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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2018
    • (edited Feb 14th 2018)

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

    Let

    𝒜dgcAlg \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 VV:

    𝒜=(Sym(V),d). \mathcal{A} = (Sym(V), d) \,.

    Consider an odd-graded element

    c𝒜 odd, c \in \mathcal{A}_{odd} \,,

    and write (c)(c) for the ideal it generates.

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

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

    2. for every element ω𝒜\omega \in \mathcal{A} there is a decomposition

      ω=ω 0+cω 1 \omega = \omega_0 + c \omega_1

      for unique ω 0,ω 1𝒜/(c)𝒜\omega_0, \omega_1 \in \mathcal{A}/(c) \hookrightarrow \mathcal{A}.

    For example if c0V odd𝒜 odd𝒜c \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 cc is the product of an odd number >1\gt 1 of odd generators, then it is not true. For example if c=c 1c 2c 3c = c_1 c_2 c_3, with c 1,c 2,c 3V odd𝒜 oddc_1, c_2, c_3 \in V_{odd} \hookrightarrow \mathcal{A}_{odd}, then for instance c(1+c 1)=c(1+c 2)=cc (1 + c_1) = c (1 + c_2) = c and so the coefficient ω 1\omega_1 is not unique.

    Is there anything useful that one can say in general?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 13th 2018
    • (edited Feb 13th 2018)

    So an equivalent way to ask this is:

    given an odd element cc in a semifree dgc-algebra 𝒜\mathcal{A} in characteristic 0, with deg(c)3deg(c) \geq 3, what are conditions that the sequence of graded modules

    𝒜c()𝒜c()𝒜 \cdots \to \mathcal{A} \overset{c \cdot (-)}{\longrightarrow} \mathcal{A} \overset{c \cdot (-)}{\longrightarrow} \mathcal{A} \to \cdots

    is exact?

    Is it sufficient that cc is not decomposable as c=αβdeg=1c = \alpha \wedge \underset{deg = 1}{\underbrace{\beta}}?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 14th 2018

    Hisham kindly points out to me that in the case that the element cc 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 cc not closed…

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeFeb 14th 2018

    So I am starting something at H-cohomology.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 17th 2018

    I have sent the question to MO, here.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeFeb 20th 2018
    • (edited Feb 20th 2018)

    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…

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 21st 2018

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

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 22nd 2018

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

    HB a bB b cB c a H \coloneqq B^a{}_b \wedge B^b{}_c \wedge B^c{}_a

    in the SpinSpin-invariant part of a free graded-commutative algebra where the generators (B ab=B ba)(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 BBs with themselves.

    This is now here, but needs scrutinization.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 22nd 2018
    • (edited Feb 22nd 2018)

    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.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 23rd 2018
    • (edited Feb 23rd 2018)

    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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