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Cohomological invariants for quadratic forms over local rings

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Abstract

Let A be local ring in which 2 is invertible and let n be a non-negative integer. We show that the nth cohomological invariant of quadratic forms is a well-defined homomorphism from the nth power of the fundamental ideal in the Witt ring of A to the degree n étale cohomology of A with mod 2 coefficients, which is surjective and has kernel the (\(\hbox {n}+1\))th power of the fundamental ideal. This is obtained by proving the Gersten conjecture for Witt groups in an important mixed-characteristic case.

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Notes

  1. The result for fields of characteristic \(p \ne 2\) is a consequence of the characteristic zero result by an easy lifting argument analogous to that of Milnor, [28, p. 344]. The author thanks Jean-Pierre Serre for sharing this insight.

  2. Morel also gave two other proofs: one in [29] for fields of characteristic zero and another which uses only \(Sq^2\) but is unpublished.

  3. See the thesis of the author titled “On the Witt groups of schemes”.

  4. For instance, injectivity follows by using the field case together with the Gersten conjecture for the Witt groups \(W(A)[\frac{1}{2}]\) with two inverted; The latter has been proved by the author for any regular excellent local ring with 2 invertible. See forthcoming work of the author titled Real cohomology and the powers of the fundamental ideal for more in this direction.

  5. Manuscript notes titled “Bloch-Ogus for the étale cohomology of certain arithmetic schemes” distributed at the 1997 Seattle algebraic K-theory conference.

  6. Thesis of the author, Theorem 4.28. One can use a transfer argument to remove the restriction that the residue field of the discrete valuation ring be infinite.

  7. In a correspondence with the author B. Kahn explained that the passage from surjectivity in the case of local rings essentially smooth over a field to this case is easy and goes back to Lichtenbaum, if you grant Gillet’s Gersten conjecture for étale cohomology. Surjectivity in the case of local rings essentially smooth over a field is known [18, 19].

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  1. Emory University, 532 Kilgo Circle, Office 413, Atlanta, GA, 30322, USA

    Jeremy Allen Jacobson

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Correspondence to Jeremy Allen Jacobson.

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The author wishes to thank Raman Parimala and Suresh Venapally for their support and encouragement at Emory University, as well as Marco Schlichting, Jean-Louis Colliot-Thélène, and Bruno Kahn, for helpful comments on an earlier draft.

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Jacobson, J.A. Cohomological invariants for quadratic forms over local rings. Math. Ann. 370, 309–329 (2018). https://doi.org/10.1007/s00208-017-1561-z

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