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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 12th 2018
   • (edited Feb 14th 2018)
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   Author: Urs
   Format: MarkdownItexI 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 1. there is an inclusion $\mathcal{A}/(c) \hookrightarrow \mathcal{A}$; 1. 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?

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

   Let

   be a differential graded-commutative algebra in characteristic zero, whose underlying graded algebra is free graded-commutative on some graded vector space :

   Consider an odd-graded element

   and write for the ideal it generates.

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

   1. there is an inclusion ;

   2. for every element there is a decomposition

      for unique .

   For example if is a generator, then these conditions are trivially true.

   On the other extreme, if is the product of an odd number of odd generators, then it is not true. For example if , with , then for instance and so the coefficient is not unique.

   Is there anything useful that one can say in general?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 13th 2018
   • (edited Feb 13th 2018)
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   Author: Urs
   Format: MarkdownItexSo an equivalent way to ask this is: given an odd element $c$ in a semifree dgc-algebra $\mathcal{A}$ in characteristic 0, with $deg(c) \geq 3$, what are conditions that the sequence of graded modules $$ \cdots \to \mathcal{A} \overset{c \cdot (-)}{\longrightarrow} \mathcal{A} \overset{c \cdot (-)}{\longrightarrow} \mathcal{A} \to \cdots $$ is exact? Is it sufficient that $c$ is not decomposable as $c = \alpha \wedge \underset{deg = 1}{\underbrace{\beta}}$?

   So an equivalent way to ask this is:

   given an odd element in a semifree dgc-algebra in characteristic 0, with , what are conditions that the sequence of graded modules

   is exact?

   Is it sufficient that is not decomposable as ?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 14th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexHisham kindly points out to me that in the case that the element $c$ in #2 is closed, the cohomology which I am asking about is sometimes called _H-cohomology_, e.g. p. 19 of * Gil R. Cavalcanti, "New aspects of the ddc-lemma" ([arXiv:math/0501406](https://arxiv.org/abs/math/0501406)) Good to have a name to attach to it, that might make talking about it easier. On the other hand, I do need it for $c$ not closed...

   Hisham kindly points out to me that in the case that the element in #2 is closed, the cohomology which I am asking about is sometimes called H-cohomology, e.g. p. 19 of

   • Gil R. Cavalcanti, “New aspects of the ddc-lemma” (arXiv:math/0501406)

   Good to have a name to attach to it, that might make talking about it easier. On the other hand, I do need it for not closed…

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeFeb 14th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexSo I am starting something at _[[H-cohomology]]_.

   So I am starting something at H-cohomology.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 17th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have sent the question to MO, [here](https://mathoverflow.net/questions/293219/vanishing-of-h-cohomology).

   I have sent the question to MO, here.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeFeb 20th 2018
   • (edited Feb 20th 2018)
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   Author: Urs
   Format: MarkdownItexThere is a simple argument in [Severa 05, p.1](https://ncatlab.org/nlab/show/H-cohomology#Severa05) (have added the reference) for the H-cohomology of graded symplectic forms. This should generalize to the case that I need by the double complex spectral sequence. But not tonight...

   There is a simple argument in Severa 05, p.1 (have added the reference) for the H-cohomology of graded symplectic forms. This should generalize to the case that I need by the double complex spectral sequence. But not tonight…

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeFeb 21st 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexSo I am getting the hang of it now. Am writing out computations in the Examples-section [here](https://ncatlab.org/nlab/show/H-cohomology#SupergravityCFieldAfterSplitting).

   So I am getting the hang of it now. Am writing out computations in the Examples-section here.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeFeb 22nd 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have now been computing (vanishing of) H-cohomology for $$ H \coloneqq B^a{}_b \wedge B^b{}_c \wedge B^c{}_a $$ in the $Spin$-invariant part of a free graded-commutative algebra where the generators $(B^{a b} = - B^{b a})$ span the bivector representation. I am trying to argue that the corresponding H-cohomology vanishes at least on the subspace of those invariant elements that don't contain contractions of the $B$s with themselves. This is now [here](https://ncatlab.org/nlab/show/H-cohomology#TheSecondSummand), but needs scrutinization.

   I have now been computing (vanishing of) H-cohomology for

   in the -invariant part of a free graded-commutative algebra where the generators span the bivector representation.

   I am trying to argue that the corresponding H-cohomology vanishes at least on the subspace of those invariant elements that don’t contain contractions of the s with themselves.

   This is now here, but needs scrutinization.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeFeb 22nd 2018
   • (edited Feb 22nd 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexKevin Sackel on MO offers a proof ([here](https://mathoverflow.net/a/293565/381)) that in fact the H-cohomology of finite sums of decomposables never vanishes. I have added that statement to the entry [here](https://ncatlab.org/nlab/show/H-cohomology#HCohomologyNeverVanishes).

   Kevin Sackel on MO offers a proof (here) that in fact the H-cohomology of finite sums of decomposables never vanishes. I have added that statement to the entry here.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 23rd 2018
    • (edited Feb 23rd 2018)
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    Author: Urs
    Format: MarkdownItexI have also taken over the proof (now [here](https://ncatlab.org/nlab/show/H-cohomology#HCohomologyNeverVanishes), with a little bit of reformatting). Pretty neat. So if an element of odd degree in an $\mathbb{N}$-graded-commutative algebra is "weakly decomposable" in that it is in the ideal generated by the degree-1 elements, then its H-cohomology is non-trivial.

    I have also taken over the proof (now here, with a little bit of reformatting). Pretty neat.

    So if an element of odd degree in an -graded-commutative algebra is “weakly decomposable” in that it is in the ideal generated by the degree-1 elements, then its H-cohomology is non-trivial.