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Witt and cohomological invariants of Witt classes. (English) Zbl 1450.11035
Let \(K\) be a field of characteristic not 2. There is a construction of M. Rost [“On the Galois cohomology of Spin(14)”, Preprint, https://www.math.uni-bielefeld.de/~rost/spin-14.htmlhttps://www.math.uni-bielefeld.de/~rost/spin-14.html] which defines a certain natural operation \(P_n:I^n(K)\rightarrow I^{2n}(K)\) by \[ P_n(\sum_{i=1}^m\varphi_i)=\sum_{1\leq i<j\leq m}\varphi_i\otimes\varphi_j. \] Composing this with the cohomological invariant \(e_{2n}:I^{2n}(K)\rightarrow H^n(K,\mu_2)\) gives rise to a cohomological invariant for \(I^n\) of degrees \(2n\). The author generalizes this to operations \(\bar{\pi}_n^d:I^n(K)\rightarrow I^{dn}(K)\) for all \(d\in\mathbb{N}\), and thus constructs similar cohomological invariants for \(I^n\) of degree \(dn\) denoted by \(f^d_n\). The author proves that \(f_n^d\) is a a set of generators for \(M(n)=\operatorname{Inv}(I^n,A)\) where \(A(K)=H^*(K,\mu_2)\) (Section 4), and provides another set of generators \(g^d_n\) (Definition 4.4), each being useful depending on the situation. The invariants \(g^d_n\) have the important property that only a finite number of them are nonzero on a fixed form (Proposition 4.7), which allows to take infinite combinations, and the author shows that any invariant of \(I^n\) is equal to such a combination (Theorem 4.9). They are also better behaved with respect to similitudes (Proposition 7.6). On the other hand, the \(f^d_n\) are preferable for handling products (Proposition 5.2 and Corollary 5.6) and restriction to \(I^{n+1}\) (Corollaries 6.3 and 6.4). The author studies the behavior with respect to residues from discrete valuations (Proposition 8.1), and establishes links to Serre’s description of invariants of isometry classes (Proposition 9.5). These invariants may be related to other various constructions on Milnor \(K\)-theory and Galois cohomology, notably by C. Vial [J. Pure Appl. Algebra 213, No. 7, 1325–1345 (2009; Zbl 1185.19002)]. The invariants defined here may be seen as lifting of Vial’s to the level of \(I^n\) (Section 10). Finally, the author adapts an idea of Rost’s (see also [S. Garibaldi, Cohomological invariants: exceptional groups and spin groups. With an appendix by Detlev W. Hoffmann. Providence, RI: American Mathematical Society (AMS) (2009; Zbl 1191.11009)]) to study invariants of Witt classes in \(I^n\) that are divisible by an \(r\)-fold Pfister form, giving a complete description for \(r=1\) (Theorem 11.4).
MSC:
11E81 Algebraic theory of quadratic forms; Witt groups and rings
12G05 Galois cohomology
19D45 Higher symbols, Milnor \(K\)-theory
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