: Nowhere dense sets Define and formulate trivial lemmas on nowhere dense sets. topology.tex | 66 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 66 insertions(+), 0 deletions(-) commit 094ef07bc845374db9df0645cdb3aa8ffb806503 Author: Aise Johan de Jong -- Hornet Date: Sun Oct 25 22:15:05 2009 -0400 Algebra: Finding roots after finite free extension We state explicitly that a monic polynomial over a ring can be split completely over a finite free extension. Then we use this in a couple of spots. algebra.tex | 42 ++++++++++++++++++++++++++---------------- 1 files changed, 26 insertions(+), 16 deletions(-) commit 851829ced1f32d4684edfd892c49de728adced7b Author: Aise Johan de Jong -- Hornet Date: Sun Oct 25 22:07:54 2009 -0400 Properties of Spaces: Cleanup of results so far We added a remark recalling some of the pertinent facts about etale morphisms of schemes in the section on points of reasonable algebraic spaces. Then we use this to shorten the proofs of the lemmas in that section. We added a lemma saying that a reasonable algebraic space covered by the spectrum of a field is the spectrum of a field. Finally, we changed the lemma stating that the topological space associated to a reasonable space is sober into stating that it is Kolmogorov. The proof was incorrect, and it will require considerably more work to prove this. spaces-properties.tex | 272 ++++++++++++++++++++++++++---------------------- topology.tex | 2 +- 2 files changed, 148 insertions(+), 126 deletions(-) commit 29972f7b94128a6f3f4994a5fea3de25963bb696 Author: Aise Johan de Jong -- Hornet Date: Fri Oct 23 09:13:15 2009 -0400 Properties of Spaces: Reorganization of material Discuss points on reasonable spaces in its own section. spaces-properties.tex | 121 +++++++++++++++++++++++++++++++++++++++--------- spaces.tex | 23 ++++++--- 2 files changed, 113 insertions(+), 31 deletions(-) commit c378ebd76bcf70cd9598d2f0cd2250f51ff0dd33 Author: Aise Johan de Jong -- Hornet Date: Thu Oct 22 20:55:38 2009 -0400 Properties of Spaces: Reasonable => sober It is a little bit rough here and there and some references need to be added. Also some of the arguments are duplicated. spaces-properties.tex | 106 +++++++++++++++++++++++++++++++++++++++++++------ 1 files changed, 94 insertions(+), 12 deletions(-) commit 2bd9d8eb1b482fd3333856c56697153505c00da1 Author: Aise Johan de Jong -- Strider Date: Thu Oct 22 17:20:55 2009 -0400 Properties of Spaces: Reasonable spaces have sober underlying |X| We have not yet completely proved this but it looks like it is going to work out. This commit has two FIXMES etale.tex | 2 +- spaces-properties.tex | 162 ++++++++++++++++++++++++++++++++++++++++++++++++- 2 files changed, 161 insertions(+), 3 deletions(-) commit 69a7f5d231f6ed2b8cb1ec9ee6fc1864b7dc2fc4 Author: Aise Johan de Jong -- Strider Date: Thu Oct 22 13:18:35 2009 -0400 Topology: Fix error in lemma on existence closed point This was noticed by Cameron Franc. Here is the email: Dear project developers, Lemma 9.5 in the chapter on topological spaces says that "every nonempty quasi-compact topological space has a closed point". This is false: consider the trivial topology on a nonempty space. One needs to assume some mild separation axiom, for instance that for any two distinct points there is a closed set containing precisely one of the two points. The book is great! Keep up the good work, -cameron franc topology.tex | 23 +++++++++++++++++------ 1 files changed, 17 insertions(+), 6 deletions(-) commit a28142de8a08c27178638a3da49d11dc6e8ec4df Author: Aise Johan de Jong -- Strider Date: Wed Oct 21 17:23:39 2009 -0400 Properties of Spaces We started to work out the suggestion in commit 2100745. In fact the suggestion was wrong and the correct notion is to require that there exists a surjective etale morphism \coprod U_i --> X such that