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put something basic into form of an algebraic group
have added some more references to logarithmic geometry and cross-linked a bit. (but there is still no genuine content)
Created p-local module.
started a minimum at analytification, mainly interested for the moment in collecting the references now given there which discuss analytification of algebraic (etc.) stacks
slightly expanded and prettified a bit at almost complex structure (nothing non-trivial, just cosmetics)
I have expanded just a little at KR-theory by giving it an actual Idea-paragraph and adding some more references.
I gather (via this nice MO comment) that
The functor that takes linear algebraic groups G to their ℝ-points G(ℝ) constitutes an equivalence of categories between compact Lie groups and ℝ-aniosotropic reductive algebraic groups over ℝ all whose connected components have ℝ-points.
For G as in this equivalence, then then complex Lie group G(ℂ) is the complexification of G(ℝ).
I have a gap in my education here and would like to fill it. What’s a good source that discusses this statement a bit more? And which one of Chevalley’s articles is this result originally due?
started a minimum at p-convex polarization
added to absolute Galois group and to Grothendieck-Teichmüller group brief pointers for the inclusion of the former into the latter (for the case of the rationals) and a pointer to
I noticed tha tthe entry separable closure existed but was effectively devoid of content. I have now copy-and-pasted the relevant paragraphs from the entry Galois theory into it.
Have given Frölicher spectral sequence an Idea-paragraph and some pointers, but it is still a stub.
I created this page: http://ncatlab.org/nlab/show/The+Mathematical+Literature
Am I right that I’m supposed to post about this here at the nForum? How big should an edit be in order to deserve a post here?
And can i link from the nForum to the nLab by doing this? The+Mathematical+Literature
I wrote a bit at heap about the empty heap (and its automorphism group, the empty group, which I put in the headline for maximum shock value).
added pointers to the section on cohomology.
Over on MO (in the comments here) Stefan Wendt kindly reminds me of an old nLab entry I once started on B1-homotopy theory. Have added a reference and hope to be adding more.
started Hodge cycle, but my battery is dying right this moment….
gave Hodge symmetry it’s own brief stand-alone entry
Larusson formulates the Oka principle homotopy-theoretically as: a complex manifold X is Oka if for every Stein manifold Σ the canonical map
Mapshol(Σ,X)→Mapstop(Σ,X)between the mapping spaces is a weak homotopy equivalence (see here).
It is natural to wonder what this looks like in terms of the cohesion of the ∞-topos ℂAnlytic∞Grpd over CplxMfd.
If we write Π:ℂAnlytic∞Grpd→∞Grpd, then up to possible technicalities to be checked, it should simply mean
Π[Σ,X]≃⟶[ΠΣ,ΠX]where [−,−] is the internal hom.
(Something close to this (but not quite the same) is what Lawvere calls the “axiom of continuity” in a cohesive topos.)
If instead we work internally and let Π:ℂAnlytic∞Grpd→ℂAnlytic∞Grpd be the shape modality, then the above is equivalently
Π[Σ,X]≃⟶♭[ΠΣ,ΠX].In either case, it is a very natural condition to ask for in general cohesive ∞-toposes. Maybe one should call it the Oka-Larusson property or something…
started analytic geometry – contents to be included as “floating table of contents”, and accordingly included it into relevant entries
I split off an entry applications of (higher) category theory from the entry nPOV.
Hopefully we find the energy to further improve this entry in various ways. For the moment I just added a 1-line intro. And a quote, which I think hits the nail of this entry on the head.
added pointers to Fornaess-Stout on complex polydiscs here
Somebody emailed me highlighting that the text green here, revision 69 of Dold-Kan correspondence does not quite parse.
I didn’t write this,though. There is a definition meant to be equivalent to that at combinatorial spectrum, but at least some indices need renamining, and it seems maybe more needs to be fixed or at least added. Not sure. Also I absolutely don’t have the leisure to look into this right now. I hope somebody finds the energy to look into it.
