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Milnor-Witt \(K\)-groups of local rings. (English) Zbl 1335.11030

Milnor-Witt \(K\)-theory of a field \(F\) is a \(\mathbb Z\)-graded ring \(K_\bullet^{\mathrm{MW}}(F)= \bigoplus_{n\in \mathbb Z}K^{\mathrm{MW}}_n(F)\) defined by an explicit presentation, due to Morel and Hopkins. Its immediate significance, as shown by F. Morel [in: Axiomatic, enriched and motivic homotopy theory. Proceedings of the NATO Advanced Study Institute, Cambridge, UK, 2002. Dordrecht: Kluwer Academic Publishers. 219–260 (2004; Zbl 1130.14019)], is that it is isomorphic to the graded ring \(\mathrm{Hom}_{SH(F)}(\mathbb{S}^0,\mathbb{G}_m^\bullet)\), where \(SH(F)\) is the stable motivic homotopy category of \(F\), at least when \(F\) is perfect of characteristic not equal to \(2\). Its precise relationship to the Milnor \(K\)-theory \(K^{\mathrm{M}}_\bullet(F)\) and the Witt ring \(W(F)\) of the field \(F\) has also been determined by F. Morel [Comment. Math. Helv. 79, No. 4, 689–703 (2004; Zbl 1061.19001)]:
There are natural surjective maps \(K_n^{\mathrm{MW}}(F)\to K_n^{\mathrm{M}}(F)\) and \(K_n^{\mathrm{MW}}(F)\to I^n(F)\) where \(I^n(F)\) is the \(n\)-th power of the fundamental ideal in \(W(F)\) (and, by definition, \(I^n(F)=W(F)\) for \(n\leq 0\)). These maps then induce an isomorphism of graded rings \(K_n^{\mathrm{MW}}(F)\cong K_n^{\mathrm{M}}(F)\times_{I^n(F)/I^{n+1}(F)}I^n(F)\).
The main result of the paper under review is to extend this latter theorem from the case of fields to that of local rings \(R\) which contain an infinite field of characteristic not \(2\). The proof uses a recent result of the first author giving an explicit presentation of the powers of the fundamental ideal of such rings [“On quadratic forms over semilocal rings”, Preprint, http://www.math.ualberta.ca/\(\sim\)gille/publikationen1.html]. This result in turn is based on a recent deep theorem due to I. Panin and K. Pimenov [Doc. Math., J. DMV Extra Vol., 515–523 (2010; Zbl 1270.11035)]. It also depends on the proof of the Milnor conjecture [D. Orlov et al., Ann. Math. (2) 165, No. 1, 1–13 (2007; Zbl 1124.14017)].
In the final section the authors deduce that if \(R\) is a regular local ring containing a field of characteristic not \(2\) and if \(K\) is the field of fractions of \(R\) then the map \(K_n^{\mathrm{MW}}(R)\to K_n^{\mathrm{MW}}(K)\) induces an isomorphism of \(K_n^{\mathrm{MW}}(R)\) with the unramified Milnor-Witt \(K\)-theory group \(K_{n,\mathrm{unr}}^{\mathrm{MW}}(\mathrm{Spec}(R))\) for all integers \(n\). By definition, \(K_{n,\mathrm{unr}}^{\mathrm{MW}}(\mathrm{Spec}(R))\) is the intersection of the kernels of the residue homomorphisms \(\partial_P:K_n^{\mathrm{MW}}(K)\to K_{n-1}^{\mathrm{MW}}(R_P/PR_P)\) as \(P\) runs through the height one primes of \(R\). This follows, via the authors’ main theorem, from the corresponding statements for Witt rings of Balmer, Gille and Panin [P. Balmer et al., Doc. Math., J. DMV 7, 203–217 (2002; Zbl 1015.19002)], and Milnor \(K\)-theory [M. Kerz and S. Müller-Stach, \(K\)-Theory 38, No. 1, 49–58 (2007; Zbl 1144.19002); M. Kerz, Invent. Math. 175, No. 1, 1–33 (2009; Zbl 1188.19002)]. The authors point out that, due to an argument of J. L. Colliot-Thélène [Proc. Symp. Pure Math. 58, 1–64 (1995; Zbl 0834.14009)], one consequence of this result is that the \(n\)th unramified Milnor-Witt \(K\)-theory group is a birational invariant of smooth and proper \(F\)-schemes for all integers \(n\) and infinite fields \(F\) of characteristic not \(2\).
One ingredient of the authors’ proof of their main theorem is the following extension to local rings of a classical result of R. Elman and T. Y. Lam for fields [J. Algebra 23, 181–213 (1972; Zbl 0246.15029)]: Over any local ring containing \(1/2\), two Pfister forms are isometric if and only if they are chain \(p\)-equivalent.

MSC:

11E81 Algebraic theory of quadratic forms; Witt groups and rings
11E12 Quadratic forms over global rings and fields
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References:

