https://ncatlab.org/jamesdolan/published/Algebraic+Geometry ]]>

There was an “Idea” at Bridgeland stability condition. I added the sections Definition, Key Results, and Examples. I also added a reference for the last key result listed. I should probably fill in some related stubs like t-structure or classical notions of stability.

]]>jet scheme to complement jet space and arc space entries; for now just a record of few references.

]]>Stub, recording the basic references: arc space.

]]>new page mixed Tate motive, mostly to record some references for now

]]>- Ludmil Katzarkov, Dmitri Orlov, Tony Pantev,
*Notes on Higgs bundles and D-branes*, (delivered as a lecture by Tony Pantev at winter school at Guanajuato 2013 link) draft pdf

added to references at Higgs bundle.

]]>I’ve created a page on the Artin-Mazur formal group.

]]>Modulo the definition, I’ve created Picard scheme. One thing I couldn’t tell, is there a standard term in nlab for the “fiber category” of a stack? I mean if $F:C\to D$ fibers $C$ over $D$ then if you pick some object $X$ from $D$ the category $C_X$ consisting of objects that go to $X$ and morphisms that go to $id_X$.

]]>Is there anything in the literature on generalizing Cartier duality to non-finite group schemes? Pointers would be welcome.

]]>New stubs tropical geometry and tropical semiring (with rig version included). Note the new book by Gross.

]]>Chevalley’s theorem on constructible sets and elimination of quantifiers. The entries are related ! The interest came partly from teaching some classical algebraic geometry these days. The related entry is also forking, though yet it is not said why; non-forking may be viewed as related to a notion of generic point, generic type (in the sense of model theory).

]]>I just learned about rigidification and decided to record it somewhere.

I’m not sure if the title is good, because there is the notion of the rigidification of quasi-categories.

Surely this notion has a higher analogue that maybe someone knows more about. Surely you could take an $n$-stack and consider the $n$-categorical fiber product to make a notion of inertia, and then rigidify with respect to some subgroup object inside…

I just added a page on the Bondal-Orlov reconstruction theorem. Feel free to edit!

]]>New entry representable morphism, in the sense of Grothendieck school. The notion is used at closed immersion of schemes where I just made some changes.

]]>I started the article Z-infinity-module. Hopefully someone here can say something more interesting about them!

]]>I changed quasicompact to quasicompact morphism though it is also about quasicompact schemes etc. as before and moved the query box here:

]]>Mike: To accord with terminological conventions, this page should probably be either “quasicompact space” or “quasicompact object.”

Zoran Skoda: I do not know what are the conventions, but it was intentional to look both at quasicompact spaces and quasicompact morphisms (which are according to the dominant point of view in algebraic geometry, more important and basic notion); and aside also for q. objects. Personally I do not understand English-language preference for noun phrases. If one is to choose, quasicompact morphism is the choice.

Toby: By the «Each definition gets its own page.» convention, I'm not even sure that this shouldn't just redirect to compact space or compact object. My impression is that assuming that ’compact’ implies Hausdorff is either (like assuming that ’ring’ implies commutative) restricted to fields where it's a common assumption or to languages (I'm thinking mostly of Bourbaki in French here) other than English. On the other hand, if it's used that way by English-writing algebraic geometers, then I would seem to be wrong (since algebraic geometers often have non-Hausdorff spaces).Zoran Skoda: Convention that ’compact’ includes Hausdorff is very common also among people working predominantly on nice spaces, particularly differetial geometers, differential topologists, people studying metric spaces and so on. But for “paracompact” the situation is more tricky: in literature, even on general topology there are also competing definitions, which are all equivalent for Hausdorff spaces. All my life I bounce in such people; my own education does not assume Hausdorffness, unless it is said in the form “compactum”. Algebraic geometers always say quasi-compact, it has nothing to do with language; but as I say for algebraic geometers the basic notion is quasi-compact. The emphasis of this entry is on the terminology and morphisms (what should be expanded on: I still did not write the deifnitions of quasi-compact MORPHISM in various setups); so redirection won’t work I think. Plus although from my point of view saying quasicompact and compact is the same for spaces; one would never say compact for the scheme; scheme is said to be quasicompact if its underlying space is (quasi)compact.