I’ve created a page on the Artin-Mazur formal group.
created Serre duality with a simple minimum of content
(I have also briefly touched a bunch of related entries on Dolbeault cohomology etc. but most of them are still in a sad state and need work)
we didn’t have any entry defining coherent cohomology, did we?
(I notice that we are lacking also an entry coherent object. That really needs to be created.)
quick stub for morphism of finite type (redirecting also morphism of locally finite type)
Added some new material to linguistics.
gave intersection theory an Idea section. Clearly, there is room for improvement, but it’s a start.
In the course of this I have created at stub for Bézout’s theorem, gave Serre intersection formula an Idea-section (please feel invited to expand further!) and cross-linked all this and a few more entries.
added some minimum of content to the old stub divisor (algebraic geometry)
started polarized algebraic variety
added the definition in complex analytic geometry with a pointer to degree of a coherent sheaf
created exponential exact sequence with the obvious basic comments. Cross-linked with Planck’s constant, with multiplicative group and circle group, and with Kummer sequence.
added to Theta characteristic two further brief Properies-paragarphs: As metaplectic and Spin-structure over (Kähler-)polarized varieties and Over intermediate Jacobians.
I created at equivariant cohomology separate subsections for, so far, Borel equivariant and Bredon equivariant cohomology.
At Bredon cohomology I added a sentence about the coefficient objects.
Since Hisham’s article came out
I briefly created a stub entry ninebrane structure and expanded the higher spin structure - table accordingly. More should be done here, but I don’t have the leisure right now.
started a minimum at Quillen Q-construction
The nLab entry Spectral Schemes has existed for a long time, now finally the article with that title exists, too. ;-) See the link there
only now realized that Zoran had an old entry moduli space of bundles. Have now vigorously cross-linked it with a bunch of related entries
created Kummer-Artin-Schreier-Witt exact sequence, but except for some references, there is no actual content yet
created a brief entry anti-modal type, since I wanted to be able to point to this. Cross-linked with localizing subcategory etc.
created some stub entries for keywords at Representability theorems
and cross-linked a bit.
no content yet besides a minimal Idea-sentence, for the moment just so as to get some pointers into place
Several related new entries: Gabriel-Rosenberg theorem, spectrum of an abelian category, local abelian category.
I added some material on arc-connected spaces to connected space.
I added also a reference to Willard’s General Topology, together with this online link to a Scribd document: Willard. Is this kosher (I am guessing this document is not “pirated”, but I’m not sure)?
Prompted by a question which I received, I went and tried to streamline the old entry Lie infinity-algebroid representation a little:
moved the pevious “Properties”-discussion of complexes of holomorphic bundles to the Examples-section;
added the example of L∞-algebra extensions
added more information to the References-section
cross-linked a bit more with infinity-action and with L-infinity algebra cohomology etc.
I started a separate page for Picard stack (which used to be just a redirect to Picard scheme), stated the general nonsense idea with a pointer to Lurie’s thesis, where this essentially appears.
(BWT, where in the DAG series did this end up? I forget.)
Of course the upshot is that it’s simply the internal hom/mapping stack Pic(X)=[X,B𝔾m]. I have a question here: it seems clear that the higher versions [X,Bk𝔾m] want to be called the higher intermediate Jacobians (their deformation theory at 0 are the Artin-Mazur formal groups). Why does nobody say this? (Or if they do, where?)
