commit 8e006a3f0ba931e1e6368b4acb3dd8146c97f9a4 Author: Aise Johan de Jong -- Strider Date: Tue Feb 2 19:38:50 2010 -0500 Groupoids: Error in proof one of the etale localization lemmas This has to be fixed before continuing the discussion in bootstrap. We can always take the formulation of this lemma from Keel and Mori, but it still seems possible a slightly more general result holds. groupoids.tex | 6 +++++- 1 files changed, 5 insertions(+), 1 deletions(-) commit 4806d0eebd5dedf7649784fca3c5dbe0c94c3358 Author: Aise Johan de Jong -- Strider Date: Tue Feb 2 10:16:43 2010 -0500 Bootstrap: Reduce a flat+fp equivalence relation to flat+fp+lqf The proof seems pretty solid and the exact same proof will later also work to show the stack associated to a flat+fp groupoid with lqf stabilizer is equivalent to the stack associated to a flat+fp+lqf groupoid I think. But everything in its own good time. bootstrap.tex | 198 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 198 insertions(+), 0 deletions(-) commit 639c2b944b66aba21e9d71f4322b6b0233aa08a1 Author: Aise Johan de Jong -- Strider Date: Mon Feb 1 21:31:03 2010 -0500 Groupoids: Useful lemma describing the locus where U' --> U/R is P If (U, R, s, t, c) is a groupoid scheme with s,t flat and locally of finite presentation, and if g : U' --> U is a morphism, then it makes sense to say that the morphism U' ---> U/R has property P for some types of properties of morphisms of schemes P. The lemma in this commit elaborates on this without actually making a formal definition (which belongs in a chapter on algebraic spaces if j is an equivalence relation, or in a chapter on algebraic stacks). groupoids.tex | 70 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 70 insertions(+), 0 deletions(-) commit d267ca3b3368bfb767e87596ce562fe47fd61b24 Author: Aise Johan de Jong -- Strider Date: Mon Feb 1 21:27:00 2010 -0500 More on Morphisms: Flat locus and base change Given X --> S and quasi-coherent F on X consider the locus in X where F is flat over S. Formation of this locue commutes with flat base change. more-morphisms.tex | 49 ++++++++++++++++++++++++++++++++++++++++++++++++- 1 files changed, 48 insertions(+), 1 deletions(-) commit 2ab11be29518147a504c79cfc7a7789890205038 Author: Aise Johan de Jong -- Strider Date: Mon Feb 1 21:24:46 2010 -0500 Morphisms: Locus where a morphism is unramified/etale and base change Formation of the locus commutes with base change under certain hypotheses. morphisms.tex | 85 ++++++++++++++++++++++++++++++++++++++++++++++++++++----- 1 files changed, 78 insertions(+), 7 deletions(-) commit 6ff954127a44c3a7a5086e6582d0e84f66f9888d Author: Aise Johan de Jong -- Strider Date: Mon Feb 1 21:21:46 2010 -0500 Algebra: Improve lemma on base change flatness The improvement is that it handles modules now too. algebra.tex | 50 ++++++++++++++++++++++++++++++++++++-------------- 1 files changed, 36 insertions(+), 14 deletions(-) commit 2000b71e7e13e30a5461ded21999193dbcfa8f41 Author: Aise Johan de Jong -- Hornet Date: Mon Feb 1 10:35:12 2010 -0500 Bootstrap: Remove unnecessary hypothesis from lemma bootstrap.tex | 13 +++---------- 1 files changed, 3 insertions(+), 10 deletions(-) commit c3d1b76488bf6e370e69a88e46f10a05a877214c Author: Aise Johan de Jong -- Cheetah Date: Sun Jan 31 20:34:20 2010 -0500 Terminology: "group space" ---> "group algebraic space" "groupoid space" ---> "groupoid in algebraic spaces" It seems more natural this way to at least three people, and in any case it prevents the possible confusion that may arise from the fact that in some places in the literature the word "space" sometimes refers to any sheaf on schemes in the fppf topology. browse.html | 2 +- chapters.tex | 2 +- spaces-groupoids.tex | 182 +++++++++++++++++++++++++------------------------- 3 files changed, 94 insertions(+), 92 deletions(-) commit 9e88016276f658066ac0d317e1d517e40e4ba874 Author: Aise Johan de Jong -- Cheetah Date: