K-functor

in algebraic geometry

An invariant of cohomology type associated with schemes in algebraic $K$- theory. More precisely, in algebraic $K$- theory one constructs a contravariant functor

$$X \ \mapsto \ K _{*} (X) \ = \ \sum _ {i \geq 0} K _{i} (X)$$

from the category of schemes into the category of graded commutative rings . The $K$- functor is related to the étale cohomology, but there is an important difference between them: $K$- theory carries global "integer" information, which is absent in étale cohomology, which has finite coefficients.

The first application of $K$- theory in algebraic geometry was in its very origin. It was a proof of a generalization (in particular, to smooth varieties of arbitrary dimension) of the classical Riemann–Roch theorem (see [2]). After higher algebraic $K$- theory had been invented, that is, the cohomology theory of the functors $K _{i}$, $i > 0$( see , [2]), its ideas began to penetrate algebraic geometry intensively. At present one can identify the following areas of research in this direction.

1) The study of algebraic cycles on algebraic varieties. Let $X$ be a smooth algebraic variety and let $\mathop{\rm CH}\nolimits (X)$ be the Chow ring of algebraic cycles on $X$ modulo rational equivalence (cf. Algebraic cycle). Then there are isomorphisms

$$\mathop{\rm CH}\nolimits (X) \otimes \mathbf Q \ \cong \ K _{0} (X) \otimes \mathbf Q ,$$

$$\mathop{\rm CH}\nolimits (X) \ \cong \ \sum _ {i \geq 0} H ^{i} (X,\ {\mathcal K} _{i} ),$$

where ${\mathcal K} _{i}$ is the sheaf (in the Zariski topology) associated with the pre-sheaf $U \rightarrow K _{i} \cdot \Gamma (U,\ {\mathcal O} _{X} )$. These facts are the basis for the study of the rings $\mathop{\rm CH}\nolimits (X)$ by methods of $K$- theory. In particular, finiteness theorems for Chow groups of $0$- cycles on arithmetic surfaces have been proved by these methods [4].

2) The values of the zeta-function and the $L$- function of an algebraic variety at integer points. There is a conjecture about the connection between the values of the zeta-functions of algebraic number fields at integer points and the orders of the torsion subgroups in the $K$- functors of their rings of integers, and also between the values of the $L$- functions of varieties over algebraic number fields at integer points and the ranks of their groups $K _{i}$ and the volumes of the lattices generated by the image of the $K$- functor in their cohomology rings (see , [9]). These conjectures have been confirmed in a number of particular cases, and they are complementary to the Birch–Swinnerton-Dyer conjecture (see Zeta-function in algebraic geometry).

3) Class field theory in higher dimensions describes the Galois group of a maximal Abelian extension of rational function fields of arithmetic schemes of dimension $i \geq 1$, and also of the corresponding local objects ( $i$- dimensional local fields , [10]). In this description, the role that is usually played in dimension 1 by the multiplicative group is filled by the Milnor groups $K _{i}$.

4) The connection between crystalline cohomology and deformation of $K$- functors (see [3]).

5) The theory of characteristic classes in algebraic $K$- theory and the Riemann–Roch–Grothendieck theorem (see [5], [11]).

6) The computation of the algebraic $K$- functor for a wide class of schemes. In particular, the $K$- functor with finite coefficients has been computed for algebraically closed fields [8].

References

 [1a] D. Quillen, "Higher algebraic -theory I" H. Bass (ed.) , Algebraic -theory I (Battelle Inst. Conf.) , Lect. notes in math. , 341 , Springer (1973) pp. 85–147 MR338129 [1b] S. Lichtenbaum, "Values of zeta-functions, étale cohomology and algebraic -theory" H. Bass (ed.) , Algebraic -theory II (Battelle Inst. Conf.) , Lect. notes in math. , 342 , Springer (1973) pp. 489–501 [2] P. Berthelot (ed.) A. Grothendieck (ed.) L. Illusie (ed.) et al. (ed.) , Théorie des intersections et théorème de Riemann–Roch (SGA 6) , Lect. notes in math. , 225 , Springer (1971) [3] S. Bloch, "Algebraic -theory and crystalline cohomology" Publ. Math. IHES , 47 (1977) pp. 187–268 MR488288 [4] J.L. Colliot-Thélène, J.J. Sansuc, C. Soulé, "Quelques théorèmes de finitude en théorie des cycles algébriques" C.R. Acad. Sci. Paris Sér. 1 , 294 (1982) pp. 749–752 (English abstract) Zbl 0572.14005 [5] H. Gillet, "Riemann–Roch theorems for higher -theory" Adv. in Math. , 440 (1981) pp. 203–289 MR624666 [6] B. Harris, G. Segal, " groups of rings of algebraic integers" Ann. of Math. , 101 : 1 (1975) pp. 20–33 MR0387379 Zbl 0331.18015 [7a] K. Kato, "A generalization of local class field theory by using -groups I" J. Fac. Sci. Univ. Tokyo Sect. 1A , 26 : 2 (1979–1980) pp. 303–376 [7b] K. Kato, "A generalization of local class field theory by using -groups II" J. Fac. Sci. Univ. Tokyo Sect 1A , 27 : 3 (1980) pp. 603–683 [8] A.A. Suslin, "On the -theory of algebraically closed fields" Invent. Math. , 73 : 2 (1983) pp. 241–245 MR714090 [9] A.A. Beilinson, "Higher regulators and values of -functions of curves" Funct. Anal. Appl. , 14 : 2 (1980) pp. 116–117 Funkts. Anal. i Primenen. , 14 : 2 (1980) pp. 46–47 MR575206 [10] A.N. Parshin, "Abelian coverings of arithmetic schemes" Soviet Math. Dokl. , 19 : 6 (1978) pp. 1438–1442 Dokl. Akad. Nauk SSSR , 243 : 4 (1978) pp. 855–858 MR0514485 Zbl 0443.12006 [11] V.V. Shekhtman, "The Riemann–Roch theorem and the Atiyah–Hirzebruch spectral sequence" Russian Math. Surveys , 35 : 6 (1980) pp. 105–106 Uspekhi Mat. Nauk , 35 : 6 (1980) pp. 179–180 MR0601773 Zbl 0491.14002 Zbl 0469.14004 [12] A.A. Suslin, "Algebraic -theory" J. Soviet Math. , 28 : 6 (1985) pp. 870–923 Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 20 (1982) pp. 71–152