Related papers: On relative grothendieck rings and algebraically c…
We define a Grothendieck ring for basic real semialgebraic formulas, that is for systems of real algebraic equations and inequalities. In this ring the class of a formula takes into consideration the algebraic nature of the set of points…
Algebraically constructible functions connect real algebra with the topology of algebraic sets. In this survey we present some history, definitions, properties, and algebraic characterizations of algebraically constructible functions, and a…
We generalize the motivic incarnation morphism from the theory of arithmetic integration to the relative case, where we work over a base variety S over a field k of characteristic zero. We develop a theory of constructible effective Chow…
In this paper we treat Grothendieck Duality for noetherian rings via rigid dualizing complexes. In particular, we prove that every ring, essentially finite type over a regular base ring, has a unique rigid dualizing complex. The rigid…
Let $k$ be a field of characteristic zero containing all roots of unity and $K=k((t))$. We build a ring morphism from the Grothendieck group of semi-algebraic sets over $K$ to the Grothendieck group of motives of rigid analytic varieties…
The relationship between spinors and Clifford (or geometric) algebra has long been studied, but little consistency may be found between the various approaches. However, when spinors are defined to be elements of the even subalgebra of some…
We consider the rational linear relations between real numbers whose squared trigonometric functions have rational values, angles we call ``geodetic''. We construct a convenient basis for the vector space over Q generated by these angles.…
The aim of this note is threefold. The first is to obtain a simple characterization of relative constructible sheaves when the parameter space is projective. The second is to study the relative Fourier-Mukai for relative constructible…
The paper studies categories of definable subassignments with some category equivalences to semi-algebraic and constructible subsets of arc spaces of algebraic varieties. These materials allow us to compare the motivic measure of…
We study the algebraic and geometric properties of the integral closure of different rings of functions on a real algebraic variety : the regular functions and the continuous rational functions.
We show that the pair given by the power set and by the "Grassmannian"(set of all subgroups) of an arbitrary group behaves very much like the pair given by a projective space and its dual projective space. More precisely, we generalize…
We clarify the relationship between Grothendieck duality \`a la Neeman and the Wirthm\"uller isomorphism \`a la Fausk-Hu-May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated…
We study monoidal categories that enjoy a certain weakening of the rigidity property, namely, the existence of a dualizing object in the sense of Grothendieck and Verdier. We call them Grothendieck-Verdier categories. Notable examples…
We consider the abelian group $PT$ generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semi-orthogonal decompositions of corresponding triangulated categories. We introduce an operation of…
We study Grothendieck rings (in the sense of logic) of fields. We prove the triviality of the Grothendieck rings of certain fields by constructing definable bijections which imply the triviality. More precisely, we consider valued fields,…
Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. Relations between operations arising from the structure definitions, however,…
This paper develops the geometry of locally bounded rational functions on non-singular real algebraic varieties. First various basic geometric and algebraic results regarding these functions are established in any dimension, culminating…
We show that on real algebraic sets algebraically constructible functions coincide with the finite sums of signs of polynomials. Then we give some applications.
The purpose of this note is to relate certain ring-theoretic properties of rings in mixed and positive characteristics that are related to each other by a tilting operation used in perfectoid geometry. To this aim, we exploit the…
We define a Grothendieck ring of pairs of complex quasi-projective varieties (that is a variety and a subvariety). We describe $\lambda$-structures and a power structure on/over this ring. We show that the conjectual symmetric power of the…