Sun Jan 31 19:51:21 2010 -0500 Conditions on algebraic spaces renamed. OK, after this commit (which is basically without mathematical content) we now have the following notions: very reasonable: this is the old notion of "reasonable" and means the space has a Zariski covering such that each piece has a quasi-compact etale covering by a scheme. reasonable: this is the old notion of "almost reasonable" and means that for every affine U and etale morphism U --> X the fibres are universally bounded. decent: this means that every point is representable by a monomorphism from the spectrum of a field and that moreover this monomorphism is quasi-compact. Each of these is a very weak notion of separation on the algebraic space. We have also defined what it means for a morphism to have those properties (in terms of "fibres"). The goal of making this change now is to prevent confusion when we start adding material later, because we think that decent/reasonable spaces will play a more important role than very reasonable spaces. spaces-morphisms.tex | 35 ++- spaces-properties.tex | 940 +++++++++++++++++++++++++----------------------- tags/tags | 2 +- 3 files changed, 516 insertions(+), 461 deletions(-) commit 1642b959ae7cbe035354935444213b1172712c2e Author: Aise Johan de Jong -- Cheetah Date: Sun Jan 31 16:50:02 2010 -0500 Terminology changes: "reasonable" ---> "very reasonable" "almost reasonable" ---> "reasonable" David Rydh suggested this change since the notion of being (what is now called) very reasonable is not a particularly good notion. On the other hand the notion of being (what is now called) reasonable behaves quite well in various situations, and it seems hard to envision results that use the assumption of being very reasonable that do not hold for reasonable spaces. Still, currently there are still some results of this form, so we need to keep the notion "very reasonable" around (of course we will always keep it around for the sake of referencing, but in the future we may delegate it to a forgotten corner). TODO (soon): Introduce decent spaces. These will be characterized by having property (gamma). spaces-morphisms.tex | 27 ++++---- spaces-properties.tex | 170 +++++++++++++++++++++++++------------------------ tags/tags | 44 ++++++------ 3 files changed, 123 insertions(+), 118 deletions(-) commit 81fadc402dc9032cd6540bf17f150aa45ad3cd41 Author: Aise Johan de Jong -- Hornet Date: Sun Jan 31 13:08:52 2010 -0500 Algebra: Local criterion for flatness, fp case Another result which did not yet have a "finitely presented" version yet. algebra.tex | 79 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++- 1 files changed, 78 insertions(+), 1 deletions(-) commit 68883e8f9ac93804d7854b47bf38cf73a1fc4764 Author: Aise Johan de Jong -- Hornet Date: Sat Jan 30 12:40:15 2010 -0500 Algebra: Grothendieck's lemma for finite presentation case We added the non-Noetherian case of what we like to call Grothendieck's lemma. It says that if R --> S is a flat and essentially of finite presentation local map of local rings, and if f is in the maximal ideal such that f maps to a nonzero divisor on S/m_RS, then f is a nonzero divisor on S and S/fS is flat over R. There is also a more general statement for modules. See EGA IV 11.3.7. For some reason we did not have this version of Grothendieck's lemma and it may be that some of the arguments in the algebra chapter may be simplified by using this lemma. algebra.tex | 153 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ groupoids.tex | 2 +- 2 files changed, 154 insertions(+), 1 deletions(-) commit 49580f90ae425ba0080800c0733ab38456cffe62 Author: Aise Johan de Jong -- Hornet Date: Fri Jan 29 09:05:39 2010 -0500 Tags: added new tags tags/tags | 48 ++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 48 insertions(+), 0 deletions(-) commit 54e462e8ce1ea2d32bea7c8398e1933b9da74b67 Author: Aise Johan de Jong -- Hornet Date: Fri Jan 29 09:04:28 2010 -0500 Bootstrap: better lemma on finding opens Pretty much optimal lemma on finding open subfunctors of U/R when you start with a groupoid scheme. bootstrap.tex | 134 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 134 insertions(+), 0 deletions(-) commit dad1b8846d872f2301d8b1c23b930fb12767b4d4 Author: Aise Johan de Jong -- Hornet Date: Thu Jan 28 20:29:13 2010 -0500 Groupoids: Fix dumb error. The error was not realizing that ``relation generated by'' is a kind of tricky operation... This basically has no further ramifications since we only used the result in situations where the the pre-relation is actually a pre-equivalence relation (which would be better named a equivalence pre-relation). groupoids.tex | 75 ++++++++++++++++++++++++++++++++++++-------------- spaces-groupoids.tex | 68 +++++++++++++++++++++++++++++++++------------ spaces.tex | 13 +++++++- 3 files changed, 115 insertions(+), 41 deletions(-) commit 2fd9732b9282f8fadabbab3eb27550c5d20ff079 Author: Aise Johan de Jong -- Strider Date: Wed Jan 27 15:51:52 2010 -0500 Groupoid schemes: slicing lemma Here is the lemma promised in commit 79c56068dee8636e2ae061dd70e3af6d8f9bd30d. The proof is really straightforward once you formulate it in the setting where you already know that s,t are Cohen-Macaulay. groupoids.tex | 157 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++- spaces.tex | 6 +- 2 files changed, 159 insertions(+), 4 deletions(-) commit 3d8ebf66d9f3fc52a27f7500c754040f7930366d Author: Aise Johan de Jong -- Strider Date: Wed Jan 27 11:42:04 2010 -0500 Restriction and equivalences We inserted a couple of lemmas describing when the restriction of a groupoid scheme gives rise to an isomorphic quotient sheaf or when it gives rise to an equivalent quotient stack. groupoids.tex | 68 ++++++++++++++++-- spaces-groupoids.tex | 188 ++++++++++++++++++++++++++++++++++++++++++++++++- spaces-topologies.tex | 17 +++++ spaces.tex | 2 +- stacks.tex | 38 ++++++++++- 5 files changed, 304 insertions(+), 9 deletions(-) commit a3db85eb31549af9fadb174afa6e5270cc140a92 Author: Aise Johan de Jong -- Strider Date: Wed Jan 27 11:39:27 2010 -0500 Groupoids: From f+lfp to CM+lfp We added a lemma that states that any groupoid with s,t flat and locally of finite presentation is ``equivalent'' to a groupoid with s,t Cohen-Macaulay and locally of finite presentation. groupoids.tex | 73 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 73 insertions(+), 0 deletions(-) commit a74ca1c560c541094b0f08135f0a065fb7808914 Author: Aise Johan de Jong -- Hornet Date: Wed Jan 27 08:23:31 2010 -0500 More morphisms: Cohen-Macaulay morphisms Cohen-Macaulay morphisms are fppf local on the target and syntomic local on the source. bootstrap.tex | 5 +- more-morphisms.tex | 154 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 157 insertions(+), 2 deletions(-) commit 1b3ed01ddac815bec147a2f8f3546e19a26a3b71 Author: Aise Johan de Jong -- Strider Date: Tue Jan 26 17:17:50 2010 -0500 Cohen-Macaulay morphisms: started section This is just preliminary discussion on the descent and ascent of the Cohen-Macaulay property under extensions of ground field. algebra.tex | 160 +++++++++++++++++++++++++++++++++++++++++---------- more-morphisms.tex | 27 +++++++++ tags/tags | 2 +- varieties.tex | 47 +++++++++++++-- 4 files changed, 197 insertions(+), 39 deletions(-) commit a7438d9089c15651c0953fff80ab158872def2bf Author: Aise Johan de Jong -- Strider Date: Tue Jan 26 11:16:40 2010 -0500 Morphisms: Changed order of sections Reason: The notion of a flat morphism was used before the section on flat morphisms. morphisms.tex | 738 ++++++++++++++++++++++++++++----------------------------- 1 files changed, 367 insertions(+), 371 deletions(-) commit 8f4eaf89140847302a73d823f726545e84d91597 Author: Aise Johan de Jong -- Hornet Date: Tue Jan 26 08:26:37 2010 -0500 Improved browse.html browse.html | 73 +++++++++++++++++++++++++--------------------------------- 