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While studying vector bundle valued differential forms, $\Omega^{\bullet}(M, E)$, or $\Omega^{\bullet}(M, \mathrm{End}(E))$ if that helps this discussion, I've come across some work in Azumaya algebras. Thinking of $\Omega$ as an $R$-module, taking values in a bundle, and reading about how Azumaya algebras can be thought of locally being a matrix algebra, in the right context, it seems there should be a connection between $\Omega$ and Azumaya algebras. Can anyone point me in the right direction or tell me why this doesn't work?

Question: Can we say that $\Omega^{\bullet}(M,E)$ is an Azumaya algebra?

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    $\begingroup$ Side remark. In characteristic $p$>0, the pushforward of the sheaf of (crystalline) differential operators on $X$ to its Frobenius twist $X^{(p)}$, has a natural Azumaya algebra structure over the structure sheaf of the cotangent bundle of $X^{(p)}$. $\endgroup$
    – Fang Hung-chien
    Commented Aug 15, 2017 at 14:02
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    $\begingroup$ You may enjoy Theorem 9.2.4 in "Central Simple Algebras and Galois Cohomology" by Philippe Gille, Tamás Szamuely, which classifies $p$-Torsion Brauer classes (in characteristic $p$!) in terms of differential forms. IIRC this theorem is originally due to Kato. I don't know a related result in characteristic zero, which is what you seem to be looking for. $\endgroup$
    – Daniel Litt
    Commented Aug 15, 2017 at 18:24
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    $\begingroup$ That theorem is due to Kato, and it is discussed in Serre's "Galois cohomology". Serre discusses the difference between fields that are of "cohomological dimension $\leq 1$" and fields that are of "dimension $\leq 1$". The second notion is a strengthening of the first notion intended to incorporate the $p$-torsion of the Brauer group. I believe Gille is writing a book about the corresponding modification of 'etale cohomology (for which "cohomological dimension $\leq 1$" is Serre's notion of "dimension $\leq 1$"). $\endgroup$
    – Jason Starr
    Commented Aug 16, 2017 at 21:36
  • $\begingroup$ About your edited question: for which algebra structure??? $\endgroup$
    – abx
    Commented Aug 18, 2017 at 14:10

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OK, I made a real hash of this the first time, so let me straighten this all out: Azumaya algebras (at least as I understand them) are algebras which are locally isomorphic to $\mathrm{End}(E)$ for $E$ a vector bundle. Note, I'm being a little vague here, since there are many contexts, many topologies, etc. where one might want to do this. There's a general yoga for understanding such algebras: there's a coordinate atlas where they are trivial, so all you need to say is the trivialization on each patch, and what the transition functions are. This is an element of Čech 1-cocycles for this cover for the sheaf $\mathrm{End}(E)^*$; you can easily work out that the isomorphic Azumaya algebras are those that come from 2-cocyles via the boundary map (you conjugate the transitions by the 2-cocycle).

So, the first sheaf homology of $\mathrm{End}(E)^*$ controls the Azumaya algebras; that's all I was trying to say; since it's a multiplicative sheaf, I think interpreting it as deRham cohomology will be trickier than I first imagined.

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    $\begingroup$ The sheaf has to be equipped with a connection, and the choice of connection matters. $\endgroup$
    – David E Speyer
    Commented Aug 15, 2017 at 14:10
  • $\begingroup$ Hmm, you're right of course. Though presumably there is some way of fixing this even for bundles without projectively flat connections. $\endgroup$
    – Ben Webster ♦
    Commented Aug 15, 2017 at 15:47
  • $\begingroup$ @BenWebster Is there some well-behaved deformation theory for curved algebras with fixed curvature? $\endgroup$
    – mme
    Commented Aug 15, 2017 at 16:04
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    $\begingroup$ Does $\text{End}(E)^*$ mean units in $\text{End}(E)$? If so I think this still isn't right; instead one wants $1$-cocycles for $PGL(E)$ (since the center of $\text{End}(E)$ acts trivially on $\text{End}(E)$ by conjugation). Then the associated Brauer class comes from the exact sequence $0\to O^*\to GL(E)\to PGL(E)\to 0$. $\endgroup$
    – Daniel Litt
    Commented Aug 15, 2017 at 18:14
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    $\begingroup$ If one wants some kind of connection, presumably one should look at Cech $H^1$ of the complex $PGL(E)\to \Omega^1(\text{End}(E))/\Omega^1$ where the arrow is given by dlog; for flat connections, one probably wants $PGL(E)\to\Omega^1(\text{End}(E))/\Omega^1\to \Omega^2(\text{End}(E))/\Omega^2$... $\endgroup$
    – Daniel Litt
    Commented Aug 15, 2017 at 18:31

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