Higher Intersection Theory on Algebraic Stacks: II

This is the second part of our work on the intersection theory of algebraic stacks. The main results here are the following. We provide an intersection pairing for all smooth Artin stacks (locally of finite type over a field) which we show reduces to the known intersection pairing on the Chow groups of smooth Deligne–Mumford stacks of finite type over a field as well as on the Chow groups of quotient stacks associated to actions of linear algebraic groups on smooth quasi-projective schemes modulo torsion. The former involves also showing the existence of Adams operations on the rational étale K-theory of all smooth Deligne–Mumford stacks of finite type over a field. In addition, we show that our definition of the higher Chow groups is intrinsic to the stack for all smooth stacks and also stacks of finite type over the given field. Next we establish the existence of Chern classes and Chern character for Artin stacks with values in our Chow groups and extend these to higher Chern classes and a higher Chern character for perfect complexes on an algebraic stack, taking values in cohomology theories of algebraic stacks that are defined with respect to complexes of sheaves on a big smooth site. As a by-product of our techniques we also provide an extension of higher intersection theory to all schemes locally of finite type over a field. As the higher cycle complex, by itself, is a bit difficult to handle, the stronger results like contravariance for arbitrary maps between smooth stacks and the intersection pairing for smooth stacks are established by comparison with motivic cohomology.     

Riemann–Roch for Algebraic Stacks: I

In this paper we establish Riemann–Roch and Lefschtez–Riemann–Roch theorems for arbitrary proper maps of finite cohomological dimension between algebraic stacks in the sense of Artin. The Riemann–Roch theorem is established as a natural transformation between the G-theory of algebraic stacks and topological G-theory for stacks: we define the latter as the localization of G-theory by topological K-homology. The Lefschtez–Riemann–Roch is an extension of this including the action of a torus for Deligne–Mumford stacks. This generalizes the corresponding Riemann–Roch theorem (Lefschetz–Riemann–Roch theorem) for proper maps between schemes (that are also equivariant for the action of a torus, respectively) making use of some fundamental results due to Vistoli and Toen. A key result established here is that topological G-theory (as well as rational G-theory) has cohomological descent on the isovariant étale site of an algebraic stack. This extends cohomological descent for topological G-theory on schemes as proved by Thomason.     

Affine Algebraic Varieties Relative to an Algebraic Theory

Affine algebraic varieties relative to an algebraic theory are introduced and described as irreducible components of affine algebraic sets. Their category is shown to be dually equivalent to the category of irreducible functional algebras.     

Algebraic stacks

This is an expository article on the theory of algebraic stacks. After introducing the general theory, we concentrate in the example of the moduli stack of vector bundles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory.     

Structure of the Algebraic Systems with the Complete Theory of Infinite Subsystems

The structure is described for the algebraic systems with the complete theory of infinite subsystems, i.e., the systems with all infinite subsystems elementarily equivalent. A theorem is proved characterizing the property of a system to have exactly the sole complete theory of infinite subsystems.     

Algebraic Graph Theory for Sparse Flexibility Matrices

An efficient algorithm is presented for the formation of suboptimal cycle bases of graphs corresponding to sparse cycle adjacency matrices, leading to the formation of highly sparse flexibility matrices. The algorithm presented employs concepts from algebraic graph theory together with a Greedy-type algorithm to select cycles with small overlaps and uses a simple graph-theoretical method for controlling the independence of the selected cycles.     

分片代数曲线交点的结式求法

本文对确定分片代数曲线的二元样条函数的整体表达式中的截断引入参数表示,给出了分片代数曲线交点的结式求法．理论与实例表明,这种算法是有效的．     

Focal Loci of Algebraic Hypersurfaces: A General Theory

The focal locus is traditionally defined for a differentiable submanifold of R ⁿ. However, since it depends essentially only on the notion of orthogonality, a focal locus can be also associated to an algebraic subvariety of the space , once we have chosen an orthogonal structure on this space. In this paper, we establish somebasic results in the theory of focal loci of algebraichypersurfaces in . Our main results concern the irreducibility of the ramification divisor of the end-point map and the dimension of the singular locus of this divisor, the birationality of the focal map and the degree of the focal locus of an algebraic hypersurface.     

On Linear Dependence Over Complete Differential Algebraic Varieties

We extend Kolchin's results from [12 Kolchin, E. (1974). Differential equations in a projective space and linear dependence over a projective variety. Contributions to analysis (A collection of papers dedicated to Lipman Bers). Cambridge, MA: Academic Press, pp. 195–214.[Crossref] , [Google Scholar]] on linear dependence over the constant points of projective algebraic varieties to linear dependence over arbitrary complete differential algebraic varieties. We show that in this more general setting, the notion of linear dependence still has necessary and sufficient conditions given by the vanishing of a certain system of differential-polynomials equations. We also discuss some conjectural questions around completeness and the catenary problem.     