1 files changed, 32 insertions(+), 41 deletions(-) commit 79c56068dee8636e2ae061dd70e3af6d8f9bd30d Author: Aise Johan de Jong -- Strider Date: Mon Jan 25 15:03:15 2010 -0500 Bootstrap: Added new chapter It seems better to discuss the bootstrap theorems in a separate chapter following the discussion of the basic material on algebraic spaces. This can then be used in the chapter on algebraic stacks when discussing presentations and what not. The goal is to prove the theorem that an fppf sheaf F for which there exists a scheme X and a morphism f : X --> F such that f is representable by algebraic spaces f is flat, surjective and locally of finite presentation is automatically an algebraic space. We have almost all the ingredients ready except for a slicing lemma -- namely somehow lemma 3.3 part (1) of Keel-Mori. Makefile | 4 +- algebraic.tex | 11 +- bootstrap.tex | 665 ++++++++++++++++++++++++++++++++++++++++++++++++++ browse.html | 40 ++- chapters.tex | 1 + preamble.tex | 1 + spaces-morphisms.tex | 305 +++++++++-------------- spaces.tex | 105 -------- tags/Makefile | 4 +- tags/tags | 24 +- 10 files changed, 835 insertions(+), 325 deletions(-) commit cccc58a13db5b2b0bafc1b46bd00f1673c4238ad Author: Aise Johan de Jong -- Strider Date: Mon Jan 25 09:08:10 2010 -0500 Tags: added new tags tags/tags | 110 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 110 insertions(+), 0 deletions(-) commit 5488a859acc1ee5c4154cd4a1fff9536f6d032c3 Author: Aise Johan de Jong -- Hornet Date: Sun Jan 24 19:57:21 2010 -0500 Groupoid spaces: More about quotient stacks Still need to add a description of [X/G]. spaces-groupoids.tex | 174 +++++++++++++++++++++++++++++++++++++++++++++++++- stacks.tex | 2 +- 2 files changed, 174 insertions(+), 2 deletions(-) commit 25a1a30dcb55d9776484b41835aec6400007fbf8 Author: Aise Johan de Jong -- Hornet Date: Sat Jan 23 16:35:44 2010 -0500 Groupoid spaces: abstract quotient stacks Some formal lemmas on properties of a quotient stack of a groupoid space. Coming up: a description of its morphisms, objects, etc. spaces-groupoids.tex | 325 +++++++++++++++++++++++++++++++++++++++++++++++--- stacks.tex | 71 +++++++++++- 2 files changed, 377 insertions(+), 19 deletions(-) commit efe73fdbf02bbac56effc3df09ce7b7745a7b3a3 Author: Aise Johan de Jong -- Cheetah Date: Fri Jan 22 23:33:29 2010 -0500 Groupoid Spaces: Added chapter to the project OK, but we still have to write the most interesting section of the chapter. Makefile | 3 +- browse.html | 28 ++++-- chapters.tex | 1 + preamble.tex | 1 + scripts/functions.py | 4 + spaces-groupoids.tex | 229 ++++++++++++++++++-------------------------------- spaces-morphisms.tex | 27 ++++++ tags/Makefile | 3 +- 8 files changed, 138 insertions(+), 158 deletions(-) commit 2313fff9ff8fb799a1900d61e294208a9fc70f76 Author: Aise Johan de Jong -- Strider Date: Fri Jan 22 17:13:58 2010 -0500 Groupoid spaces: Towards stackify What we have to do now is stackify a groupoid space. Hence we have to rewrite (the easy parts of) the chapter on groupoid spaces, and then add some material on stackifying groupoid spaces. This is all preliminary before starting to discuss presentations of algebraic stacks. groupoids.tex | 2 +- spaces-groupoids.tex | 1473 ++++++++++++++++++++++++++++++++++++++++++++++++++ tags/tags | 2 +- 3 files changed, 1475 insertions(+), 2 deletions(-) commit 0436365cb84154b47b159fcb45714352b8827dcb Author: Aise Johan de Jong -- Strider Date: Wed Jan 20 16:53:18 2010 -0500 Algebraic stacks: Deligne-Mumford with trivial inertia is a space To prove this for a general algebraic stack we will first prove a characterization of a DM stack as an algebraic stack whose inertia is formally unramified, or equivalently diagonal is formally unramified. Before we do this it makes sense to change the notion of unramified as suggested by David Rydh (see documentation/todo-list). We also added the proof of the statement that the property of being