## Remarks on zeta functions and $$K$$-theory over $${\mathbb F}_1$$.(English)Zbl 1173.14004

This short article splits into two sections, both of which concern the arithmetic of schemes over the field of one element, or $${\mathbb F}_1$$-schemes. The first section is concerned with the definition of the zeta function of an $${\mathbb F}_1$$-scheme, the second with the definition of the $$K$$-theory of an affine $${\mathbb F}_1$$-scheme. Both sections are clear and well written, with the concepts demonstrated by well-chosen examples.
Let $$X$$ be a scheme of finite type over the ring of integers $$\mathbb Z$$. A definition of the zeta function $$\zeta_{X|{\mathbb F}_1}$$ was given by C. Soulé [Mosc. Math. J. 4, No. 1, 217–244 (2004; Zbl 1103.14003)]. That definition imposes the condition that there exist a polynomial $$N$$ in one variable, with integer coefficients, such that $$\#X({\mathbb F}_{p^n}) = N(p^n)$$ for all $$p$$ and $$n$$. In the present article, the author relaxes this condition to suppose only that $$\#X({\mathbb F}_q) = N(q)$$ for every prime power $$q$$ with $$(q-1,e)=1$$, where $$e$$ is a natural number depending on $$X$$. If the scheme $$X$$ is “defined over $${\mathbb F}_1$$”, as described by A. Deitmar [in: Number fields and function fields – two parallel worlds. Progr. Math. 239, 87–100. Boston, MA: Birkhäuser (2005; Zbl 1078.11002)], then Theorem 1 shows that this condition is always satisfied, and so the definition of the zeta function applies to all such schemes.
The second topic of the article is the definition of $$K$$-theory for affine $${\mathbb F}_1$$-schemes, or equivalently for monoids. Following D. Quillen [“Higher algebraic $$K$$-theory. I”, Lect. Notes Math. 341, 85–147 (1973; Zbl 0292.18004)], the author describes two constructions, the $$+$$-construction and the Q-construction. The first, the $$+$$-construction, is given by defining the groups $$\text{GL}_n(A)$$ for a monoid $$A$$, and then defining $$K^+_j(A)$$ to be the homotopy group $$\pi_j(BGL^+ (A))$$. In particular, this construction gives $$K^+_j({\mathbb F}_1)$$ to be the $$j$$th stable homotopy group of the spheres.
The second construction, the Q-construction, is described in greater detail. Given a monoid $$A$$, the author considers the category $${\mathcal P}$$ of finitely generated pointed projective $$A$$-modules; together with the class $${\mathcal E}$$ of sequences which are strong exact in the category of all modules, this forms a quasi-exact category. Given a quasi-exact category $$({\mathcal C}, {\mathcal E})$$, the author shows how to define a new category $$Q{\mathcal C}$$, and its classifying space $$BQ{\mathcal C}$$. Theorem 4 shows that the fundamental group $$\pi_1(BQ{\mathcal C})$$ is canonically isomorphic to the Grothendieck group $$K_0({\mathcal C}, {\mathcal E})$$. The $$K$$-groups of $$({\mathcal C}, {\mathcal E})$$ may therefore be defined as $$K_i({\mathcal C}, {\mathcal E}) = \pi_{i+1} (BQ{\mathcal C})$$. For our monoid $$A$$, we now define $$K_i^Q A$$ to be $$K_i({\mathcal P}, {\mathcal E})$$.
Theorem 5 shows, following D. Grayson [“Higher algebraic $$K$$-theory: II.”, Lect. Notes Math. 551, 217–240 (1976; Zbl 0362.18015)], that the two constructions give the same K-theories when $$A$$ is a group. An example is given to show that, for a general monoid, the two theories differ.

### MSC:

 14A15 Schemes and morphisms 11G25 Varieties over finite and local fields 14G15 Finite ground fields in algebraic geometry 19D06 $$Q$$- and plus-constructions 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects) 11S40 Zeta functions and $$L$$-functions 11S70 $$K$$-theory of local fields

### Keywords:

field of one element; zeta function; K-theory

### Citations:

Zbl 1103.14003; Zbl 1078.11002; Zbl 0292.18004; Zbl 0362.18015
Full Text:

### References:

 [1] A. Deitmar, Schemes over $$\mathbf{F}_1$$, in: Number fields and function fields – Two parallel worlds (eds. G. van der Gee, B. J. J. Moonen, and R. Schoof), Prg. Math., vol.239, Birkhäuser, Boston, 2005. · Zbl 1098.14003 · doi:10.1007/0-8176-4447-4_6 [2] A. Deitmar, Homological algebra over belian categories and cohomology of F1-schemes. http://arxiv.org/abs/math.NT/0508642. · Zbl 1161.11346 [3] D. Grayson, Higher algebraic $$K$$-theory. II (after Daniel Quillen), in Algebraic $$K$$-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), pp.217-240. Lecture Notes in Math., 551, Springer, Berlin, 1976. [4] K., Kato, Toric singularities. Amer. J. Math. 116 (1994), no.5, 1073-1099. · Zbl 0832.14002 · doi:10.2307/2374941 [5] B. Kurokawa, H. Ochiai and A. Wakayama, Absolute derivations and zeta functions. Doc. Math. 2003 (2003), Extra vol., 565-584. · Zbl 1101.11325 [6] N. Kurokawa, Zeta functions over $$F_ 1$$. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no.10, 180-184 (2006). · Zbl 1141.11316 · doi:10.3792/pjaa.81.180 [7] Y. Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa), Astérisque No.228 (1995), 4, 121-163 (1995). · Zbl 0840.14001 [8] S. Priddy, On $$\Omega ^{\infty }S^{\infty }$$ and the infinite symmetric group, in Algebraic topology ( Proc. Sympos. Pure Math., Vol.XXII, Univ. Wisconsin, Madison, Wis. , 1970), 217-220. Amer. Math. Soc., Providence, R.I., 1971. · Zbl 0242.55011 [9] D. Quillen, Higher algebraic $$K$$-theory. I, in Algebraic $$K$$-theory , I Higher $$K$$-theories ( Proc. Conf., Battelle Memorial Inst., Seattle, Wash. , 1972), 85-147. Lecture Notes in Math., 341, Springer, Berlin, 1973. · Zbl 0292.18004 [10] C. Soulé, Les variétés sur le corps à un élément, Mosc. Math. J. 4 (2004), no.1, 217-244, 312. · Zbl 1103.14003 [11] J. Tits, Sur les analogues algébriques des groupes semi-simples complexes, in Colloque d’algèbre supérieure, Bruxelles du 19 au 22 décembre 1956, 261-289, Centre Belge de Recherches Aathématiques Établissements Ceuterick, Louvain; Librairie Gauthier-Villars, Paris.
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