for each i the two projection morphisms U_i \times_X U_i --> U_i are quasi-compact. We are calling such an algebraic space ``reasonable''. If you do not like this please complain soon. Sofar the only interesting observation is that points on reasonable spaces are represented by monomorphisms from spectra of fields. We also expect that valuative criteria will work well for reasonable algebraic spaces. spaces-properties.tex | 235 +++++++++++++++++++++++++++++++++++++++++-------- 1 files changed, 199 insertions(+), 36 deletions(-) commit 6805851344ed263499af80ae84bf6e212a8880b8 Author: Aise Johan de Jong -- Strider Date: Wed Oct 21 17:21:55 2009 -0400 Algebraic spaces: properties do not change when changing base We added a lemma that, basically, will tell us that local properties of an algebraic space X over S hold if and only if the hold for X thought of as an algebraic space over Z spaces.tex | 65 ++++++++++++++++++++++++++++++++++++++++++++++++++--------- 1 files changed, 55 insertions(+), 10 deletions(-) commit 8ff45116c0675990d21ccf8b0e52e586d22cd9d0 Author: Aise Johan de Jong -- Strider Date: Wed Oct 21 17:19:32 2009 -0400 Sites: Descrition j_! when topology weaker than canonical For some reason this lemma was still not part of the discussion in the section on localization. It says that in the description of j_! we do not have to sheafify if the topology on our site is weaker than the canonical topology. A condition which holds for the etale, fppf sites of a scheme for example. sites.tex | 43 +++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 43 insertions(+), 0 deletions(-) commit 5c1185b45e3e745603838903f31db05e5110794d Author: Aise Johan de Jong -- Strider Date: Wed Oct 21 10:34:08 2009 -0400 Spaces: Change of base scheme An algebraic space over a base scheme S can be base changed to any scheme lying over S. Moreover, given S --> S' then it can be viewed as an algebraic space over S' sites.tex | 33 +++++++++++++++--- spaces.tex | 109 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++- 2 files changed, 136 insertions(+), 6 deletions(-) commit 7ad17478ac531b6ced3f73a7463a6c5e05e973fe Author: Aise Johan de Jong -- Hornet Date: Tue Oct 20 17:03:35 2009 -0400 Sets: Two typos fixed Pointed out by Tobi Lehman sets.tex | 4 ++-- 1 files changed, 2 insertions(+), 2 deletions(-) commit 21007451409ce2b417e470b9b53f981efe408c7e Author: Aise Johan de Jong -- Strider Date: Tue Oct 20 16:40:19 2009 -0400 Morphisms of spaces: Characterizing universally closed morphisms Namely, to see that X --> Y is universally closed, it suffices to check that the base change by any affine scheme mapping into Y is universally closed Future idea: We might consider the collection of algebraic spaces X with the property that there exist a set of schemes U_i and a surjective, etale surjection \coprod U_i --> X such that each U_i \to X is quasi-compact. Note that this includes all schemes, it includes all quasi-separated algebraic spaces, as well as all Zariski locally quasi-separated algebraic spaces... But it does not contain [A^1/Z]. spaces-morphisms.tex | 85 ++++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 85 insertions(+), 0 deletions(-) commit 8d6af9626415a0afe36b599d52f95e54144006af Author: Aise Johan de Jong -- Strider Date: Tue Oct 20 16:39:35 2009 -0400 Properties of spaces: etale morphisms are open spaces-properties.tex | 14 ++++++++++++++ 1 files changed, 14 insertions(+), 0 deletions(-) commit 9bbd7e8d07792b01315186f5a1e738ed90b3e362 Author: Aise Johan de Jong -- Hornet Date: Tue Oct 20 09:36:13 2009 -0400 Makefile: Changed format log.log file It may be useful for the people visiting the website to be able to read the blurbs describing the changes in each commit. Makefile | 2 +- 1 files changed, 1 insertions(+), 1 deletions(-) commit 03910d037486ec1841a621de0326935715f4230c