Moduli Stacks of Permutation Classes of Pointed Stable Curves

The notion of m/Γ-pointed stable curves is introduced. It should be viewed as a generalization of the notion of m-pointed stable curves of a given genus, where the labels of the marked points are only determined up to the action of a group of permutations Γ. The classical moduli spaces and moduli stacks are generalized to this wider setting. Finally, an explicit construction of the new moduli stack of m/Γ-pointed stable curves as a quotient stack is given. Received: February 2008     

Directed Algebraic Topology,Categories and Higher Categories

Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A ‘directed space’ has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundamental n-categories (replacing the fundamental n-groupoids of the classical case). On the other hand, directed homotopy can give geometric models for lax higher categories. Applications have been mostly developed in the theory of concurrency. Unexpected links with noncommutative geometry and the modelling of biological systems have emerged. Work partially supported by MIUR Research Projects.     

Approximation to Real Numbers by Cubic Algebraic Integers I

In 1969, H. Davenport and W. M. Schmidt studied the problemof approximation to a real number by algebraic integers ofdegree at most 3. They did so, using geometry of numbers, byresorting to the dual problem of finding simultaneous approximationsto and ² by rational numbers with the same denominator. Inthis paper, we show that their measure of approximation forthe dual problem is optimal and that it is realized for a countableset of real numbers . We give several properties of these numbersincluding measures of approximation by rational numbers, byquadratic real numbers and by algebraic integers of degree atmost 3. 2000 Mathematics Subject Classification 11J04 (primary),11J13, 11J82 (secondary).     

Algebraic Operations and Generic Effects

Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal V-category C with cotensors and a strong V-monad T on C, we investigate axioms under which an Ob C-indexed family of operations of the form _x :(Tx) ^v (Tx) ^w provides semantics for algebraic operations on the computational -calculus. We recall a definition for which we have elsewhere given adequacy results, and we show that an enrichment of it is equivalent to a range of other possible natural definitions of algebraic operation. In particular, we define the notion of generic effect and show that to give a generic effect is equivalent to giving an algebraic operation. We further show how the usual monadic semantics of the computational -calculus extends uniformly to incorporate generic effects. We outline examples and non-examples and we show that our definition also enriches one for call-by-name languages with effects.     

Algebraic cocycles on normal, quasi-projective varieties

This paper extends to quasi-projective varieties earlier work by the author and H. Blaine Lawson concerning spaces of algebraic cocycles on projective varieties. The topological monoid C_r(Y) (U) of effective cocycles on a normal, quasi-projective variety U with values in a projective variety Y consists of algebraic cycles on U×Y equi-dimensional of relative dimension r over U. A careful choice of topology enables the establishment of various good properties: the definition is essentially algebraic, the group completion Z_r (Y) (U) has 'sensible' homotopy groups, the construction is contravariant with respect to U, convariant with respect to Y, and there is a natural 'quality map" to the topological group of cycles on U×Y. The fundamental theorem presented here is the extension of Friedlander-Lawson duality to this context: the duality map Z_r (Y) (U) to Z_{_r+m} (U × Y) is a homotopy equivalence provided that both U and Y are smooth (where m=dim U). Various application are given, especially the determination of the homotopy types of certain topological groups of algeb raic morphisms.     

实分片代数曲线的某些研究

分片代数曲线作为二元样条函数的零点集合是经典代数曲线的推广. 利用代数的基本知识, 本文对实分片代数曲线的基本性质进行了初步讨论, 并且将实分片代数曲线与相应的二元样条分类进行讨论. 最后, 对实分片代数曲线上的孤立点进行了研究.     

代数系数幂级数在超越数上值的代数无关性

本文证明了一类具有代数系数的幂级数在超越数上值约代数无关性．     

Intersection Quasi-Orderings: An Alternative Proof

This note provides an alternative proof of a known result on the existence of collections of orderings generating intersection quasi-orderings. Instead of proving the result from first principles, it is illustrated how it can be obtained by making use of an analogous relationship between partial orders and linear orders.     

多项式的重根判别式和代数基本定理的简证

运用齐次线性方程组的理论研究实数域上多项式根的问题,给出了n次实系数多项式在复数域上存在某种特殊非零重根的判别公式,同时给出了代数基本定理的一个简洁的代数证明.     

剩余交和一般剩余交的几个性质

本文给出了剩余交和一般剩余交的几个性质，主要讨论了剩余交的ＧＣＭ性和在形变下的变化情况，并讨论了一般剩余交的ＧＣＭ性，ＣＭ性和可光滑性等．     

On Homotopy Groups of Real Algebraic Varieties and their Complexifications

Let X ₀ be a topological component of a nonsingular real algebraic variety and i:X → X _C is a nonsingular projective complexification of X. In this paper, we will study the homomorphism on homotopy groups induced by the inclusion map i:X ₀ → X _C and obtain several results using rational homotopy theory and other standard tools of homotopy theory.