Toric varieties, monoid schemes and cdh descent (Q486427)

The paper develops the theory of monoid schemes from first principles up to a theory of cdh descent. The main result, which relates descent over classical schemes to descent over monoid schemes, is motivated by an application to the cyclotomic trace map in \(K\)-theory. \textbf{Monoid schemes} Monoid schemes, whose origins are attributed to [\textit{K. Kato}, Am. J. Math. 116, No. 5, 1073--1099 (1994; Zbl 0832.14002)], form one of several competing models for algebraic geometry over the field with one element \(\mathbb F_1\). See [\textit{A.\ Deitmar}, Betr. Algebra Geom. 49, 517--525 (2008; Zbl 1152.14001)] for some prior work in this direction and [\textit{J. López Peña} and \textit{O. Lorscheid}, in: Noncommutative geometry, arithmetic, and related topics. Proceedings of the 21st meeting of the Japan-U.S. Mathematics Institute (JAMI) held at Johns Hopkins University, Baltimore, MD, USA, March 23--26, 2009. Baltimore, MD: Johns Hopkins University Press. 241--265 (2011; Zbl 1271.14003)] for an overview over other approaches. The present paper encompasses a systematic study of monoid schemes and develops many of the notions prominent in algebraic geometry. How closely the treatment parallels classical introductions to algebraic geometry is perhaps best illustrated by the following extract from the table of contents: {\parindent=6mm \begin{itemize} \item[1.] Monoids (affine monoid schemes) \item [2.] Monoid schemes \item [3.] Basechange and separated morphisms \item [6.] Normal and smooth monoid schemes \item [7.] MProj and blowups \item [8.] Proper morphisms \item [10.] Birational morphisms. \end{itemize}} For any commutative ring \(k\), there is a realization functor taking monoid schemes to \(k\)-schemes (section 5). When \(k\) is a field, the usual functor from fans to toric \(k\)-varieties factors through this realization, with essential image the category of \textit{toric} monoid schemes (section~4). The precise relation between fans and toric monoid schemes is spelt out with great care in Theorem~4.4, clarifying related claims in the literature. \textbf{Main result (cdh descent)} The main result, Theorem 14.3, concerns cdh descent for presheaves of spectra on the subcategory of \textit{partially cancellative torsion free (pctf)} monoid schemes introduced in section 9. Easier to state is the following corollary (Corollary 14.4), highlighted in the introduction. Let \(\mathrm{Sch}/k\) denote the category of separated schemes essentially of finite type over a commutative regular noetherian ring \(k\) containing an infinite field. A presheaf of spectra \(\mathcal F\) on \(\mathrm{Sch}/k\) that satisfies the Mayer-Vietoris property for Zariski covers, finite abstract blow-up squares, and blow-ups along regularly embedded subschemes satisfies the Mayer-Vietoris property for all abstract blow-up squares of toric \(k\)-schemes obtained from subdividing a fan. Monoid schemes enter into the proof of this theorem as follows: By a result of [\textit{E. Bierstone} and \textit{P. D. Milman}, J. Algebr. Geom. 15, No. 3, 443--486 (2006; Zbl 1120.14009)], the singularities of any toric variety can be resolved by a sequence of blow-ups along certain smooth equivariant centers. However, such a blow-up need not itself by a toric variety. Thus, the authors are forced to work with a larger class of schemes, and they show that (realizations of) monoid schemes are a convenient such class. \textbf{Application\ motivation (cyclotomic trace)} The main motivation of the authors, included at the end of the final section, is the following application to \(K\)-theory: Fix a commutative regular ring \(k\) of characteristic \(p\), and consider the cyclotomic trace \(\mathcal K(X) \to \{TC^\nu(X,p)\}_\nu\) as a map of pro-presheaves of spectra on \(\mathrm{Sch}/k\). The homotopy fibre of this map satisfies the Mayer-Vietoris property for all abstract blow-up squares of toric schemes. Although this is not a direct consequence of the main result, the techniques used in its proof can be adapted to this setting. The assumptions stated in Corollary 14.4 have in this case been verified in [\textit{T. Geisser} and \textit{L. Hesselholt}, Invent. Math. 166, No. 2, 359--395 (2006; Zbl 1107.19002); Math. Ann. 348, No. 3, 707--736 (2010; Zbl 1203.19001)].