Author: Aise Johan de Jong -- Hornet Date: Tue Oct 20 09:32:55 2009 -0400 Morphisms of spaces: Universally closed + valuative criteria Started writing a little more on universally closed morphisms of algebraic spaces and (possibly) valuative criteria. This seems to be already slightly tricky in our general setting. spaces-morphisms.tex | 102 +++++++++++++++++++++++++++++++++++++++++++++++++- 1 files changed, 101 insertions(+), 1 deletions(-) commit b3afd44dfe5c81097a4eae61f1e7ab728bd7f767 Author: Aise Johan de Jong -- Hornet Date: Tue Oct 20 09:30:38 2009 -0400 Properties of Spaces: Closed subsets of spaces <-> closed subspaces We started writing the section on reduced algebraic spaces and the corresponding material on closed algebraic subspaces. Not done yet. spaces-properties.tex | 66 +++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 66 insertions(+), 0 deletions(-) commit 68695d2600cfbfaddb1ec963c54db15699faafe3 Author: Aise Johan de Jong -- Hornet Date: Tue Oct 20 09:26:22 2009 -0400 Spaces: Etale locally representable map sometimes representable Given an algebraic space X and a sheaf F and a tranformation F --> X which after an etale covering becomes representable and lies in a class of morphisms P which satisfy effective descent for etale coverings, then F is an algebraic space, F --> X is representable and has P. spaces.tex | 61 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 61 insertions(+), 0 deletions(-) commit 705573e4aeb32d6bf8784709579fa8bb295dad0d Author: Aise Johan de Jong -- Hornet Date: Tue Oct 20 09:24:16 2009 -0400 Descent: Closed immersions satisfy descent w.r.t. etale coverings Of course this follows immediately from the already proven case of affine morphisms. We only add it for convenient reference. Maybe we should add to the conclusion that the resulting morphism is a closed immersion as well? descent.tex | 16 ++++++++++++++++ 1 files changed, 16 insertions(+), 0 deletions(-) commit 4ac058b3644ffc49a8aa7e5522fade6cb30d7a8e Author: Aise Johan de Jong -- Strider Date: Mon Oct 19 14:12:57 2009 -0400 Topology + Morphisms: Fix error in topology lemma + fallout The error was in thinking that generalizations lift along any open map of topological spaces, which they do not. An example can be constructed relatively easily. It turns out that we never used this anywhere in the rest of the chapters, which is a lucky break. We fixed a comment about this in the chapter on morphisms of schemes to be a little more clear, and we added a lemma stating explicitly that generalizations do lift along flat morphisms of schemes. morphisms.tex | 27 +++++++++++++++++++-------- topology.tex | 31 +++++++++++++++++++++++++++++-- 2 files changed, 48 insertions(+), 10 deletions(-) commit a9d7807b691d50ac90286e28b1d4af1690467e35 Author: Aise Johan de Jong -- Hornet Date: Sun Oct 18 16:00:58 2009 -0400 Tags: Added new tags tags/tags | 47 +++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 47 insertions(+), 0 deletions(-) commit 6bee4798f27397fd5a22a4969f4b486173805612 Author: Aise Johan de Jong -- Hornet Date: Sun Oct 18 15:57:46 2009 -0400 Groupoids + More on Morphisms: Switched order of chapters Two lemmas on etale localization of groupoids need material from the section on etale localization in More on Morphisms Makefile | 2 +- browse.html | 16 ++++++++-------- chapters.tex | 2 +- preamble.tex | 2 +- tags/Makefile | 2 +- 5 files changed, 12 insertions(+), 12 deletions(-) commit a37c06ee99e00232dc9abf06af408e0feacd63f5 Author: Aise Johan de Jong -- Hornet Date: Sun Oct 18 15:46:20 2009 -0400 More on Morphisms: Improved readability of proof Stein factorization Only in the general case. Also changed designation into theorem from lemma. more-morphisms.tex | 105 ++++++++++++++++++++++++++++++++++----------------- 1 files changed, 70 insertions(+), 35 deletions(-)