Intersection theories on a non singular variety

Summary In this paper we show a relationship between the Chow ringA(X) of a non singular varietyX and its graded Grothendieck ringGK(X) by constructing explicitly a surjective ring homomorphism fromA(X) toGK(X). It is a new proof of a well-known theorem of Grothendieck.

---

Riassunto In questa nota si dimostra una relazione tra l’anello di ChowA(X) di una varietà non singolareX ed il suo anello graduato di GrothendieckGK(X), costruendo esplicitamente un omomorfismo suriettivo di anelliA(X)→GK(X). è questa una nuova dimostrazione di un noto teorema di Grothendieck.

     

The Chow ring of a singular surface

We construct the Chow ringCH*(X) =CH ⁰ (X)⊕CH ¹ (X)⊕CH ² (X) of a reduced, quasi-projective surfaceX, together with Chern class mapsc _i :K ₀ (X) → CH ⁱ (X), with the usual properties. As a consequence, we show that the cycle mapCH ² (X)→ F ₀ K ₀ (X) is an isomorphism. Our treatment is greatly influenced by an unpublished 1983 preprint of Levine’s, but is much simpler, since we deal only with surfaces.     

Generators and Defining Relations for the Ring of Differential Operators on a Smooth Affine Algebraic Variety

For the ring of differential operators on a smooth affine algebraic variety X over a field of characteristic zero a finite set of algebra generators and a finite set of defining relations are found explicitly. As a consequence, a finite set of generators and a finite set of defining relations are given for the module Der_K(O (X)){\rm Der}_K({\cal O} (X)) of derivations on the algebra O (X){\cal O} (X) of regular functions on the variety X. For the variety X which is not necessarily smooth, a set of natural derivations der_K(O (X)){\rm der}_K({\cal O} (X)) of the algebra O (X){\cal O} (X) and a ring \mathfrakD (O (X))\mathfrak{D} ({\cal O} (X)) of natural differential operators on O (X){\cal O} (X) are introduced. The algebra \mathfrakD (O (X))\mathfrak{D} ({\cal O} (X)) is a Noetherian algebra of Gelfand–Kirillov dimension 2dim (X). When X is smooth then der_K(O (X))=Der_K(O (X)){\rm der}_K({\cal O} (X))={\rm Der}_K({\cal O} (X)) and \mathfrakD (O (X))=D (O (X))\mathfrak{D} ({\cal O} (X))={\cal D} ({\cal O} (X)). A criterion of smoothness of X is given when X is irreducible (X is smooth iff \mathfrakD (O (X))\mathfrak{D} ({\cal O} (X)) is a simple algebra iff O (X){\cal O} (X) is a simple \mathfrakD (O (X))\mathfrak{D} ({\cal O} (X))-module). The same results are true for regular algebras of essentially finite type. For a singular irreducible affine algebraic variety X, in general, the algebra of differential operators D (O (X)){\cal D} ({\cal O} (X)) needs not be finitely generated nor (left or right) Noetherian, it is proved that each term D (O (X))_i{\cal D} ({\cal O} (X))_i of the order filtration D (O (X))=è_{i 3 0}D (O (X))_i{\cal D} ({\cal O} (X))=\cup_{i\geq 0}{\cal D} ({\cal O} (X))_i is a finitely generated left O (X){\cal O} (X)-module.     

The occupation time of Brownian motion in a ball

LetW be a Wiener process of dimensiond3, starting from 0, and letX(t) be the total time spent byW in the ball centered at 0 with radiust. We give an affirmative answer to a conjecture of Taylor and Tricot⁽¹⁶⁾ on the tail distribution ofX(t). Lévy's lower functions ofX(t) are characterized by an integral test.     

The exceptional set for the sum of a prime and a square

Let E(X) denote the number of natural numbers not exceeding X which cannot be written as a sum of a prime and a square. In this paper we show that for sufficiently large X we have E(X)<< X^0.982. This revised version was published online in June 2006 with corrections to the Cover Date.     

The space of maps from a locally compact space to a Banach space

The space of continuous maps from a topological spaceX to topological spaceY is denoted byC(X,Y) with the compact-open topology. In this paper we prove thatC(X,Y) is an absolute retract ifX is a locally compact separable metric space andY a convex set in a Banach space. From the above fact we know thatC(X,Y) is homomorphic to Hilbert spacel ₂ ifX is a locally compact separable metric space andY a separable Banach space; in particular,C(R ⁿ,R^m) is homomorphic to Hilbert spacel ₂. This research is supported by the Science Foundation of Shanxi Province's Scientific Committee     

The lattice of ideals of a polynomial semiring

We show that for a semiringR, the following statements are equivalent: (1)R is a ring, (2) every left ideal ofR[X], the semiring of polynomials overR, is subtractive, (3) the lattice of left ideals ofR[X] is modular.Presented by R. W. Quackenbush.     

Spaces for which the first uncountable ordinal space is a remainder

A remainder of a locally compact, non-compact Hausdorff spaceX is anyX – X whereX is a Hausdorff compactification ofX. LetK(X) be the lattice of compactifications ofX. Conditions onK(X) and an internal condition are obtained which characterize when the first uncountable ordinal space is a remainder ofX.     

