Let $A$ be a commutative ring with unit. A non-degenerate symmetric bilinear form $\phi$ on a finitely generated projective $A$-module $P$ is called metabolic if there is a direct summand $L$ of $P$ such that the adjoint map of $\phi$ $$ \phi' \colon P \to P^\ast = Hom(P,A) $$ restricts to $0$ on $L$ and $L = L^\perp$, where $L^\perp = \ker(P \stackrel{\phi'}{\to} P^\ast \to L^\ast)$. In different terms $L^\perp$ is the submodule of $P$ consisting of elements $p$ such that $\phi(p,L)=\{0\}$. In this case $(P,\phi)$ is isometric to $L\oplus L^\ast$ with bilinear form of the type $((l,\lambda),(k,\kappa)) \mapsto \kappa(l) + \lambda(k) + \gamma(\lambda,\kappa)$, where $\gamma$ is a symmetric (possibly degenerate) bilinear form on $L^\ast$.

Now $(P,\phi)$ is called stably metabolic if there is a metabolic module $(N,\nu)$ such that the orthogonal sum $(P,\phi)\perp(N,\nu)$, which is just $(P \oplus N,\phi\oplus \nu)$, is metabolic. The stably metabolic modules are precisely the ones that become $0$ in the Witt-group $W(A)$ of symmetric bilinear forms over $A$.

My question: When is a stably metabolic module in fact metabolic? Are there conditions I can put on $A$ to make this true?

(As usual $2$ being a unit in $A$ will probably make everything much easier but I'm trying to avoid this.)


I think there is no general answer to this question, and partial answers can only be obtained in sufficiently low dimensions. Very similar to the question when a stably free projective module is actually free (which doesn't have a complete answer). I explain below my favourite way of thinking about this kind of problem: take the topological approach and algebraize. This favourite way has the side-effect that I have to work in characteristic $\neq 2$ and therefore replace metabolic by hyperbolic throughout.

We can first look at the topological situation. For even $n$, the tangent bundle of $S^n$ is an example of an $SO(n)$-principal bundles which is stably trivial but non-trivial, detected by the Euler class. Now take one such bundle and complexify, then you get an $SO(n;\mathbb{C})$-principal bundle over the smooth affine quadric $Q_m=V(\sum_{i=0}^{m}x_i^2-1)$. The homotopy classification results (and the fact that $SO(n)$ is a deformation retract of its complexification $SO(n;\mathbb{C})$) tell us that this principal $SO(n;\mathbb{C})$-bundle is stably trivial but non-trivial. Under the Serre-Swan theorem, this corresponds to a stably hyperbolic but non-hyperbolic quadratic form on the function ring $\mathbb{C}[x_0,\dots,x_{m}]/(\sum x_i^2-1)$.

The lesson learned from this example is that even for smooth affine varieties over $\mathbb{C}$, the question of stably hyperbolic vs. hyperbolic is subtle. In general, one needs more information than just characteristic classes. The topological way of answering this type of question is obstruction theory; this requires knowledge of homotopy groups of orthogonal groups, which restricts us to "low dimension".

Algebraic analogues of the topological results can be obtained by means of $\mathbb{A}^1$-homotopy theory. This, however, requires some assumptions: we need to work over an infinite field $k$ of characteristic $\neq 2$, and consider smooth affine $k$-schemes $X=\operatorname{Spec}(A)$. This leads a bit away from the actual question which specifically asks for results also applicable in characteristic $2$, but I would expect that results in characteristic $2$ or for non-smooth rings will likely be even more complicated to obtain.

As an example of what can be proved using $\mathbb{A}^1$-homotopy (and how) let me explain a simple case: the symmetric bilinear form analogue of Serre's splitting theorem for projective modules. Recall that Serre's result states that if $R$ is a noetherian ring of Krull dimension $d$, then any projective $R$-module of rank $>d$ splits off a free rank one summand. Over a field, the analogue for symmetric bilinear forms is just Witt's cancellation theorem: a stably hyperbolic form is hyperbolic. (Note that the assumption about characteristic $\neq 2$ means that metabolic is the same as hyperbolic.)