There is an additional reason for that: one can consider a nonsingular variety over complexes which is quasicompact, and which itself is not compact in complex topology (under GAGA). But in the same considerations it is often useful to have some arguments in Zariski and some in complex topology; one of the reasons for word quasicompact is that sometimes we have the “same” example which we are used to think as of noncompact space but it is (quasi)compact in Zariski topology. When an algebraic geometer thinks of the difference between compact and quasicompact for complex varieties he has that in mind; in more general setups about Hausdorff vs nonHausdorff. In the same time, when talking about objects in derived categories of qcoh sheaves, even algebaric geometers use moreoften term compact than quasicompact; thus redirecting to compact object and saying this is for algebraic geometry won’t do for all the 3 notions in this entry (on the contrary side, nobody says compact morphism as far as I could confirm, but quasicompact morphism).

Toby: Ah, so when you've got both Zariski and complex topologies around, you can easily distinguish the former by the prefix ’quasi’; that's cute. Anyway, perhaps we'll move this to quasicompact morphism if you write mostly about that, but I won't try to move anything for now.

More at field with one element, after creating person entry Christophe Soulé about the creator. By the way the Soulé has different encoding in n-Forum than in nlab so the link does not access the right page from here. See redirect Christophe Soule.

]]>Let $\mathcal{Z}$ be the Zariski topos, in the sense of the classifying topos for local rings. I was wondering whether there might be any connection between $\mathbf{Sh}(\operatorname{Spec} \mathbb{Z})$ and $\mathcal{Z}$. Certainly, there is a geometric morphism $\mathcal{Z} \to \mathbf{Sh}(\operatorname{Spec} \mathbb{Z})$, and there’s also a geometric inclusion $\mathbf{Sh}(\operatorname{Spec} \mathbb{Z}) \to \mathcal{Z}$. On the other hand, there’s no chance of $\mathcal{Z}$ itself being localic, since it has a proper class of (isomorphism classes of) points. Let’s write $L \mathcal{Z}$ for the localic reflection of $\mathcal{Z}$; the first geometric morphism I mentioned then corresponds to a locale map $L \mathcal{Z} \to \operatorname{Spec} \mathbb{Z}$. But what is $L \mathcal{Z}$ itself?

The open objects in $\mathcal{Z}$ can be identified with certain saturated cosieves on $\mathcal{Z}$ in the category of finitely-presented commutative rings, and so may be identified with certain sets of isomorphism classes of finitely-presented commutative rings. If I’m not mistaken, every finitely-presented commutative ring gives rise to an open object in $\mathcal{Z}$. This suggests that $L \mathcal{Z}$ might be some kind of (non-spatial) union of all isomorphism classes of affine schemes of finite type over $\mathbb{Z}$, which is an incredibly mind-boggling thing to think about. It’s not clear to me whether other kinds of open objects exist. For example, does every not-necessarily-affine open subset of $\operatorname{Spec} A$, for every finitely-presented ring $A$, also show up…?

]]>I addede a paragraph at Poincaré duality about the generalizations, and created the entry (so far only descent bibliography) Grothendieck duality; the list of examples expanded at duality. All prompted by seeing the today’s arXiv article of Drinfel’d and Boyarchenko.

]]>New stubs absolute de Rham cohomology, L-function, prompted by one answer to my MathOverflow question and having just basic links. By the way, the link to the pdf file of a Kapranov’s article listed at de Rham complex does not seem to work.

]]>I made a very, very brief start to K3 surfaces

]]>New entry descent of affine schemes: the fibered category of affine morphisms (SGA I.8.2 th.2.1) satisfies effective descent along any fpqc morphism. This fact is harder than the descent for quasicoherent sheaves of $\mathcal{O}_X$-modules.

]]>Recently I’ve been looking a lot at the derived category of coherent sheaves on a scheme (specifically of Calabi-Yau threefolds). I’ve been told that it is “better” to consider this as a dg-category for whatever reason. I’m quite interested in exploring this, since I have no idea why this would be. For instance, the great book Fourier-Mukai Transforms in Algebraic Geometry by Huybrechts develops tons of theory and has lots of great beautiful theorems, but the phrase “dg-category” never once appears in the whole book. Do you really get much more from this point of view?

Anyway, I’m interested in writing notes here as I try to figure out what is going on here. I’m trying to take inventory of what pages already exist on this topic. I’ve found derived algebraic geometry, dg-category, derived category. I can’t find anything more on topic than those, so am I right in assuming that there seems to be very little about the derived category of sheaves or am I missing it somewhere?

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