Relative Riemann-Zariski spaces

In this paper we study relative Riemann-Zariski spaces associated to a morphism of schemes and generalizing the classical Riemann-Zariski space of a field. We prove that similarly to the classical RZ spaces, the relative ones can be described either as projective limits of schemes in the category of locally ringed spaces or as certain spaces of valuations. We apply these spaces to prove the following two new results: a strong version of stable modification theorem for relative curves; a decomposition theorem which asserts that any separated morphism between quasi-compact and quasiseparated schemes factors as a composition of an affine morphism and a proper morphism. In particular, we obtain a new proof of Nagata’s compactification theorem.     

Läuchli’s Completeness Theorem from a Topos-Theoretic Perspective

We prove a variant of Läuchli’s completeness theorem for intuitionistic predicate calculus. The formulation of the result relies on the observation (due to Lawvere) that Läuchli’s theorem is related to the logic of the canonical indexing of the atomic topos of \(\mathbb{Z}{\text{ - sets}}\). We show that the process that transforms Kripke-counter-models into Läuchli-counter-models is (essentially) the inverse image of a geometric morphism. Completeness follows because this geometric morphism is an open surjection.     

A Friedrichs–Maz'ya inequality for functions of bounded variation

The aim of this short note is to give an alternative proof, which applies to functions of bounded variation in arbitrary domains, of an inequality by Maz'ya that improves Friedrichs inequality. A remarkable feature of such a proof is that it is rather elementary, if the basic background in the theory of functions of bounded variation is assumed. Nevertheless, it allows to extend all the previously known versions of this fundamental inequality to a completely general version. In fact the inequality presented here is optimal in several respects. As already observed in previous proofs, the crucial step is to provide conditions under which a function of bounded variation on a bounded open set, extended to zero outside, has bounded variation on the whole space. We push such conditions to their limits. In fact, we give a sufficient and necessary condition if the open set has a boundary with σ‐finite surface measure and a sufficient condition if the open set is fully arbitrary. Via a counterexample we show that such a general sufficient condition is sharp.     

THE EXPONENTIABLE MORPHISMS IN KELLEY ARE THE OPEN MAPS

Abstract

We show that in the category Kelley of Hausdorff k-spaces a map is exponentiable if and only if it is open and that any open surjection is an effective descent morphism.     

Pure Morphisms are Effective for Modules

Yet another proof of the result asserting that a morphism of commutative rings is an effective descent morphism for modules if and only if it is pure is given. Moreover, it is shown that this result cannot be derived from Moerdijk’s descent criterion.     

Morphisms between complete Riemannian pseudogroups

We introduce the concept of morphism of pseudogroups generalizing the étalé morphisms of Haefliger. With our definition, any continuous foliated map induces a morphism between the corresponding holonomy pseudogroups. The main theorem states that any morphism between complete Riemannian pseudogroups is complete, has a closure and its maps are C^∞ along the orbit closures. Here, completeness and closure are versions for morphisms of concepts introduced by Haefliger for pseudogroups. This result is applied to approximate foliated maps by smooth ones in the case of transversely complete Riemannian foliations, yielding the foliated homotopy invariance of their spectral sequence. This generalizes the topological invariance of their basic cohomology, shown by El Kacimi-Alaoui-Nicolau. A different proof of the spectral sequence invariance was also given by the second author.     

Saddle points and overdetermined problems for the Helmholtz equation

In a seminal 1971 paper, James Serrin showed that the only open, smoothly bounded domain in ⁿ on which the positive Dirichlet eigenfunction of the Laplacian has constant (nonzero) normal derivative on the boundary, is then-dimensional ball. The positivity of the eigenfunction is crucial to his proof. To date it is an open conjecture that the same result is true for Dirichlet eigenvalues other than the least. We show that for simply connected, plane domains, the absence of saddle points is a condition sufficient to validate this conjecture. This condition is also sufficient to prove Schiffer's conjecture: the only simply connected planar domain, on the boundary of which a nonconstant Neumann eigenfunction of the Laplacian can take constant value, is the disc.     

Analyticity of the Stokes semigroup in spaces of bounded functions

The analyticity of the Stokes semigroup with the Dirichlet boundary condition is established in spaces of bounded functions when the domain occupied with fluid is bounded or more generally admissible which admits a special estimate for the Helmholtz decomposition. The proof is based on a blow-up argument. This is the first proof of the analyticity in spaces of bounded functions which was left open more than thirty years.     

