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Some time ago I started a stub characteristic variety to record few references, mainly in D-module context. Regarding that the related notion of a characteristic ideal also appears in the treatment of Iwasawa polynomial and Alexander polynomial which Urs wants to understand from the point of view of connections and differential refinements of cohomology, maybe we should do some effort to make some pages which will connect various notions of characteristic ideals and their loci across various subjects. I just recorded
at characteristic ideal for the version in the context of Iwasawa theory.
gave Langlands correspondence an actual Idea-section.
(Am in a rush and on a horrible wifi connection. Need to proof-read and add more links later.)
added the following story to the Properties-section of Dedekind eta function and also to the Examples-section of functional determinant and zeta function of an elliptic differential operator:
For a complex torus (complex elliptic curve) equipped with its standard flat Riemannian metric, then the zeta function of the corresponding Laplace operator is
The corresponding functional determinant is
where is the Dedekind eta function.
Recorded the basic idea at Artin L-function
Also: added the following paragraph to the Idea-section at Langlands correspondence (below the numbered two items stating the conjectured correspondence between Galois representations and automorphic representations):
Moreover, to each such automorphic representation is associated an L-function – the automorphic L-function – and in generalization of Artin reciprocity the conjecture is that the Artin L-function associated with the given Galois representation is equal to the automorphic L-function of the corresponding automorphic representation.
I have added the following sentence to class number formula and to all the other entries that the sentence links to:
Given a number field , the Dedekind zeta function of has a simple pole at . The class number formula says that its residue there is proportional the product of the regulator with the class number of
In particular I have also created regulator of a number field and cross-linked it with Beilinson regulator, which I have renamed to higher regulator.
gave zeta function regularization its own entry and expanded a bit, added more pertinent references
just to clean up entries, I have given torsion module its own entry (the keyword used to have non-overlapping discussion at torsion subgroup and at torsion approximation)
added to zeta function of an elliptic differential operator also some first minimum comments on the functional determinant and zeta-function regularization
added the definition of idele class group to group of ideles, with a brief pointer to its role in the moduli stack of line bundles. Added both then to the function field analogy – table
started a minimum at Euler product, the main point being for the moment to mention briefly how Euler product forms naturally appears from the point of view of adelic integration/Iwasawa-Tate theory.
I have renamed the entry pretriangulated dg-category to stable dg-category. I think this is a logical move that emphasizes the close relationship with the notions of stable (infinity,1)-category and stable model category. If anyone disagrees I would be interested to hear why. I have also edited the body of the page to give an exposition more in line with modern references like Cisinski-Tabuada. However I have adopted the term dg-presheaf for what is usually called (right) dg-module. This seems much more natural to me, but again I am open to hearing any arguments against it. When I get a minute I will also update the entry dg-category to match the conventions of this page.
Also, I never liked the term “quasi-equivalence”. I think that this should just be called equivalence of dg-categories, or maybe Dwyer-Kan equivalence if this is too ambiguous. Any thoughts?
@Urs: I do not quite agree with the sentence “This is unrelated to other notions of monads” in Beilinson monad.
One can indeed view the Beilinson monad as the monad of an adjoint equivalence between (interpreted as the heart of and some category of linear complexes over an exterior algebra (the Koszul dual of the Cox ring of ).
created a minimum at adelic integration, for the moment this is just a glorified pointer to Fesenko 08, section 3
At differential cohesion there used to be the statement that every object canonically has a “spectrum” given by , but the (simple) argument that indeed satisfies the axioms of a structure sheaf used to be missing. I have now added it here.
added references.
Any book that develops a bit of algebraic geometry of non-unital commutative rings or one that discusses what would be hte major things that break?
started a wee bit at nonunital Ek-algebra
started augmented A-infinity algebra
I added a clarifying clause to infinity-field so it now reads
The Morava K-theory A-∞ rings are essentially the only -fields. See at Morava K-theory – As infinity-Fields, where and we define as .
This is from Lurie’s lectures. What precisely does he mean? He says in lecture 24 that for any field that its E-M spectrum is an infinity-field, so the “essentially” is doing some work. Is the idea that all infinity-fields are -modules (cor 10, lecture 25), so the essentially cover things?
On another point, would there be a higher form of the rational/p-adic fracturing of , involving the ?
collected some introductions and surveys
started completion of a module, for the moment mainly so as to record a bunch of basic definitions and facts about completion of -modules from DAG12
I have given the notion of canonical transformation as used in Hamiltonian mechanics its own brief page.
So in particular I removed the redirect of that term to canonical morphism and instead added disambiguation lines on the top of both entries. I think this is justified: the term “canonical transformation” has been standard since ancient times in Hamiltonian mechanics and is in each and every textbook on the matter. On the other hand the same term as referring to canonical morphisms was mainly the proposal of one single person in category theory, and never caught up much, I think. (Also I find the term ill-motivated in category theory in the first place).
Therefore, while the disambiguation redirects ensure that both notions still can be found, I think it is clear that the default meaning must be that in Hamiltonian mechanics.
started a table-for-inclusion arithmetic cohesion – table and included it into relevant entries.
In the course of this I started a minimum at adic residual.
created a minimum at higher local field and higher arithmetic geometry and added disambiguation with “E-∞ arithmetic geometry”.
I have worked on the entry synthetic differential infinity-groupoid;
added a brief remark in the Idea section;
spelled out statement and proof that is totally -connected over ;
began some discussion on how the induced relative fundamental -groupoid functor is : the infinitesimal path -groupoid functor, such that is the de Rham space of and a morphism an -stack of D-modules on . But this deserves more discussion.
Concerning the writeup of the second point I had myself confused about the direction of one of the arrows for a while. Hope I got it right now.
Look at hausdorff locally convex space.
created an entry for torsion approximation, mainly to be able to refer to this concept by a link.
Unfortunately, I am lacking chocolat medals (as well as the authority to award them), but thanks to the author (presumably Todd) who graced Karoubi envelope with the proof that smooth manifolds result from open sets by idempotent splitting.
I have added a reference to Lawvere’s Perugia notes where this appeared as an exercise.
Entre parenthèses: it appears to me that it’d be better to have the proof at the page for smooth manifolds and to mention the result only at Karoubi envelope as I think this is kind of a butterfly at Karoubi though a beautiful one but an important result for manifolds.
polished and expanded somewhat the entry groupoid object in an (infinity,1)-category
created a bare minimum at Fermat quotient.
I created generalized uniform structures - table in the style of all of those tables that Urs makes and included it on most of the relevant pages. (I left the pages on the simplest concepts, the binary relations.)
I hope that the headers “monad on an object” and “monad on a pro-object” are accurate. These should be objects in an equipment, I think. Perhaps Mike can help me figure out what equipments are relevant here.
I have greatly expanded the basic definition at prometric space to show other ways to look at the concept.
Started adjoint lifting theorem. For now, it only includes a version for lifting left adjoints (I still haven’t read Johnstone’s 1975 paper for the case of right adjoints). I hope there is no substantial error in the appliaction for cocompleteness.
New entry movable singularity (for ODE’s with complex time) redirecting also fixed singularity.
started stub on Tannaka duality for geometric stacks, but need to interrupt now.
The theorem there can be read as justifying the point of view of derived noncommutative geometry to regard the 2-algebra as a valid replacement for the 1-algebra .
I edited group scheme and scheme a bit.
Fumbling around Jonsson-Tarski topos I’ve created entries on Thompson group and Higman’s theorem. Basically some links to further material. As Jónsson-Tarski seems to bring together a lot of stuff from different fields, I think we should have also some material on Tom’s work on self-similarity here.
I created ordinal subdivision to get rid of a grey link in subdivision, but it just gives a reference to the paper by Phil Ehlers and myself. I need to check at ordinal sum to see what was put there before continuing.
I have expanded on ETCC and introduced a section on ET2CC which I could occasionally fill with Mike’s ideas from MO in case all n-categorists are lying on the beach. I would also propose to replace the current ’idea’ section with what is right now called ’overview’ or some reworking of that.
I think it best to keep everything in a single entry given that these ideas on ETCC with or without 2 meet only a limited enthusiasm in the HOTT times.
I added a section on Lawvere’s definition to adjoint functor and also made an article for Functorial Semantics of Algebraic Theories.
created some bare minimum at mod p Whitehead theorem
created stub for
with just some references and
with just some pointers, cross-linked with
To be expanded…
I suppose we were lacking an entry on p-localization (?)
I’ve created a page for the Witt vectors. It seems that even with all that I wrote here (don’t worry I had a set of about 10 blog entries I wrote a few months ago that I just condensed, so I didn’t write this whole thing tonight) there are all sorts of things still missing here. The Witt functor is mention at Lambda-ring and there seems to be connections to the field with one element (?!). I just needed to refer to Witt vectors in the next few pages I want to make, so I decided this had to come first. Dieudonne module will need it and obviously Witt cohomology will need it.
added to étale topos some basics in the section Properties – Base change and sheaf cohomology
killed a spam page, now called spam
I felt like starting a table infinitesimal and local - table and included it into the relevant entries. So far it reads as follows:
| first order infinitesimal object | infinitesimal | formal = arbitrary order infinitesimal | local = stalkwise | finite | |||
|---|---|---|---|---|---|---|---|
| derivative | Taylor series | germ | function | ||||
| tangent vector | jet | germ of curve | curve | ||||
| Lie algebra | formal group | local Lie group | Lie group | ||||
| Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |
Can be further expanded, clearly.
The entry Borel-Weil theorem mentions extensions of the theorem to quantum groups, without however giving a reference. I just got an email asking for these.
The statement dates from August 19, 2009, due to Zoran.
created a bare minimum at branched cover of Riemann sphere, just to record the fact that every compact connected Riemann surface admits this structure.
I began adding proofs of Lemma 1-4 to the page transfinite construction of free algebras. The layout of the two array environment has to be fixed; proof of 3-4 to be added.
Any help/suggestion is extremely appreciated!
gave p-adic complex numbers an entry
Someone (anonymous) has created an empty page oon finite dimensional vector spaces.
created a minimum at global field
needed to point to restricted product, so I created a bare (and unsophisticated) minimum
I mostly wanted to record the correct meaning of this term. Then maybe later I can use this as a reference to fix Wikipedia (^_^). But there's a bit more here too.
gave Cartesian space a TOC and added some statements and references.
Edited biholomorphic function to follow the same format as diffeomorphism. In particular, this means that I qualified biholomorphic function to refer only to maps between complex manifolds. Is there a more general definition of holomorphic functions between complex analytic spaces?