[1] Arason, J., Der Wittring projektiver Räume, Math. Ann., 253, 205-212 (1980) · Zbl 0431.10011
[2] Arason, J.; Elman, R., Powers of the fundamental ideal in the Witt ring, J. Algebra, 239, 150-160 (2001) · Zbl 0990.11021
[3] Asok, A.; Fasel, J., Splitting vector bundles outside the stable range and \(A^1\)-homotopy sheaves of punctured affine spaces, J. Amer. Math. Soc., 28, 1031-1062 (2015) · Zbl 1329.14045
[4] Balmer, P., Witt cohomology, Mayer-Vietoris, homotopy invariance and the Gersten conjecture, K-Theory, 23, 15-30 (2001) · Zbl 0987.19002
[5] Balmer, P.; Gille, S.; Panin, I.; Walter, Ch., The Gersten conjecture for Witt groups in the equicharacteristic case, Doc. Math., 7, 203-217 (2002) · Zbl 1015.19002
[6] Balmer, P.; Walter, Ch., A Gersten-Witt spectral sequence for regular schemes, Ann. Sci. Éc. Norm. Supér. (4), 35, 127-152 (2002) · Zbl 1012.19003
[7] Barge, J.; Morel, F., Groupe de Chow des cycles orientés et classe d’Euler des fibrés vectoriels, C. R. Acad. Sci. Paris Sér. I Math., 328, 191-196 (1999)
[8] Barge, J.; Morel, F., Cohomologie des groupes linéares, \(K\)-théorie de Milnor et groupes de Witt, C. R. Acad. Sci. Paris Sér. I Math., 330, 287-290 (2000)
[9] Colliot-Thélène, J.-L., Birational invariants, purity and the Gersten conjecture, \((K\)-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras. \(K\)-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, Santa Barbara, CA, \(1992. K\)-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras. \(K\)-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, Santa Barbara, CA, 1992, Proc. Sympos. Pure Math., vol. 58 (1995), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 1-64 · Zbl 0834.14009
[10] Elman, R.; Lam, T., Pfister forms and \(K\)-theory of fields, J. Algebra, 23, 181-213 (1972) · Zbl 0246.15029
[11] Fasel, J., The Chow-Witt ring, Doc. Math., 12, 275-312 (2007) · Zbl 1169.14302
[12] Fasel, J., Groupes de Chow-Witt, Mém. Soc. Math. Fr. (N.S.), 113, 1-197 (2008) · Zbl 1190.14001
[13] Gille, S., On quadratic forms over semilocal rings (2015), preprint, available at
[14] Hutchinson, K.; Tao, L., Homology stability for the special linear group of a field and Milnor-Witt \(K\)-theory, Extra Volume: Andrei A. Suslin Sixtieth Birthday. Extra Volume: Andrei A. Suslin Sixtieth Birthday, Doc. Math., 267-315 (2010) · Zbl 1216.19005
[15] Kerz, M., The Gersten conjecture for Milnor \(K\)-theory, Invent. Math., 175, 1-33 (2009) · Zbl 1188.19002
[16] Kerz, M., Milnor \(K\)-theory of local rings with finite residue fields, J. Algebraic Geom., 19, 173-191 (2010) · Zbl 1190.14021
[17] Kerz, M.; Müller-Stach, S., The Milnor-Chow homomorphism revisited, K-Theory, 38, 49-58 (2007) · Zbl 1144.19002
[18] Knebusch, M., Symmetric bilinear forms over algebraic varieties, (Conference on Quadratic Forms - 1976, Proc. Conf.. Conference on Quadratic Forms - 1976, Proc. Conf., Queen’s Univ., Kingston, Ont., 1976. Conference on Quadratic Forms - 1976, Proc. Conf.. Conference on Quadratic Forms - 1976, Proc. Conf., Queen’s Univ., Kingston, Ont., 1976, Queen’s Papers in Pure and Appl. Math., vol. 46 (1977), Queen’s Univ.: Queen’s Univ. Kingston, Ont.), 103-283
[19] Lam, T., Introduction to Quadratic Forms over Fields (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1068.11023
[20] Milnor, J., Algebraic \(K\)-theory and quadratic forms, Invent. Math., 9, 318-344 (1969/1970) · Zbl 0199.55501
[21] Morel, F., Sur les puissances l’idéal fondamental de l’anneau de Witt, Comment. Math. Helv., 79, 689-703 (2004) · Zbl 1061.19001
[22] Morel, F., On the motivic \(\pi_0\) of the sphere spectrum, (Axiomatic, Enriched and Motivic Homotopy Theory. Axiomatic, Enriched and Motivic Homotopy Theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131 (2004), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 219-260 · Zbl 1130.14019
[23] Morel, F., \(A^1\)-Algebraic Topology over a Field, Lecture Notes in Math., vol. 2052 (2012), Springer: Springer Heidelberg · Zbl 1263.14003
[24] Neshitov, A., Framed correspondences and the Milnor-Witt \(K\)-theory (2014), preprint, available at · Zbl 1407.14016
[25] Nesterenko, Y.; Suslin, A., Homology of the full linear group over a local ring and Milnor’s \(K\)-theory, Math. USSR Izv., 34, 121-145 (1990) · Zbl 0684.18001
[26] Orlov, D.; Vishik, A.; Voevodsky, V., An exact sequence for \(K_\ast^M / 2\) with applications to quadratic forms, Ann. of Math. (2), 165, 1-13 (2007) · Zbl 1124.14017
[27] Panin, I.; Pimenov, K., Rationally isotropic quadratic spaces are locally isotropic: II, Extra Volume: Andrei A. Suslin Sixtieth Birthday. Extra Volume: Andrei A. Suslin Sixtieth Birthday, Doc. Math., 515-523 (2010) · Zbl 1270.11035
[28] Scharlau, W., Quadratic and Hermitian Forms (1985), Springer: Springer Berlin · Zbl 0584.10010
[29] Schlichting, M., Euler class groups, and the homology of elementary and special linear groups (2015), preprint, available at
[30] Suslin, A., Torsion in \(K_2^M\) of fields, K-Theory, 1, 5-29 (1987) · Zbl 0635.12015
[31] Voevodsky, V., Motivic cohomology with \(Z / 2\)-coefficients, Publ. Math. Inst. Hautes Études Sci., 98, 59-104 (2003) · Zbl 1057.14028
[32] Witt, E., Theorie der quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math., 176, 31-44 (1937) · JFM 62.0106.02
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