Now if $(P,\phi)$ is a stably hyperbolic symmetric bilinear form, then the base change $(P,\phi)\otimes_AK$ (where $K$ is the field of fractions of $A$) is hyperbolic by Witt cancellation as above. In particular, $(P,\phi)$ can be seen as a rationally trivial torsor under the split orthogonal group. In this case, there is a representability theorem in $\mathbb{A}^1$-homotopy theory to identify isomorphism classes of rationally hyperbolic quadratic forms on $A$ with $\mathbb{A}^1$-homotopy classes $[\operatorname{Spec}A,\operatorname{B}G]_{\mathbb{A}^1}$ where $G$ is the split form of the respective orthogonal group $\operatorname{Aut}(\phi)$. (see here for the paper on the arXiv)

Now the set of $\mathbb{A}^1$-homotopy classes can be studied using $\mathbb{A}^1$-obstruction theory. For example, the quotient $\operatorname{SO}(n,n)/\operatorname{SO}(n-1,n-1)$ is a smooth affine variety. This variety is a fiber bundle over a quadric, whose fiber is also a quadric. In $\mathbb{A}^1$-homotopy, these quadrics are motivic spheres (obvious in odd dimension, true for even dimensions by work of Asok-Doran-Fasel). Then Morel's theorem on connectivity of motivic spheres shows that $\operatorname{SO}(n,n)/\operatorname{SO}(n-1,n-1)$ is $(n-2)$-$\mathbb{A}^1$-connected. The $\mathbb{A}^1$-obstruction theory then tells you that a stably hyperbolic quadratic form of rank $2n$ on $A$ splits off a hyperbolic plane if $d\leq n-1$.

The first case where there is a nontrivial obstruction is the one corresponding to the Euler class for real vector bundles. The obstruction to lifting a map to $\operatorname{B}\operatorname{SO}(n,n)$ to $\operatorname{B}\operatorname{SO}(n-1,n)$ lives in the first non-vanishing homotopy group of the quotient $\operatorname{SO}(n,n)/\operatorname{SO}(n-1,n)$, which is the $2n-1$-dimensional smooth affine quadric. This is $\pi_{n-1}^{\mathbb{A}^1}(Q_{2n-1})\cong \mathbf{K}^{MW}_n$. The relevant obstruction group is $\operatorname{H}^n_{\operatorname{Nis}}(X,\mathbf{K}^{MW}_n)\cong \widetilde{\operatorname{CH}}^n(X)$. So there is a class in the Chow-Witt group which detects if we can reduce the structure group to $\operatorname{SO}(n-1,n)$. Similarly, for the next reduction to $\operatorname{SO}(n-1,n-1)$, the relevant homotopy group is $\pi_{n-1}^{\mathbb{A}^1}(Q_{2n-2})\cong\mathbf{K}^{MW}_{n-1}$, and the obstruction class lives in $\operatorname{H}^n_{\operatorname{Nis}}(X,\mathbf{K}^{MW}_{n-1})$. The latter group vanishes for quadratically closed fields, so the obstruction to splitting off a hyperbolic plane from a stably hyperbolic quadratic form of rank $2n$ where $n$ is equal to the dimension $d=\dim X$ is an Euler class in $\widetilde{\operatorname{CH}}^n(X)$.

The lesson learned is that the topological obstruction theory arguments work in algebraic geometry, provided we assume what we need to handle things the $\mathbb{A}^1$-homotopy way. Any topological theorem stating that stably trivial real vector bundles are trivial under some cohomological conditions can be made algebraic provided that the corresponding $\mathbb{A}^1$-homotopy groups (and cohomology with coefficients in these homotopy groups) can be computed. The latter requirement is fairly serious; if you want to see how complicated things can get check the papers of Asok and Fasel on $\mathbb{A}^1$-classification of projective modules.

Finally, we should probably say something about actual examples. Of course, the map $\operatorname{SO}(n,n)/\operatorname{SO}(n-1,n-1)\to \operatorname{B}\operatorname{SO}(n-1,n-1)$ induced by the above quotient classifies an $\operatorname{SO}(n-1,n-1)$-bundle which becomes trivial upon adding a hyperbolic plane; and it is the universal example of this kind. In some sense, this is a quadratic form version of the standard example, the tangent bundle of $S^2$.

Further examples (of more arithmetic nature) can be furnished by realizing the obstruction classes mentioned above in $\widetilde{\operatorname{CH}}^n(X)$ (even the Chow groups $\operatorname{CH}^n(X)$ can be fairly complicated for $n$-dimensional smooth affine varieties over $\mathbb{C}$).

Finally, for the cases of quadratic forms of low rank over varieties of small dimension, it is probably possible to use the exceptional isomorphisms $\operatorname{SL}_2=\operatorname{SO}(3)$, $\operatorname{SL}_2\times \operatorname{SL}_2=\operatorname{SO}(4)$, $\operatorname{Sp}_2=\operatorname{SO}(5)$ and $\operatorname{SL}_4=\operatorname{SO}(6)$ to do more explicit computations, but I guess this answer is long enough as it is.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.