Hans Schoutens, City University of New York Schemic Grothendiek rings and motivic series Denef and Loeser used QE for valued fields, due to Pas, to extend Konsevitch's motivic integration, and then used it to show that the rationality of the Igusa zeta function is motivic, meaning that there is a rational function over (a localization of) the Grothendieck ring of the complex numbers which specializes to the classical Igusa zeta series. This result piqued the interest of model-theorists and, since then, Grothendieck rings of various first-order theories have been studied. There are many other generating (zeta) functions in algebra and geometry, and one may ask whether their rationality is motivic as well. However, to even formulate this question, say for the classical Hilbert series, one needs a finer invariant than the Grothendieck ring of an algebraically closed field K, as the latter is generated by classes of (reduced) varieties only, whereas we need classes of arbitrary (non-reduced) schemes to even define the motivic Hilbert series. The solution I propose is to work in the schemic Grothendieck ring, obtained using only pp-formulae (in the language of K-algebras) modulo the theory of finite dimensional K-algebras. We can then associate to each scheme of finite type over K an element in the schemic Grothendieck ring. To define a geometric structure on the complement of an open subscheme, we need however certain infinitary formulae, called formularies; these then correspond to the formal scheme structure on the complement. |
