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Rigidity for linear framed presheaves and generalized motivic cohomology theories. (English) Zbl 1393.14018

A theorem of Suslin and Gillet-Thomason (which is a particular case of the famous Gabber rigidity theorem [O. Gabber, Contemp. Math. 126, 59–70 (1992; Zbl 0791.19002)]) states that if \(\mathcal{O}^h_{X,x}\) is the Henselization of the local ring of a smooth variety \(X\) over a field \(k\) at a rational point \(x\) and \(n\) is an integer invertible in \(k\), then there is a canonical isomorphism \(K_i(\mathcal{O}^h_{X,x},\mathbb{Z}/n)\simeq K_i(k,\mathbb{Z}/n)\) of algebraic \(K\)-theory with mod \(n\) coefficients. In this article, the authors establish some similar rigidity results in \(\mathbb{A}^1\)-motivic homotopy theory, using the theory of framed correspondences developed by G. Garkusha and I. Panin [“Framed motives of algebraic varieties (after V. Voevodsky)”, Preprint, arXiv:1409.4372]. The main result is the following (Theorem 6.3): let \(X\) be a smooth variety over a field \(k\) with \(x\) a closed point with residue field separable over \(k\) and \(n\) be an integer invertible in \(k\). Let \(\mathcal{F}\) be a homotopy invariant presheaf over smooth schemes over \(k\) which has transfers with respect to framed correspondences and is stable (Definition 4.1 (1)), such that \(\mathcal{F}\) is \(nh\)-torsion where \(h\) is the framed correspondence that corresponds to the hyperbolic form (if \(k\) has characteristic \(2\), \(\mathcal{F}\) is required to be \(n\)-torsion). Then there is a canonical isomorphism \[ \mathcal{F}(\text{Spec } \mathcal{O}^h_{X,x})\simeq\mathcal{F}(k(x)) \] where the left hand side is defined as the limit of values of \(\mathcal{F}\) along all étale neighborhoods of \(x\) in \(X\). In particular, this result applies to any (bigraded) cohomology theory which is representable in the stable motivic homotopy category \(SH(k)\), since any such theory always has framed transfers. The proof of the main result reduces to a statement about framed correspondences on smooth curves over a Henselian local ring (Theorem 6.1), and then uses geometric tools to give a detailed study of explicit framed correspondences.

MSC:

14F42 Motivic cohomology; motivic homotopy theory
19E20 Relations of \(K\)-theory with cohomology theories

Citations:

Zbl 0791.19002
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References:

[1] Ananyevskiy, A.; Garkusha, G.; Panin, I., Cancellation theorem for framed motives of algebraic varieties · Zbl 1471.14050
[2] Bachmann, T., Motivic and real étale stable homotopy theory, Compos. Math., 154, 5, 883-917, (2018) · Zbl 1496.14019
[3] Gabber, O., K-theory of henselian local rings and Henselian pairs, Contemp. Math., 126, 59-70, (1992) · Zbl 0791.19002
[4] Garkusha, G.; Panin, I., Framed motives of algebraic varieties (after V. Voevodsky)
[5] Garkusha, G.; Panin, I., Homotopy invariant presheaves with framed transfers · Zbl 1453.14066
[6] Garkusha, G.; Neshitov, A.; Panin, I., Framed motives of relative motivic spheres · Zbl 1484.14048
[7] Gillet, H.; Thomason, R., The K-theory of strict hensel local rings and a theorem of Suslin, J. Pure Appl. Algebra, 34, 241-254, (1984) · Zbl 0577.13009
[8] Grothendieck, A.; Dieudonné, J., Éléments de géométrie algébrique IV. étude locale des schémas et des morphismes de schémas (troisième partie), Publ. Math. Inst. Hautes Études Sci., 28, (1967) · Zbl 0153.22301
[9] Hartshorne, R., Algebraic geometry, (1977), Springer-Verlag New York · Zbl 0367.14001
[10] Hornbostel, J.; Yagunov, S., Rigidity for Henselian local rings and \(\mathbb{A}^1\)-representable theories, Math. Z., 255, 437-449, (2007) · Zbl 1190.14010
[11] Jardine, J. F., Motivic symmetric spectra, Doc. Math., 5, 445-552, (2000) · Zbl 0969.19004
[12] Milne, J., Étale cohomology, (1980), Princeton University Princeton · Zbl 0433.14012
[13] Morel, F., An introduction to \(\mathbb{A}^1\)-homotopy theory, (Contemporary Developments in Algebraic K-theory, ICTP Lect. Notes, (2004)), 357-441 · Zbl 1081.14029
[14] Morel, F., On the Friedlander-Milnor conjecture for groups of small rank, (Contemporary Developments in Algebraic K-theory, Current Developments in Mathematics, vol. 2010, (2011)), 45-93 · Zbl 1257.55007
[15] Morel, F., \(\mathbb{A}^1\)-algebraic topology over a field, Lect. Notes in Math., vol. 2052, (2012), Springer-Verlag · Zbl 1263.14003
[16] Morel, F.; Voevodsky, V., \(\mathbb{A}^1\)-homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci., 90, 45-143, (1999) · Zbl 0983.14007
[17] Panin, I.; Yagunov, S., Rigidity for orientable functors, J. Pure Appl. Algebra, 172, 49-77, (2002) · Zbl 1056.14027
[18] Röndigs, O.; Østvaer, P., Rigidity in motivic homotopy theory, Math. Ann., 341, 651-675, (2008) · Zbl 1180.14014
[19] Stavrova, A., Transfers for non-stable \(\operatorname{K}_1\)-functors of classical type
[20] Suslin, A., On the K-theory of algebraically closed fields, Invent. Math., 73, 2, 241-245, (1983) · Zbl 0514.18008
[21] Suslin, A.; Voevodsky, V., Singular homology of abstract algebraic varieties, Invent. Math., 123, 1, 61-94, (1996) · Zbl 0896.55002
[22] Voevodsky, V., \(\mathbb{A}^1\)-homotopy theory, Doc. Math., Extra Vol. I, 579-604, (1998) · Zbl 0907.19002
[23] V. Voevodsky, Notes on framed correspondences, 2001, math.ias.edu/vladimir/files/framed.pdf; V. Voevodsky, Notes on framed correspondences, 2001, math.ias.edu/vladimir/files/framed.pdf
[24] Yagunov, S., Rigidity II: non-orientable case, Doc. Math., 9, 29-40, (2004) · Zbl 1056.14029
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