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- Discussion Type
- discussion topicH-cohomology
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 1 day ago

I am running into the following simple question and am wondering if there is anything useful to be said.

Let

$\mathcal{A} \in dgcAlg_\mathbb{Q}$be a differential graded-commutative algebra in characteristic zero, whose underlying graded algebra is free graded-commutative on some graded vector space $V$:

$\mathcal{A} = (Sym(V), d) \,.$Consider an odd-graded element

$c \in \mathcal{A}_{odd} \,,$and write $(c)$ for the ideal it generates.

In this situation I’d like to determine whether it is true that

there is an inclusion $\mathcal{A}/(c) \hookrightarrow \mathcal{A}$;

for every element $\omega \in \mathcal{A}$ there is a decomposition

$\omega = \omega_0 + c \omega_1$for

*unique*$\omega_0, \omega_1 \in \mathcal{A}/(c) \hookrightarrow \mathcal{A}$.

For example if $c \neq 0 \in V_{odd} \hookrightarrow \mathcal{A}_{odd} \hookrightarrow \mathcal{A}$ is a generator, then these conditions are trivially true.

On the other extreme, if $c$ is the product of an odd number $\gt 1$ of odd generators, then it is not true. For example if $c = c_1 c_2 c_3$, with $c_1, c_2, c_3 \in V_{odd} \hookrightarrow \mathcal{A}_{odd}$, then for instance $c (1 + c_1) = c (1 + c_2) = c$ and so the coefficient $\omega_1$ is not unique.

Is there anything useful that one can say in general?

- Discussion Type
- discussion topicsupersymmetry breaking
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

added a few lines and original references to

*supersymmetry breaking*.

- Discussion Type
- discussion topicGruppenpest
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by David_Corfield
- Last Active 1 day ago

- Discussion Type
- discussion topicPoisson summation
- Category Latest Changes
- Started by Todd_Trimble
- Comments 4
- Last comment by Todd_Trimble
- Last Active 2 days ago

Took a stab at a general formulation of Poisson summation formula, although the class of functions to which it is supposed to apply wasn’t nailed down (yet).

(Some of the ingredients of Tate’s thesis are currently on my mind.)

- Discussion Type
- discussion topicsplitting principle
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 2 days ago

started something at

*splitting principle*(wanted to do more, but need to interrupt now)

- Discussion Type
- discussion topichigher gauge theory
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by David_Corfield
- Last Active 2 days ago

I have spent some minutes starting to put some actual expository content into the Idea-section on

*higher gauge theory*. Needs to be much expanded, still, but that’s it for the moment.

- Discussion Type
- discussion topicHiggs field
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 3 days ago

added a brief historical comment to

*Higgs field*and added the historical references

- Discussion Type
- discussion topicsurjective geometric morphism
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by IngoBlechschmidt
- Last Active 4 days ago

at surjective geometric morphism I have spelled out in detail most of the proof of the various equivalent characterizations, and all of the proof of the statement that geometric surjections are comonadic.

- Discussion Type
- discussion topic5d supergravity
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 4 days ago

I have added the Fierz identities that give the $S^2$-valued supercocycle in 5d here.

Added this briefly also at

*Fierz identity*: here

- Discussion Type
- discussion topicorthogonal factorization system
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Mike Shulman
- Last Active 4 days ago

I have added to

*orthogonal factorization system*in the Definition-section three equivalent explicit formulations of the definition;

in the Properties-section the statement of the cancellability property.

Wanted to add more (and to add the proofs). But have to quit now. Maybe later.

- Discussion Type
- discussion topiclattice (disambiguation)
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Todd_Trimble
- Last Active 4 days ago

This was long overdue: I have created a page

*lattice (disambiguation)*and added corresponding warnings on terminology to the relevant entries.In the course of this I have created stubs for

*lattice in a vector space*,*integral lattice*and*modular integral lattice*.

- Discussion Type
- discussion topicball
- Category Latest Changes
- Started by Urs
- Comments 52
- Last comment by DavidRoberts
- Last Active 5 days ago

created ball

- Discussion Type
- discussion topicformal group
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by DavidRoberts
- Last Active 7 days ago

I have touched formal group a bit, but don’t have time to do anything substantial.

I need to adjust some of the terminology that I had been setting up at cohesive (infinity,1)-topos related to

*infinitesimal cohesion*: the abstract notion currently called “$\infty$-Lie algebroid” there should be called “formal cohesive $\infty$-groupoid”. The actual L-infinity algebroids are (just) the first order formal smooth $\infty$-groupoids.While on the train I started expanding some other entries on this point, but I need to quit now and continue after a little interruption.

- Discussion Type
- discussion topicworldline formalism
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Feb 10th 2018

created

*worldline formalism*to go with this Physics.SE answer

- Discussion Type
- discussion topicposet-valued sets
- Category Latest Changes
- Started by Mike Shulman
- Comments 5
- Last comment by Mike Shulman
- Last Active Feb 10th 2018

Created a stubby poset-valued set.