an algebraic stack is invariant under equivalences. algebraic.tex | 219 +++++++++++++++++++++++++++++++----- categories.tex | 112 ++++++++++++++++++- documentation/todo-list | 5 +- stacks.tex | 285 ++++++++++++++++++++++++++++++++++++++++++++--- 4 files changed, 568 insertions(+), 53 deletions(-) commit 8fc748044a4c34fd12b060b8f95221a66b881cb5 Author: Aise Johan de Jong -- Hornet Date: Wed Jan 20 09:58:28 2010 -0500 Webpage: added link to blog on main webpage. stacks-git.html | 4 +++- 1 files changed, 3 insertions(+), 1 deletions(-) commit 77e06f25007f2e5e1fb058063884a026eabd3eb5 Author: Aise Johan de Jong -- Hornet Date: Tue Jan 19 09:40:55 2010 -0500 Algebraic stacks: properties of morphisms representable by spaces Finally we have enough material so that our definition of algebraic stacks makes sense. Phew! algebraic.tex | 286 +++++++++++++++++++++++++++++++++++++++++++++++++++++++-- spaces.tex | 13 ++- 2 files changed, 286 insertions(+), 13 deletions(-) commit 20857fc2a204c515d0c547a2987125842ce7e9ed Author: Aise Johan de Jong -- Hornet Date: Sun Jan 17 21:12:07 2010 -0500 Descent for Spaces: Immersions are fppf local on target The proof currently uses that separated, locally quasi-finite morphisms are representable. It implies that being "locally separated" is fppf local on the target too. Also added: Short section on monomorphisms of algebraic spaces. spaces-descent.tex | 79 ++++++++++++++++++++++++- spaces-morphisms.tex | 152 +++++++++++++++++++++++++++++++++++++++++++++++++ spaces-topologies.tex | 14 +++++ 3 files changed, 242 insertions(+), 3 deletions(-) commit aeaa96993f65763342a2314e27a8ad5423197064 Author: Aise Johan de Jong -- Strider Date: Sun Jan 17 12:51:06 2010 -0500 Tags: added new tags tags/tags | 83 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 files changed, 83 insertions(+), 0 deletions(-) commit 794fe5fc475cdf76685f6f2aefaf5ddb9bf2c595 Author: Aise Johan de Jong -- Hornet Date: Sun Jan 17 12:44:39 2010 -0500 Descent on Spaces: Done with fpqc local properties of morphisms This time we added: Integral Finite Quasi-finite Syntomic Smooth Etale unramified Finite locally free Monomorphism The only one that is missing is "locally of finite type of relative dimension d". This would be easy to add, but it seems wrong to do so without discussing in some detail the notion of dimension on algebraic spaces (which might be a little tricky do "correctly" for general algebraic spaces (since specializations of points already caused some trouble). spaces-descent.tex | 245 +++++++++++++++++++++++++++++++++++++++++++++++++ spaces-morphisms.tex | 62 +++++++++++++ spaces-topologies.tex | 14 +++ 3 files changed, 321 insertions(+), 0 deletions(-) commit 451b3b95b5a8b873ee4afab9cc82700f9f8eb507 Author: Aise Johan de Jong -- Hornet Date: Sun Jan 17 11:39:21 2010 -0500 Morphisms of Spaces: Uniform latex labels for lemmas Only coding changes, except for moving a lemma in spaces-descent.tex to its proper place. spaces-descent.tex | 44 ++++++++++++++++++++++---------------------- spaces-morphisms.tex | 22 +++++++++++----------- tags/tags | 12 ++++++------ 3 files changed, 39 insertions(+), 39 deletions(-) commit bb61741faa5e9228001c4fcc6dd42ea8df1ff10e Author: Aise Johan de Jong -- Strider Date: Fri Jan 15 15:54:24 2010 -0500 Descent: morphisms of schemes satisfy fpqc descent We edited the descent chapter to improve our exposition, prompted by a question of Thanos D. Papaïoannou, namely "Do morphisms of algebraic spaces satisfy fpqc descent?" Here is one answer: Suppose X, Y are algebraic spaces over an affine base S. Suppose that S' --> S is a surjective flat morphism of affine schemes. Finally suppose that a' : X_{S'} --> Y_{S'} is a morphism of algebraic spaces over S' which is compatible with the canonical descent data. Now you want to know if a' descends to a morphism a : X --> Y over S. If X, Y are representable, then this is Descent, Lemma Tag 02W0. See also