In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields is given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this date. A new feature of this algorithm is that the role of the algebraic part (the dependence on the degrees of the input polynomials) and the combinatorial part (the dependence on the number of polynomials) are separated. Another new feature is that the degrees of the polynomials in the equivalent quantifier-free formula that is output, are independent of the number of input polynomials. As special cases of this algorithm, new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields, are obtained.

We develop representation theory of the rational Cherednik algebra Hc associated to a finite Coxeter group W in a vector space h, and a parameter “c. ” We use it to show that, for integral values of “c, ” the algebra Hc is simple and Morita equivalent to D(h)#W, the cross product of W with the algebra of polynomial differential operators on h. O. Chalykh, M. Feigin, and A. Veselov [CV1], [FV] introduced an algebra, Qc, of quasi-invariant polynomials on h, such that C[h] W ⊂ Qc ⊂ C[h]. We prove that the algebra D(Qc) of differential operators on quasi-invariants is a simple algebra, Morita equivalent to D(h). The subalgebra D(Qc) W ⊂ D(Qc) of W-invariant operators turns out to be isomorphic to the spherical subalgebra eHce ⊂ Hc. We show that D(Qc) is generated, as an algebra, by Qc and its “Fourier dual ” Q ♭ c, and that D(Qc) is a rank-one projective (Qc ⊗ Q ♭ c)-module (via multiplication-action on D(Qc) on opposite sides).

We prove a version of the Atiyah-Segal completion theorem for proper actions of an infinite discrete group G. More precisely, for any finite proper G-CW-complex X, K (EG\Theta GX) is the completion of K G (X) with respect to a certain ideal. We also show, for such G and X, that KG (X) can be defined as the Grothendieck group of the monoid of G-vector bundles over X. Key words: K-theory, proper actions, vector bundles 1991 mathematics subject classification: primary 55N91, secondary 19L47 Let G be any discrete group. For such G, a G-CW-complex is a CW-complex with G- action which permutes the cells, such that an element g 2 G sends a cell to itself only by the identity map. A G-CW-complex X is proper if all of its isotropy subgroups have finite order, and is finite if it is made up of finitely many orbits of cells. A G-CW-pair is a pair of G-spaces (X; A), where X is a G-CW-complex and A is a G-invariant subcomplex. The main results of this paper are Theorems 3.2 and 4.3 below...

Notes for a class taught at the University of Kaiserslautern 2002/2003

In the work is constructed new acyclic resolutions of quasicoherent sheaves. These resolutions is connected with multidimensional local fields. Then the obtained resolutions is applied for a construction of generalization of the Krichever map to algebraic varieties of any dimension. This map gives in the canonical way two k-subspaces B ⊂ k((z1))... ((zn)) and W ⊂ k((z1))... ((zn)) ⊕r from arbitrary algebraic n-dimensional Cohen-Macaulay projective integral scheme X over a field k, a flag of closed integral subschemes X = Y0 ⊃ Y1 ⊃...Yn (such that Yi is an ample Cartier divisor on Yi−1, and Yn is a smooth k-point on all Yi), formal local parameters of this flag in the point Yn, a rank r vector bundle F on X, and a trivialization F in the formal neighbourhood of the point Yn, where the n-dimensional local field k((z1))... ((zn)) is associated with the flag Y0 ⊃... ⊃ Yn. In addition, the constructed map is injective, i. e., it is possible to reconstruct uniquely all the original geometrical data. Besides, from the subspace B is written explicitly a complex, which calculates cohomology of the sheaf OX on X; and from the subspace W is written explicitly a complex, which calculates cohomology of F on X. 1

Abstract. We develop a theory of quasi–coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of Laumon and Moret-Bailly. We study basic cohomological properties of such sheaves, and prove stack–theoretic versions of Grothendieck’s Fundamental Theorem for proper morphisms, Grothendieck’s Existence Theorem, Zariski’s Connectedness Theorem, as well as finiteness Theorems for proper pushforwards of coherent and constructible sheaves. We also explain how to define a derived pullback functor which enables one to carry through the construction of a cotangent complex for a morphism of algebraic stacks due to Laumon and Moret–Bailly. 1.1. In the book ([LM-B]) the lisse-étale topos of an algebraic stack was introduced, and a theory of quasi–coherent and constructible sheaves in this topology was developed. Unfortunately, it was since observed by Gabber and Behrend (independently) that the lisse-étale topos is not functorial as asserted in (loc. cit.), and hence the development of the theory of sheaves in this book is not satisfactory “as is”. In addition, since the publication of the book ([LM-B]), several new results have been obtained such as finiteness of coherent and étale cohomology ([Fa], [Ol]) and various other consequences of Chow’s Lemma ([Ol]).

In this thesis we present new algorithms to solve several very general problems of semi-algebraic geometry. These are currently the best algorithms for solving these problems. In addition, we have proved new bounds on the topological complexity of real semi-algebraic sets, in terms of the parameters of the polynomial system defining them, which improve some old and widely used results in this field. In the first part of the thesis we describe new algorithms for solving the decision problem for the first order theory of real closed fields and the more general problem of quantifier elimination. Moreover, we prove some purely mathematical theorems on the number of connected components and on the existence of small rational points in a given semi-algebraic set. The second part of this thesis deals with connectivity questions of semialgebraic sets. We develop new techniques in order to give a...

. It is a well-known fact that if K is a eld, then the Hilbert series of a quotient of the polynomial ring K[x1 ; : : : ; xn ] by a homogeneous ideal is of the form q(t) (1 t) n ; we call the polynomial q(t) the Hilbert numerator of the quotient algebra. We will generalize this concept to a class of non-nitely generated, graded, commutative algebras, which are endowed with a surjective \co-ltration" of nitely generated algebras. Then, although the Hilbert series themselves can not be dened (since the subvector -spaces involved have innite dimension), we get a sequence of Hilbert numerators qn (t), which we show converge to a power series in Z[[t]]. This power series we call the (generalized) Hilbert numerator of the non-nitely generated algebra. The question of determining when this power series is in fact a polynomial is the topic of the last part of this article. We show that quotients of the ring R 0 by homogeneous ideals that are generated by nitely many monomials hav...

1.1. Motivation. This paper is largely concerned with constructing quotients by étale equivalence relations. We are inspired by questions in classical rigid geometry, but to give satisfactory answers in that category we have to first solve quotient problems within the framework of Berkovich’s k-analytic spaces. One source of motivation is the relationship between algebraic spaces and analytic spaces over C, as follows. If X

this article is that the inverse limit of a surjective inverse system (in the sense of [2]) of UFDs is a c-topological UFD: