# Cohomological interpretations of quadratic form invariants over rings?

The standard approach to classifying of quadratic forms over $\Bbb Q$ is to use the Hasse (local-global) principle together with a system of standard invariants of quadratic forms over the local fields $\Bbb Q_p$ and $\Bbb R$. These local invariants over $\Bbb Q_p$ consist of a triple $(n,d,c)$ given by the dimension, determinant, and Hasse invariant. These local invariants can be interpreted cohomologically as invariants respectively coming from the cohomology groups $H^i(\Bbb Q_p, \Bbb Z/2\Bbb Z)$, where $i\leq 2$ (since the cohomological dimension of a $\Bbb Q_p$ is two). (These invariants are described in the recent book "Cohomological invariants in Galois cohomology" by Garibaldi, Merkurjev, Serre, and also in "The Algebraic and Geometric Theory of Quadratic Forms" by Elman, Karpenko, Merkurjev.)

In contrast, if one considers the classification of quadratic forms over $\Bbb Z$, then the local-global principle does not work as well (leading to the notion of a "genus" of forms). However it is known that for $p>2$ that a complete system of invariants can be given in terms of the field invariants for each of the (uniquely determined) $p$-power scalings of unimodular forms, know as the Jordan block decomposition. (One can also say something for $p=2$, but it's not important here.) My question is if there is a good cohomological framework for understanding this result, using some cohomology theory defined over $\Bbb Z_p$? (and if so, similarly for $\Bbb Z$?)

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