Generalized solution to a semilinear hyperbolic system with a non-Lipshitz nonlinearity

Let

Some Identities for Elements of a Symmetric Matrix

Let Sym (n) be the space of n-dimensional real symmetric matrices and let X ∈ Sym (n). The matrices E, X, X²,... , X^{n − 1} may be considered as vectors of the Euclidean space of dimension n². Denote by V(E, X,... , X^{n − 1}) the volume of the parallelepiped spanned by these vectors. It is proved that

The category of a map and the grade of a module

Letf: Y → X be a continuous map between connectedCW complexes. The homologyH _* (F) of the homotopy fibre is then a module over the loop space homologyH _* (ΩX). Theorem:If H _* (F; R) and H _* (ΩX; R) are R-free (R a principal ideal domain) then for some H _* (ΩX; R)-projective module P=P_>0 and for some m ≤ cat f: Some applications are also given. Research partially supported by a NATO travel grant. Research partially supported by an NSERC operating grant. U.A. au CNRS 751.     

On the Asymptotic Behaviour of the Solutions of a System of Functional Equations

In this paper we study the asymptotic behaviour of the solutions of the functional equation

Decomposition of a class of functionals

LetX(t) be a symmetric conservative Hunt process. In this paper, we decompose the functionalu(X(t), t) into a sum of a square integrable martingale and a regular 0-quadratic variation process. This paper is based on the author's dissertation, presented for the degree of Doctor of philosophy in Institute of Applied Mathematics, Academia Sinica under the supervision of Prof. Yan Jia-an.     

On the Existence of a Compact Generator on the Derived Category of a Noetherian Formal Scheme

In this paper, we prove that for a noetherian formal scheme \mathfrak X\mathfrak X, its derived category of sheaves of modules with quasi-coherent torsion homologies D_qct(\mathfrak X)\boldsymbol{\mathsf{D}}_\mathsf{qct}(\mathfrak X) is generated by a single compact object. In an Appendix we prove that the category of compact objects in D_qct(\mathfrak X)\boldsymbol{\mathsf{D}}_\mathsf{qct}(\mathfrak X) is skeletally small.     

On the abelian fundamental group scheme of a family of varieties

Let S be a connected Dedekind scheme and X an S-scheme provided with a section x. We prove that the morphism between fundamental group schemes π ₁(X, x) ^ab → π ₁(Alb _X/S , 0_AlbX/S{0_{{\rm{Al}}{{\rm{b}}_{X/S}}}}) induced by the canonical morphism from X to its Albanese scheme Alb _X/S (when the latter exists) fits in an exact sequence of group schemes 0 → (NS _X/S ^τ )^⋎ → π ₁(X, x) ^ab → π ₁(Alb _X/S , 0_AlbX/S{0_{{\rm{Al}}{{\rm{b}}_{X/S}}}}) → 0, where the kernel is a finite and flat S-group scheme. Furthermore, we prove that any finite and commutative quotient pointed torsor over the generic fiber X _η of X can be extended to a finite and commutative pointed torsor over X.     

An $ \mathfrak{X} $-crown of a finite soluble group

Let G be a finite soluble group and F_\mathfrakX(G) {\Phi_\mathfrak{X}}(G) an intersection of all those maximal subgroups M of G for which G

On the sum of a prime and a k-th power of prime in short intervals

Let H_k\mathcal{H}_{k} denote the set {n∣2|n, n\not o 1 (mod p)n\not\equiv 1\ (\mathrm{mod}\ p) ∀ p>2 with p−1|k}. We prove that when X^{\frac1120(1-\frac12k) +e}\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{2k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n ? \allowbreak H_k ?(X, X+H]n\in\allowbreak {\mathcal{H}_{k} \cap (X, X+H]} can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when X^{\frac1120(1-\frac1k) +e}\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n∈(X,X+H] can be represented as the sum of a prime and a k-th power of integer for k≧3.     

Ergodic decomposition of a topological space

Given a positive contraction,P, onC(X) we define the conservative and dissipative parts ofP and establish divergence of ΣP ⁿf(x) on the conservative part ofX.     

Zero-divisor graph of C(X)

Summary In this article the zero-divisor graph Γ(C(X)) of the ring C(X) is studied. We associate the ring properties of C(X), the graph properties of Γ(C(X)) and the topological properties of X. Cycles in Γ(C(X)) are investigated and an algebraic and a topological characterization is given for the graph Γ(C(X)) to be triangulated or hypertriangulated. We have shown that the clique number of Γ(C(X)), the cellularity of X and the Goldie dimension of C(X) coincide. It turns out that the dominating number of Γ(C(X)) is between the density and the weight of X. Finally we have shown that Γ(C(X)) is not triangulated and the set of centers of Γ(C(X)) is a dominating set if and only if the set of isolated points of X is dense in X if and only if the socle of C(X) is an essential ideal.     

Combinatorial fiber bundles and fragmentation of a fiberwise PL homeomorphism

With a compact PL manifold X we associate a category . The objects of are all combinatorial manifolds of type X, and morphisms are combinatorial assemblies. We prove that the homotopy equivalence