Wittringhomologie

by Manfred Schmid (doctoral thesis, 1998, 101 pages)

For varieties over a field of characteristic different from 2, the text defines cycle complexes, similar to the cycle complexes studied in [M. Rost, Chow Groups with Coefficients, Doc. Math. 1 (1996), 319-393], however with the coefficient system given by the Wittgroup of quadratic forms. Special care has to be taken in positive characteristic. The complexes can be twisted by line bundles. Basic functorial properties of the complexes and homotopy invariance are established.

The homology groups for projective spaces are computed: All of them vanish, except possibly in the extreme dimensions, depending on the parity of the dimension and the twisting line bundle.

Full text: [pdf]

Related:

• Rost, M., Chow Groups with Coefficients, Doc. Math. J. DMV 1 (1996) 319-393. MR 1418952, Zbl 864.14002.
• The Rost-Schmid Complex of a Strongly A¹-Invariant Sheaf in: Fabien Morel, A¹-Algebraic Topology over a Field, Lecture Notes in Mathematics 2052 (2012). MR 2934577, Zbl 1263.14003.

Addendum

(added on Nov 5, 1998)

In Satz 3.1.5 of M. Schmid's thesis it is stated but not proved that the Gersten type localizations sequences for Witt groups are indeed complexes. This fact can be shown in a similar way as the corresponding statement for the localization sequences for Milnor's K-theory. There are two possibilities:

• Along the lines of Theorem 2.3 in [Ro]. Here one reduces first to the localization of the affine plane in 0 and then uses the following fact [Mi]: If an element of W(k(t)) is unramified everywhere on the affine line, then it is in W(k). The arguments in [Ro] for Milnor's K-theory are completely formal and easily transfer to the framework of Witt groups. However they work only for schemes of finite type over a field.
• Along the lines of [Ka]. Here one reduces first to the spectrum of the ring of power series in two variables and then uses the sum formula for the projective line [GHKS], [Mo]. This approach works for very general schemes. However, transferring Kato's arguments from Milnor's K-theory to Witt groups is not completely formal.

References

[GHKS]

Geyer, W.-D., Harder, G., Knebusch, M., Scharlau, W., Ein Residuensatz für symmetrische Bilinearformen, Invent. Math. 11 (1970) 319-328. MR 283005.

[Ka]

Kato, K., Milnor K-theory and the Chow group of zero cycles, in: Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Part I, Proceedings, Contemp. Math. 55 (1986) 241-253. MR 862638.

[Mi]

Milnor, J., Algebraic K-theory and quadratic forms, Invent. Math. 9 (1970) 318-344. MR 260844.

[Mo]

Motscha, A., Zur Funktorialität der Randabbildung beim Wittring und der Milnor-Scharlau-Sequenz, Diplomarbeit, Universität Regensburg, 1992

[Ro]

Rost, M., Chow Groups with Coefficients, Doc. Math. J. DMV 1 (1996) 319-393. MR 1418952, Zbl 864.14002.

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