Over in the thread on “Picard infinity-stack” we turned to discussion of Brauer stack. Just for completeness I should probably make this a separate thread here: I had created Brauer stack for the moment only with the following Idea-section
It is traditional to speak, for a suitable scheme X, of its Picard group and of its Brauer group. Moreover, it is a classical fact that under suitable conditions the former admits itself a canonical geometric structure that makes it the Picard scheme of X. Still well known, if maybe less commonly highlighted, is that this is just the 0-truncation of the Picard stack of X, which is simply the mapping stack [X,B𝔾m] into the delooping of the multiplicative group. In this form this applies immediately also to more general context such as E-∞ geometry ("spectral geometry") and gives a concept of Picard ∞-stack ("derived Picard stack"). Given this and the relation of the Brauer group to étale cohomology it is clear that the Brauer group similarly arises as the torsion subgroup of the 0-truncation of the ∞-stack which ought to be called the Brauer stack, given as the mapping stack
Br(X)≔[X,B2𝔾m]into the second delooping of the multiplicative group (modulating line 2-bundles). Indeed, just as the Picard stack turns under Lie integration (evaluation on infinitesimally thickened points) and 0-truncation into what is commonly called the formal Picard group, so this Brauer ∞-stack similarly gives what is commonly called the formal Brauer group.
However, while therefore the terminology "Brauer stack" is the evident continuation of a traditional pattern (which in the other direction continues with the group of units and the mapping scheme [X,𝔾,]), it seems that this terminology has never been introduced in the literature (at time of this writing). (?)
at holomorphic vector bundle I have started a section titled As complex vector bundles with holomorphically flat connections.
This deserves much more discussion (and maybe in a dedicated entry), but for the moment I have there the following paragraphs (with lots of room for further improvement):
+– {: .num_theorem #KoszulMalgrangeTheorem}
Holomorphic vector bundles over a complex manifold are equivalently complex vector bundles which are equipped with a holomorphic flat connection. Under this identification the Dolbeault operator ˉ∂ acting on the sections of the holomorphic vector bundle is identified with the holomorphic component of the covariant derivative of the given connection.
The analogous statement is true for generalization of vector bundles to chain complexes of module sheaves with coherent cohomology.
=–
For complex vector bundles over complex varieties this statement is due to Alexander Grothendieck and (Koszul-Malgrange 58), recalled for instance as (Pali 06, theorem 1). It may be understood as a special case of the Newlander-Nirenberg theorem, see (Delzant-Py 10, section 6), which also generalises the proof to infinite-dimensional vector bundles. Over Riemann surfaces, see below, the statement was highlighted in (Atiyah-Bott 83) in the context of the Narasimhan–Seshadri theorem.
The generalization from vector bundles to coherent sheaves is due to (Pali 06). In the genrality of (∞,1)-categories of chain complexes (dg-categories) of holomorphic vector bundles the statement is discussed in (Block 05).
+– {: .num_remark}
The equivalence in theorem \ref{KoszulMalgrangeTheorem} serves to relate a fair bit of differential geometry/differential cohomology with constructions in algebraic geometry. For instance intermediate Jacobians arise in differential geometry and quantum field theory as moduli spaces of flat connections equipped with symplectic structure and Kähler polarization, all of which in terms of algebraic geometry directly comes down moduli spaces of abelian sheaf cohomology with coefficients in the structure sheaf (and/or some variants of that, under the exponential exact sequence).
=–
started Narasimhan–Seshadri theorem, for the moment just to collect references.
I have just deleted a large number of dollar \ , dollar from the bottom of Blakers-Massey theorem. The effect of such is to add a large ammount of blank space at the end of the page. Was this intentional extra space for something? If not, what is causing it? I should add that I have found similar blank space before and deleted that as well.
added some minimum to Hodge cohomology
since it turned out to be hard to find, I gave the discussion of filtered homotopy colimits at combinatorial model category its own entry, and cross-linked:
started a minimum at Calabi-Yau cohomology.
This is an obvious idea that must have been studied before (for n≥2) but I have had no luck with finding much detail so far.
;-). I found a typo ‘gorup’ and did a search on the n-Lab…. great fun! It is good to know others have disobient fingers!
am starting an entry F/M-theory on elliptically fibered Calabi-Yau 4-folds
So far there is mainly an Idea-section.
created a survey-table-for-inclusion
and included it into the relevant entries