publication . Article . Preprint . 2017

K-groups of reciprocity functors

Ivorra, Florian; Rülling, Kay;
Open Access English
  • Published: 01 Jan 2017
  • Publisher: HAL CCSD
  • Country: France
Abstract
In this work we introduce reciprocity functors, construct the associated K-group of a family of reciprocity functors, which itself is a reciprocity functor, and compute it in several different cases. It may be seen as a first attempt to get close to the notion of reciprocity sheaves imagined by B. Kahn. Commutative algebraic groups, homotopy invariant Nisnevich sheaves with transfers, cycle modules or K\"ahler differentials are examples of reciprocity functors. As commutative algebraic groups do, reciprocity functors are equipped with symbols and satisfy a reciprocity law for curves.
Subjects
arXiv: Mathematics::Category TheoryMathematics::K-Theory and HomologyMathematics::Algebraic Topology
free text keywords: [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], [ MATH.MATH-KT ] Mathematics [math]/K-Theory and Homology [math.KT], [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], [MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT], Mathematics - Algebraic Geometry, Mathematics - K-Theory and Homology, Homotopy, Algebraic number, Pure mathematics, Invariant (mathematics), Mathematics, Functor, Reciprocity (social psychology), Reciprocity law, Mathematical analysis, Commutative property
39 references, page 1 of 3

1.1.5. Lemma. Let X, Y, Z be in Reg61 and [V ] ∈ Cor(X, Y ), [W ] ∈ Cor(Y, Z) be elementary correspondences. Let T be an irreducible component of V ×Y W . Then ˜ W / X 4.2.9. Corollary. Let M1, . . . , Mn−1 be reciprocity functors. Then θX,U (φ ⊗ ψ ⊗ (f × idY ) ◦ h) =

1. Théorie des topos et cohomologie étale des schémas. tome 3, Lecture Notes in Mathematics, Vol. 305, Springer-Verlag, Berlin, 1973, Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat.

2. Reza Akhtar, Milnor K-theory of smooth varieties, K-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of K-Theory in the Mathematical Sciences 32 (2004), no. 3, 269-291.

3. H. Bass and J. Tate, The Milnor ring of a global field, Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972), Springer, Berlin, 1973, pp. 349-446. Lecture Notes in Math., Vol. 342.

4. Spencer Bloch and Hélène Esnault, The additive dilogarithm, Documenta Mathematica (2003), no. Extra Vol., 131-155 (electronic), Kazuya Kato's fiftieth birthday.

5. Spencer Bloch and Kazuya Kato, p-adic étale cohomology, Institut des Hautes Études Scientifiques. Publications Mathématiques (1986), no. 63, 107-152. [OpenAIRE]

6. Philippe Elbaz-Vincent and Stefan Müller-Stach, Milnor K-theory of rings, higher Chow groups and applications, Inventiones Mathematicae 148 (2002), no. 1, 177-206.

7. William Fulton, Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)

8. Ofer Gabber, Letter to Bruno Kahn, (1998).

9. Thomas Geisser and Marc Levine, The K-theory of fields in characteristic p, Inventiones Mathematicae 139 (2000), no. 3, 459-493.

10. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Institut des Hautes Études Scientifiques. Publications Mathématiques (1965), no. 24, 231.

11. Toshiro Hiranouchi, Somekawa's K-groups and additive higher Chow groups, http://arxiv.org/abs/1208.6455 (2012). [OpenAIRE]

12. Annette Huber and Bruno Kahn, The slice filtration and mixed Tate motives, Compos. Math. 142 (2006), no. 4, 907-936. MR 2249535 (2007e:14034)

13. Reinhold Hübl and Ernst Kunz, On algebraic varieties over fields of prime characteristic, Arch. Math. 62 (1994), no. 1, 88-96.

14. Luc Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Annales Scientifiques de l'École Normale Supérieure. Quatrième Série 12 (1979), no. 4, 501-661.

39 references, page 1 of 3
Powered by OpenAIRE Research Graph
Any information missing or wrong?Report an Issue