the explanation in the remark following that lemma. The proof is kind of long since I tried to prove somehow more general statements in that section. The key is Descent Lemma Tag 0241. It mainly relies on the fact that the representable presheaf associated to a scheme satisfies the sheaf condition for the fpqc topology (which is Descent, Lemma Tag 023Q). The analogue for this in the category of algebraic spaces is Properties of Spaces, Lemma Tag 03WB but there is a condition, namely that the algebraic space is Zariski locally quasi-separated. So I do not know how to prove this descent in general, and it may not be true. But it is true for Zariski locally quasi-separated spaces. Eventually we will state this descent property explicitly in the stacks project. Any algebraic space in the literature, say published before year 2000, is quasi-separated (since there are basically no references which deal with more general ones). Hence fpqc descent for morphisms of algebraic spaces is true for any algebraic spaces which you can find in these articles/books. descent.tex | 243 +++++++++++++++++++++++++++++-------------------- spaces-properties.tex | 2 +- spaces.tex | 10 +- topologies.tex | 15 +++ 4 files changed, 167 insertions(+), 103 deletions(-) commit fd5d30488ce724c9616c14a131b84dadaaa4dd78 Author: Aise Johan de Jong -- Hornet Date: Thu Jan 14 21:08:34 2010 -0500 Descent on Spaces: More fpqc descent of properties of morphisms Namely: Affine Closed immersion Separated Quasi-affine Quasi-compact immersion descent.tex | 2 +- spaces-descent.tex | 198 ++++++++++++++++++++++++++++++++++++++++++++++++++- 2 files changed, 195 insertions(+), 5 deletions(-) commit 3e3e6a677cf8eda5feeba06ebc41013c08e3bb0f Author: Aise Johan de Jong -- Strider Date: Thu Jan 14 17:07:39 2010 -0500 Descent on Spaces: More fpqc descent of properties of morphisms Namely: Universally open Universally closed Surjective Radicial Locally finite type Locally finite presentation Finite type Finite presentation Proper Flat Open immersion Isomorphism Todo soon: separated locally separated (for fppf topology) spaces-descent.tex | 407 ++++++++++++++++++++++++++++++++++++++++++---------- 1 files changed, 333 insertions(+), 74 deletions(-) commit 5b3f915e2c6edc8f0c99af3853242bc188f9a226 Author: Aise Johan de Jong -- Strider Date: Thu Jan 14 14:10:11 2010 -0500 Morphisms of Spaces: homogenizing lemma-P-local This finishes the work uniformizing the assertions in the lemmas with latex labels ``lemma-P-local'' started in commit ef7137879bb7ba18d2038feb459a262181bbcb0a. Of course we should fill in the proofs. spaces-morphisms.tex | 274 ++++++++++++++++++++++++++++++++++++++++++++------ 1 files changed, 244 insertions(+), 30 deletions(-) commit 4e7ab974e92cea377d8a70c9270c1eeb7d53a81d Author: Aise Johan de Jong -- Strider Date: Thu Jan 14 13:38:34 2010 -0500 Etale cohomology: typo in tag etale-cohomology.tex | 8 ++++---- tags/tags | 2 +- 2 files changed, 5 insertions(+), 5 deletions(-) commit ef7137879bb7ba18d2038feb459a262181bbcb0a Author: Aise Johan de Jong -- Strider Date: Thu Jan 14 12:10:55 2010 -0500 Morphisms of Spaces: properties are Zariski local on target Also added: types of coverings in Topologies of Spaces spaces-morphisms.tex | 343 ++++++++++++++++++++++++++++++++++++------------- spaces-topologies.tex | 217 ++++++++++++++++++++++++++++++-- 2 files changed, 459 insertions(+), 101 deletions(-) commit 3116a18811a07408eaf0af9df8b8530d191eb0dc Author: Aise Johan de Jong -- Strider Date: Wed Jan 13 17:04:24 2010 -0500 Descent on Spaces: Lemma on types of morphisms fpqc local on the base spaces-descent.tex | 53 +++++++++++++++++++++++++++++++++++++++++++++++++ spaces-properties.tex | 4 +- 2 files changed, 55 insertions(+), 2 deletions(-) commit 607fa33f316cfd03f53053f5e36990c19c41fe40 Author: Aise Johan de Jong -- Cheetah Date: Sun Jan 10 11:15:54 2010 -0500 Morphisms of Spaces: Submersive morphisms Also changed: Lemma