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What is the earliest possible reference for definition and basic properties of Clifford algebras associated to quadratic modules over a ringed space? The ringed space does not need to be locally ringed, but I am willing to assume that $2$ is invertible in the sheaf of rings if that helps. Searching mathematical reviews revealed a somewhat related paper:

  • B. Auslander. The Brauer group of a ringed space. J. Alg. 4 (1966), 220-273.

This paper develops some of the relevant math, but does not talk about the Clifford algebra. I would expect that there is some paper from around the same period defining the Clifford algebra for quadratic modules over a ringed space, but I could not find anything.

I would also be interested if there are papers writing down the Clifford invariant mapping from the Witt group to the Brauer group in the general setting of ringed spaces.

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  • $\begingroup$ You're asking for the earliest possible reference. Is there any? (This is a serious question, I'd really like to see a reasonable construction. The scheme-version doesn't generalise immediately, I believe.) $\endgroup$ – Ben Aug 29 '14 at 6:17
  • $\begingroup$ @BenA. This is mostly the motivation for asking the question. I was convinced that there should be something from around a couple of years after EGA, exactly because I found this paper of Auslander where the Brauer group is discussed in full generality of ringed spaces... $\endgroup$ – Matthias Wendt Aug 29 '14 at 8:36
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    $\begingroup$ If you want to try, the definition in users.math.yale.edu/~auel/papers/docs/ABB_quadrics.pdf, (sorry, somehow the link is screwed) section 1.5 is independent of scheme language and the constructed sheaves fulfil universal properties. This is why I'm sure it will work out well in the case you start with a sufficiently nice module. For example, I expect that the construction behaves well with the "locally projective modules of finite type" of Auslander. $\endgroup$ – Ben Sep 4 '14 at 8:34
  • $\begingroup$ @BenA. Right, my initial feeling was that the universal properties plus the stuff in Auslander's paper should be mostly enough to get construction and basic properties of Clifford algebras, so I asked the question. I am surprised that there seem to be no papers from the seventies setting up the basics for Clifford algebras... Still, a lot of interesting references turned up in the answers. $\endgroup$ – Matthias Wendt Sep 4 '14 at 18:32
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Maybe Asher Auel's thesis? But maybe it's still schemes, not sure! (but that thesis is still awesome)

http://users.math.yale.edu/~auel/papers/docs/AUEL_thesisSS.pdf

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    $\begingroup$ There is an even more detailed note on the history of quadratic forms and clifford algebras on schemes in http://users.math.yale.edu/~auel/papers/docs/AUEL_lineclifford.pdf. It also sketches all the aforementioned different approaches. $\endgroup$ – Ben Aug 29 '14 at 6:07
  • $\begingroup$ The references to Auel's work are very good. The reference provided by @BenA. also contains a lot of further references to earlier work, maybe I can find something following these pointers... $\endgroup$ – Matthias Wendt Aug 29 '14 at 8:38
  • $\begingroup$ @MatthiasWendt: why exactly do you care about it being a general ringed space? Is it because you want to generalize to algebraic spaces/stacks, or to satisfy some other curiosity? If it's for general curiosity then maybe you should develop a theory of Clifford algebras for any ringed topos! (actually, now that I think of it, I think I've seen people define Clifford algebras in MONOIDAL categories, that might be some more googling for you... :) $\endgroup$ – bananastack Aug 29 '14 at 13:03
  • $\begingroup$ @user125763: Concerning your remark about more general settings, I found a preprint of Vezzosi on Clifford algebras in derived algebraic geometry: dma.unifi.it/~vezzosi/papers/derivedClifford.pdf. $\endgroup$ – Matthias Wendt Aug 29 '14 at 13:51
  • $\begingroup$ @MatthiasWendt: nice. $\endgroup$ – bananastack Aug 29 '14 at 15:00
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You will probably find the article "Quadratic Forms with Values in Line Bundles" (by Bichsel and Knus) rather useful. Here is the link:

http://www.math.ethz.ch/~knus/papers/bk.pdf

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  • $\begingroup$ Thanks for the reference. However, it seems that most of the paper is about the situation where the ringed space is actually a scheme. What about the general case? $\endgroup$ – Matthias Wendt Jul 28 '14 at 14:02

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