- Discussion Type
- discussion topicpower series / Taylor series
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Todd_Trimble
- Last Active Feb 9th 2018

wanted to record Borel’s theorem on Taylor series expansion, so created stubs for power series and Taylor series

- Discussion Type
- discussion topiccosmological constant
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Feb 9th 2018

I have added a comment and collected some references on the renormalization freedom in the cosmological constant: here

I have cross-linked this with related entries:

*renormalization*,*perturbative quantum gravity*and*stress-energy tensor*

- Discussion Type
- discussion topicS-matrix
- Category Latest Changes
- Started by Urs
- Comments 48
- Last comment by Urs
- Last Active Feb 9th 2018

added to

*S-matrix*a useful historical comment by Ron Maimon (see there for citation)

- Discussion Type
- discussion topicconditional expectation
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 8
- Last comment by Urs
- Last Active Feb 8th 2018

- The observation that the conditional expectation enjoys a universal property inspired me to write some ''random text''.

- Discussion Type
- discussion topicpushforward measure
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 8th 2018

In order to un-gray a link at

*conditional expectation*I created a minimum for*pushforward measure*.

- Discussion Type
- discussion topicA First Idea of Quantum Fields
- Category Latest Changes
- Started by Urs
- Comments 82
- Last comment by Urs
- Last Active Feb 8th 2018

At

*field (physics)*I am beginning to write an actual introduction to the topic, now in a new section titled “A first idea of quantum fields”.This means to introduce the concept with precise detail, but in a simple context (trivial and bosonic field bundles over Minkowski spacetime, perturbatively quantized) that allows to get a quick idea of the idea of the concept of (quantum) fields as such, without being distracted by other details.

So far I made it up to the derivation of the EOMs. Discussion of (deformation) quantization is to follow (maybe by tonight, depending on how much trouble I have with the trains) and I plan to sprinkle in the detailed example from

*scalar field*in parallel with the abstract discussion.

- Discussion Type
- discussion topicKLT relations
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 7th 2018

at

*KLT relations*I have expanded the list of references. I added also references for the generalization of these relations that is known these days as “gravity is Yang-Mills squared” or similar (eventually this might want to be a separate entry).In this course I also expanded the list of references at quantum gravity – As a perturbative quantum field theory

- Discussion Type
- discussion topicrenormalization
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Feb 7th 2018

at

*renormalization*I made the Definition-section have three different subsections, a new one of which is now*Definition – Of theories in BV-CS form*on Kevin Costello’s apprach. On the other hand, so far this contains mostly just a pointer to his article.

- Discussion Type
- discussion topicquantum probability
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by David_Corfield
- Last Active Feb 7th 2018

I started a bare minimum at

*quantum probability*(redirecting*noncommutative probability space*etc.)Some entries have long been secretly referencing such an entry, and I have cross-linked accordingly, for instance from

*von Neumann algebra*and*quantum computing*.I had the feeling somewhere we already had a detailed account of probability theory dually in terms of von NNeumann algebras, but if we do I didn’t find it(?)

- Discussion Type
- discussion topicHowTo
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Feb 7th 2018

Have added to HowTo a description for how to label equations

In the course of this I restructured the section “How to make links to subsections of a page” by giving it a few descriptively-titled subsections.

- Discussion Type
- discussion topicwave function collapse
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 6th 2018

I have added to

*wave function collapse*its relation to the expression for conditional expectation values in quantum probability: here (e.g. Kuperberg 05, section 1.2, Yuan 12)

- Discussion Type
- discussion topicanomalous magnetic moment
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 6th 2018

there is an old article (Berends-Gastman 75) that computes the 1-loop corrections due to perturbative quantum gravity to the anomalous magnetic moment of the electron and the muon. The result turns out to be independent of the choice of (“re”-)normalization (hence what they call “finite”).

I have added a remark on this in the $(g-2)$-entry here and also at

*quantum gravity*here.

- Discussion Type
- discussion topicdifferential renormalization
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Feb 5th 2018

started something at

*differential renormalization*. More later

- Discussion Type
- discussion topicBraided monoidal category
- Category Latest Changes
- Started by John Baez
- Comments 9
- Last comment by Richard Williamson
- Last Active Feb 3rd 2018

- The definition of braided monoidal category was wrong or at least nonstandard, because it left out one of the hexagon axioms and included a 'compatibility with the unit object' law which follows from the usual definition.

I changed it to the usual definition.

If the nonstandard definition is equivalent to the usual one, I'd love to know why! But I don't see how you get two hexagons from one, even given compatibility with the unit object.

(Of course for a*symmetric*monoidal category we just need one hexagon.)

I also beefed up the definition at symmetric monoidal category so the poor reader doesn't need to run back to braided monoidal, then monoidal.

- Discussion Type
- discussion topicNice objects/nice category
- Category Latest Changes
- Started by DavidRoberts
- Comments 4
- Last comment by zskoda
- Last Active Feb 3rd 2018

At dichotomy between nice objects and nice categories I added a quote from Deligne about allowing awful schemes gave a nice category of schemes. I can’t find the page I was thinking of where this dichotomy is also mentioned along with attribution of the general idea to Grothendieck. I wanted to add it there as well.