which states that a surjective quasi-compact flat morphism of algebraic spaces is (universally) submersive. Previously we have only proved this for representable morphisms. spaces-descent.tex | 2 +- spaces-morphisms.tex | 59 ++++++++++++++++++++++++++++++++++++++++++------- 2 files changed, 51 insertions(+), 10 deletions(-) commit ae8ec19da1a72611e843afef1f5861093370b23e Author: Aise Johan de Jong -- Hornet Date: Sat Jan 9 12:35:05 2010 -0500 Specializing, Generizing and Submersive morphisms Introduced the terminology listed above, as suggested by David Rydh. In EGA I (new edition) they use g\'en\'erisant and I guess the english form is indeed generizing, but I still have to get used to it. Also added: Lemma stating that every open morphism of schemss is generizing which was missing (David Rydh). algebra.tex | 23 +++++++++++++++++++++ morphisms.tex | 62 +++++++++++++++++++++++++++++++++++++++++++++++++++++--- topology.tex | 34 +++++++++++++++++++++++++----- 3 files changed, 109 insertions(+), 10 deletions(-) commit 9082eb9c3784dabde96631889fc9249589af7283 Author: Aise Johan de Jong -- Hornet Date: Fri Jan 8 21:03:07 2010 -0500 Documentation: Typos and additional possible todo documentation/git-howto | 2 +- documentation/todo-list | 12 ++++++++---- 2 files changed, 9 insertions(+), 5 deletions(-) commit 9d33b483a730e045d5727aef6f235106a852104a Author: Aise Johan de Jong -- Hornet Date: Fri Jan 8 20:08:11 2010 -0500 Morphisms of Spaces: Fix Lemma 03L1 This lemma says that morphisms of type beta, gamma, or delta are preserved under compositions. This finishes (hopefully) the series of fixes due to the mistake fixed in commit 617b8ab5d804f770a0e58d5a4a4b9355ee8ea201 spaces-morphisms.tex | 9 ++++++--- 1 files changed, 6 insertions(+), 3 deletions(-) commit 66f6fc5b6d9febcbd5f5101b48327601f33e9cf3 Author: Aise Johan de Jong -- Hornet Date: Fri Jan 8 19:47:45 2010 -0500 Properties of Spaces: Fix Lemma 03KF This lemma says that if X --> Y is representable and Y has property \beta, \gamma, \delta, \epsilon or \eta, then also X has that property. spaces-properties.tex | 51 +++++++++++++++++++++++++++++++----------------- 1 files changed, 33 insertions(+), 18 deletions(-) commit 99943e5da707bfa703b567f28a305cc1d0dbce5f Author: Aise Johan de Jong -- Hornet Date: Fri Jan 8 19:37:06 2010 -0500 Properties of Spaces: Added another condition to Lemma 03JS As mentioned in commit 2e77361c58f0a32c168a8e5fdc6c67b91d5038cd spaces-properties.tex | 28 +++++++++++++++++++++++++--- 1 files changed, 25 insertions(+), 3 deletions(-) commit 26d443bd339514bc9ebc1df5e04e3abf524c719d Author: Aise Johan de Jong -- Strider Date: Fri Jan 8 16:25:01 2010 -0500 Properties of Spaces: Improve statement of claim in Proposition 03JI CLAIM: Suppose you have a groupoid scheme (U, R, s, t,...) which defines an etale equivalence relation with s, t finite (!). Then the union of the R-invariant affine opens of U is topologically dense in U. Here is a discussion of the proof which may be clearer than what is now actually in the stacks project: If U is separated this is easy because a finite set of generic points of U is contained in an affine open, and generic points are dense. Hence given a generic point of U you take an affine W containing its orbit $O$ (which is a finite set of generic points of U), and then you find an invariant affine inside W containing this orbit $O$ by standard arguments. The point (in the proof of the Proposition) is that such an argument also works if U is arbitrary by the silly trick which says that any quasi-compact scheme has a dense open which is separated... spaces-properties.tex | 5 +++-- 1 files changed, 3 insertions(+), 2 deletions(-) commit 1b4acf27c1e68941284bf1ede15bfbc349c27dc1 Author: Aise Johan de Jong -- Strider Date: Fri Jan 8 16:02:58 2010 -0500 Todo list: Application of ZMT to add later documentation/todo-list | 9 +++++++++ 1 files changed, 9 insertions(+), 0 deletions(-)