Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: H-cohomology

Bottom of Page

1 to 10 of 10

  1.  
    • 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

    𝒜∈dgcAlg ℚ \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

    c∈𝒜 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 ω∈𝒜\omega \in \mathcal{A} there is a decomposition

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

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

    For example if c≠0∈V odd↪𝒜 odd↪𝒜c \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 3∈V odd↪𝒜 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

    ⋯→𝒜⟶c⋅(−)𝒜⟶c⋅(−)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=α∧β⏟deg=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

    H≔B a b∧B b c∧B 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 ℕ\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.

1 to 10 of 10

  1.