9
$\begingroup$

If $G$ is an algebraic group, then one can show $\operatorname{H}^*(BG,k)$ and $\operatorname{H}_*(G,k)$ are Koszul dual dg algebras, e.g. Drinfeld and Gaitsgory, "Finiteness questions for algebraic stacks", (7.2). Basically you write $G$ as a pullback of two copies of the trivial map $\mathrm{pt}\to BG$ and use sheaf theory and base change. For this the group should be locally of finite type and characteristic zero.

E.g. the cohomology of $B\mathbf{G}_m$ and $\mathbf{G}_m$ are freely generated in degrees $2$ and $1$, respectively.

Is there a concise proof of this fact in topology (where I think it was first discovered)? Something that is different from the sheaf theory argument. For instance, does it follow more or less formally from writing $\operatorname{H}^*(BG,\mathbf{Q})=\operatorname{Maps}(BG,H\mathbf{Q})$?

Second, for what sort of topological groups is this theorem true? There are lots of topological spaces not coming from the above sorts of algebraic groups, e.g. mapping spaces $\operatorname{Maps}(X,G)$ for $X$ a topological space.

Improve this question
$\endgroup$
7
  • 3
    $\begingroup$ One useful reference on this is "Equivariant cohomology, Koszul duality, and the localization theorem" by Goresky-Kottwitz-Macpherson. They seem to say this was first written down by Ginzburg in "Equivariant cohomologies and Kähler's geometry". I don't think their perspective is especially different from the sheaf theory argument though, though maybe phrased as the spectral sequence in homology for the map EG -> BG. $\endgroup$
    – Ben Webster ♦
    Commented Feb 11, 2023 at 23:32
  • 4
    $\begingroup$ In topology, this basically boils down to the observation that the derived tensor product $k \otimes_{C_*(G; k)} k$ can be identified with $C_*(BG; k)$, so its $k$-linear dual $\mathrm{End}_{C_*(G: k)}(k) = C^*(BG; k)$. This works even if $k$ is not a field. $\endgroup$
    – skd
    Commented Feb 11, 2023 at 23:49
  • 1
    $\begingroup$ This does have some hypotheses (like connectivity) - eg it’s false if G is finite and k has characteristic prime to |G| $\endgroup$
    – David Ben-Zvi
    Commented Feb 12, 2023 at 0:37
  • 1
    $\begingroup$ It’s also an “affineness” result - it says the trivial rep of G (constant sheaf on BG) generates $\endgroup$
    – David Ben-Zvi
    Commented Feb 12, 2023 at 0:49
  • 3
    $\begingroup$ That "sheaf theory" you're referring to is usually called Eilenberg-Moore spectral sequence for cohomology of a fiber, which degenerates for stupid reasons in your case. $C_*(\Omega X)$ and $C^*(X)$ are always Koszul dual. Original paper by Adams ("On the cobar construction", 1956) is quite readable, decent modern reference is arxiv.org/abs/1910.08455. After that you notice that both $G$ and $BG$ are formal in case where $G$ is an algebraic group, so duality remains after passing to (co)homology. I see little reason for that to be true in case where either $G$ or $BG$ are not formal. $\endgroup$
    – Denis T
    Commented Feb 12, 2023 at 1:48

1 Answer 1

Reset to default
3
$\begingroup$

I am posting this as an answer because it is a bit long for a comment. Questions close to this one have appeared before on MathOverflow. You ask specifically about Koszul duality, but for $k=\mathbb{Q}$, Koszul duality is a consequence of the more precise statement that $H_*(G;\mathbb{Q})$ with its Pontrjagin product is a free exterior $\mathbb{Q}$-algebra on finitely many generators in odd degrees, and the associated free symmetric $\mathbb{Q}$-algebra on those generators shifted in degree by one is $H^*(BG;\mathbb{Q})$.

This statement goes back to Borel and to Chern, with other important work by Bott in his "Bott spectral sequence."

Armand Borel
Topology of Lie groups and characteristic classes
Bull. Amer. Math. Soc. 61 (1955), 397-432
https://www.ams.org/journals/bull/1955-61-05/S0002-9904-1955-09936-1
Theorem (A) on p. 410.

As you indicate, one formulation of the result follows from the statement that the Leray spectral sequence of $EG\to BG$ has a special form, with each of the generators transgressive. This is what Chern addresses with his Chern–Weil theory. The approach of Becker–Gottlieb to the Adams conjecture also proves this result (the Adams conjecture was originally proved by Dennis Sullivan, from whom I learned everything in this "comment / answer"). There is also a very short proof in the following book of Félix, Oprea and Tanré.  It is "Proof 2" on pp. 39 – 40 of Algebraic models in geometry.

Improve this answer
$\endgroup$
2
  • 1
    $\begingroup$ More canonically the cohomology of BG and cohomology of G relate by this transgression - to identify these generators with those of the homology of G requires an invariant form (the former are functions on shifted forms of the Cartan, the latter on the dual Cartan) $\endgroup$
    – David Ben-Zvi
    Commented Feb 12, 2023 at 14:20
  • 1
    $\begingroup$ [Weyl invariant functions in all three cases] $\endgroup$
    – David Ben-Zvi
    Commented Feb 12, 2023 at 14:36

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .