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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?
So 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 $c = \alpha \wedge \underset{deg = 1}{\underbrace{\beta}}$?
Hisham 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
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…
So I am starting something at H-cohomology.
I have sent the question to MO, here.
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…
So I am getting the hang of it now. Am writing out computations in the Examples-section here.
I 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, but needs scrutinization.
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 $\mathbb{N}$-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.
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