Filtered Ends, Proper Holomorphic Mappings of Kähler Manifolds to Riemann Surfaces, and Kähler Groups

The main result of this paper is that a connected bounded geometry complete K?hler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. As an application, it is also proved that any properly ascending HNN extension with finitely generated base group, as well as Thompson’s groups V, T, and F, are not K?hler. The results and techniques also yield a different proof of the theorem of Gromov and Schoen that, for a connected compact K?hler manifold whose fundamental group admits a proper amalgamated product decomposition, some finite unramified cover admits a surjective holomorphic mapping onto a curve of genus at least 2. Received: January 2006, Revision: November 2006, Accepted: March 2007     

Foundations of the theory of bounded cohomology

In this paper we give a new approach to the theory of bounded cohomology. The ideas of relative homological algebra, modified so that they are based on a natural seminorm in the bounded cohomology, play a central role in this approach. Moreover, a new proof is given of the vanishing theorem in the bounded cohomology of simply connected spaces, and also an analog of Leray's theorem on coverings in the theory of bounded cohomology.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 143, pp. 69–109, 1985.     

Extensions and renormalized traces

It has been shown by Nistor (Doc Math J DMV 2:263–295, 1997) that given any extension of associative algebras over \mathbb C{\mathbb C}, the connecting morphism in periodic cyclic homology is compatible, under the Chern–Connes character, with the index morphism in lower algebraic K-theory. The proof relies on the abstract properties of cyclic theory, essentially excision, which does not provide explicit formulas a priori. Avoiding the use of excision, we explain in this article how to get explicit formulas in a wide range of situations. The method is connected to the renormalization procedure introduced in our previous work on the bivariant Chern character for quasihomomorphisms Perrot (J Geom Phys 60:1441–1473, 2010), leading to “local” index formulas in the sense of non-commutative geometry. We illustrate these principles with the example of the classical family index theorem: we find that the characteristic numbers of the index bundle associated to a family of elliptic pseudodifferential operators are expressed in terms of the (fiberwise) Wodzicki residue.     

Remarks on a Theorem of Amitsur

Amitsur proved the following theorem for an uncountable field k: Let A be a k-algebra, then any finite-dimensional nil subspace of A has bounded index, and any finite dimensional algebraic subspace of A has bounded degree. Here we give a new proof of Amitsur's theorem, a proof which is more elementary and direct, than Amitsur's original proof.     

Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices

For any irrational cut-and-project setup, we demonstrate a natural infinite family of windows which gives rise to separated nets that are each bounded distance to a lattice. Our proof provides a new construction, using a sufficient condition of Rauzy, of an infinite family of non-trivial bounded remainder sets for any totally irrational toral rotation in any dimension.     

Reg ouvert et complexe cotangent

For a J-ring A, each deviation is bounded on Spec A and the morphism of A into the infinite product IIA has almost the contangent complex of a smooth morphism.     

Existence of solutions to the nonhomogeneous steady Magnetohydrodynamic equations

We study the nonhomogeneous boundary value problem for the steady Magnetohydrodynamic equations in a two-dimensional bounded domain with multiply connected boundary. We prove that this problem has an admissible solution in an admissible domain if the boundary value is admissible. The proof of the main result uses some property for a weak solution to the transport equations in an admissible domain.     

Approximation par éléments finis de problèmes elliptiques d'optimisation de forme

We consider a problem of elliptic optimal design. The control is the shape of the domain on which the Dirichlet problem for the Laplace equation is posed. In dimension n=2, S?veràk proved that there exists an optimal domain in the class of all open subsets of a given bounded open set, whose complements have a uniformly bounded number of connected components. The proof (J. Math. Pures Appl. 72 (1993) 537–551) is based on the compactness of this class of domains with respect to the complementary-Hausdorff topology and the continuous dependence of the solutions of the Dirichlet Laplacian in H¹ with respect to it. In this Note we consider a finite-element discrete version of this problem and prove that the discrete optimal domains converge in that topology towards the continuous one as the mesh-size tends to zero. The key point of the proof is that finite-element approximations of the solution of the Dirichlet Laplacian converge in H¹ whenever the polygonal domains converge in the sense of that topology. To cite this article: D. Chenais, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Every surface is a leaf

FOR EACH integer n≥3, there are open, connected n-manifolds that cannot be realized as leaves in any compact, C⁰-foliated (n+1)-manifold [8] (for the C² case, cf. [10]). For n=2, the corresponding question has been thought to be hard. Here we give a proof, announced in [7], that all open surfaces are realizable. This answers a basic question posed in [14].In the following, L denotes an open, connected 2-manifold, M a closed, connected 3-manifold.     

Free Factorizations

Given any morphism, we construct extensions of the original category in which this morphism admits certain factorizations, in particular a (retraction, section)-factorization. To this end, we solve the word problem for a certain type of systems of generators and relations for categories. This also enables us to prove preservation properties for the said extensions, e.g. preservation of a pair of diagonalizing classes of epimorphisms and monomorphisms.Iterating such extension processes, we obtain factorizable extensions of categories; in particular, we construct a free proper factorization structure on a given category, which leads to a characterization of preimages of proper factorization structures under full embeddings. As a further application, we characterize an absoluteness property regarding factorizations of functorial images of a morphism.