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In Grothendieck's Brauer group papers, he uses deformation theory to bootstrap the theory of central simple algebras over a field to the theory of Azumaya algebras over rings (and schemes). I am surprised to be unable to find any parallel discussion of octonion algebras in the literature, and wonder if someone may know a reference. The lack of associativity is what seems to make it hard to adapt Grothendieck's calculations.

Is it known (better: proved somewhere citable?) that any two octonion algebras over an artin local ring with algebraically closed residue field (of any characteristic) are necessarily isomorphic?

The non-associativity is very disorienting to me for dealing with the cohomological obstruction. I'd prefer not to reinvent a useful non-associative wheel.


Some remarks to avoid confusion:

Remark 1: An octonion algebra over a scheme $S$ is a rank-8 non-degenerate quadratic space $(V,q)$ equipped with a (not necessarily associative) $O_S$-algebra structure admitting 2-sided identity $e \in V(S)$ such that $q(xy) = q(x)q(y)$ (which forces $q(e)=1$). It can be deduced from this that for the perfect bilinear $B_q(x,y) := q(x+y) - q(x)-q(y)$ and the linear form $t = B_q(\cdot,e)$, the operation $x \mapsto x^{\ast} := t(x)-x$ is an anti-involution of the algebra, $x x^{\ast} = x^{\ast} x = q(x)$, $B_q(x,y) = t(x y^{\ast})$, and every algebra automorphism automatically preserves $q$ (hence also $t$ and the conjugation).

Remark 2: There is a recent paper developing a low-degree cohomology/obstruction theory for left alternative algebras (which has a hypothesis of alg. closed ground field of char. 0 that is never used in the main proofs). In that theory, vanishing of ${\rm{H}}^2$ is equivalent to triviality of all deformations as an alternative algebra, but it isn't clear that this framework should be helpful adequate for dealing with octonion algebras. It seems unlikely, since in the same paper it is shown that the "left alternative" ${\rm{H}}^2$ for a $2 \times 2$ matrix algebra is nonzero, so that matrix algebra has nontrivial deformations as an alternative algebra (which are irrelevant if one cares about deforming it as an associative algebra).

Remark 3: By computing derivations over a field, we know that the automorphism scheme of an octonion algebra over a field is smooth, even connected semisimple of type ${\rm{G}}_2$. By a trick with Dedekind domains I know that the automorphism scheme of an octonion algebra over a Dedekind domain is actually flat, so (by the theory over fields) it is a semisimple group scheme of type ${\rm{G}}_2$. I do not know what happens for a more general base ring.

The question above is equivalent to asking for smoothness of the Isom-scheme between a pair of octonion algebras over a general base (and it is equivalent to flatness as well, without which it isn't clear how to show the Isom-scheme is a torsor for the automorphism scheme of either of the given octonion algebras). It is also equivalent to asking that every octonion algebra becomes "split" etale-locally on the base, which is a classical fact for fields but I have no idea if it is true over the dual numbers over a field.

For example, in lieu of knowing a positive answer to my question, it isn't clear (to me) that the study of octonion algebras is completely controlled by the study of ${\rm{G}}_2$-bundles.

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Why not provide an actual link to the paper in the arXiv? –  Mariano Suárez-Alvarez Jul 5 '13 at 6:17
    
@Mariano: (i) because I don't know how, and (ii) to compensate for (i) it seems entirely sufficient that I gave the title in full so that anyone who wishes can find it (or just type "octonion algebra", "deformation theory" into Google, which is how I found it). If you'd like to edit in the link, please feel free to do so. –  user61789 Jul 5 '13 at 6:39
    
That preprint has become a paper, removing the Left from the title: ashdin.com/journals/jglta/2011/G110105.pdf –  v08ltu Jul 5 '13 at 6:40
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well, I am of a tradition which values making things easier for the reader. –  Mariano Suárez-Alvarez Jul 5 '13 at 6:43
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Dear Daniel: Before I posed the question I was well aware that constructing a suitable cohomology theory is a natural approach and would do the job -- that is the entire reason that I wrote Remark 2. I don't think we're getting anywhere with this discussion (as it isn't telling me anything I didn't know before I posted the question). Regards, –  user61789 Jul 5 '13 at 14:48

1 Answer 1

up vote 5 down vote accepted

I can answer my own question, upon finding the 1959 paper "The arithmetics of octaves and the group ${\rm{G}}_2$" by van der Blij and Springer in the library this morning (pp. 406-418 in volume 21 of Indag. Math.). They work over a field, but their style of calculation on pages 407-412 (ignoring everything about "radical", and ignoring the top half of page 410, up to and including 2.3(i)) can be adapted to work over any local ring. The upshot is that one can prove the following theorem:

Theorem: Let $A$ be an octonion algebra over a local ring $R$, with quadratic norm form $q:A \rightarrow R$. If the non-degenerate quadratic space $(A,q)$ is split then $(A,q)$ is a "split" octonion algebra (i.e., isomorphic to a certain explicit construction built from the vector cross product on $R^3$ and the determinant on $3 \times 3$ matrices).

Since we can split any non-degenerate quadratic space of even rank over a ring by passing to an etale cover (OK for odd rank if 2 is a unit, but we need rank 8), it follows from the above Theorem (and "spreading out" from local rings) that any octonion algebra over a ring becomes split over an etale cover.

Thus, any two octonion algebras over a scheme become isomorphic etale-locally over the base, and the automorphism scheme of any such algebra is flat (hence even semisimple of type ${\rm{G}}_2$) due to comparison with the split octonion algebra that begins life over the Dedekind base Spec($\mathbf{Z}$). Hence, isomorphism classes of octonion algebras over a scheme $S$ are classified by isomorphism classes of ${\rm{G}}_2$-torsors over $S$.

So that affirmatively answers everything I was wondering about, and happily without any need to develop an octonionic cohomological deformation theory. And a posteriori, we see it is the same as the deformation theory of ${\rm{G}}_2$-torsors.

It is impractical to get into the gritty details of the argument over local rings here, but it is relatively straightforward if one reads the first two sections of the paper of van der Blij and Springer.


For anyone who wants to look at the above paper and carry out the same exercise I did of adapting their arguments to work over a local ring, let me make some additional remarks, using notation as in that paper.

  1. To fill in various details that van der Blij and Springer omit (over a field), one has to bring in the two additional Moufang identities which they don't discuss (but which can be proved exactly as in the theory over a field; see the proof of Prop. 1.4.1 in the book of Springer and Veldkamp) in order to see that the associator is alternating. This underlies their computation of $x_0(x_1x_2)$ and $(x_1x_2)y_1$ on page 411, and at the bottom of page 411 their formula for the multiplication law in the split octonions should have the minus signs replaced with plus signs (but leave the sign in their formula for the norm form $Q$). [EDIT: actually, all of the necessary identities are explained in the early sections of the Springer--Veldkamp book on Octonions and Jordan algebras.]

  2. The role of being over a local ring is that at certain steps one can use unit constructions over the residue field to check certain constructions in the local ring are units, and more importantly one can carry out their direct sum decompositions of the octonion algebra on the top of page 411 (denoted as $F \oplus G$ in their paper) without having bases, so one gets the existence of the required bases of the modules $F, G, F_0, G_0$ they build due to the fact that a direct summand of a finite free module over a local ring is itself finite free.

  3. In section 1.8 of the book of Springer and Veldkamp there is a very short proof over fields that an octonion algebra with isotropic norm form is split. That proof is too field-specific, and does not seem to adapt to working over local rings. It is the proof in the paper of van der Blij and Springer that is sufficiently robust that it adapts to working over local rings.

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