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.


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
Full Text: DOI arXiv Euclid


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