Mixed Artin-Tate motives over number rings

This paper studies Artin-Tate motives over bases , for a number field F. As a subcategory of motives over S, the triangulated category of Artin-Tate motives is generated by motives , where ? is any finite map. After establishing the stability of these subcategories under pullback and pushforward along open and closed immersions, a motivic t-structure is constructed. Exactness properties of these functors familiar from perverse sheaves are shown to hold in this context. The cohomological dimension of mixed Artin-Tate motives () is two, and there is an equivalence .     

The doubles of a numerical semigroup

Let S be a numerical semigroup and let p be a positive integer. Then the quotient is also a numerical semigroup. When p=2 we say that is half of the numerical semigroup S. Dually, we say that S is a double of the numerical semigroup . We characterize the set of all doubles of a numerical semigroup. We also give some alternative proofs and improvements for some results that we find in previous papers.     

Homology of artinian and Matlis reflexive modules, I

Let R be a commutative local noetherian ring, and let L and L^′ be R-modules. We investigate the properties of the functors and . For instance, we show the following:

(a)

if L and L^′ are artinian, then is artinian, and is noetherian over the completion ;

(b)

if L is artinian and L^′ is Matlis reflexive, then , , and are Matlis reflexive.

Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.     

Vanishing of derived limits of non-standard inverse systems

It is proved that for certain non-standard versions of inverse systems G of R-modules, we have for n>0.The result is applied to define a reasonably canonical non-standard resolution for an arbitrary inverse system G of R-modules, such that the application of the inverse limit functor yields a complex whose cohomology groups are isomorphic to the derived limits, limn. Furthermore, the non-standard resolution gives also the maps limng, and the connecting homomorphisms to a reasonable degree.We also prove that for miscellaneous types of inverse systems H of modules, the system H is a direct summand of .     

On the Brauer group of a semisimple algebraic group

Let k be a field with algebraic closure , G a semisimple algebraic k-group, and a maximal torus with character group X(T). Denote Λ the abstract weight lattice of the roots system of G, and by and the n-torsion subgroup of the Brauer group of k and G, respectively. We prove that if chark does not divide n and n is prime to the order of Λ/X(T) then the natural homomorphism is an isomorphism.     

Exact rates in log law for positively associated random variables

Let be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set , Mn=maxk?n|Sk|, n?1. Suppose . In this paper, we study the exact convergence rates of a kind of weighted infinite series of , and as ε↘0, respectively.     

Families of low dimensional determinantal schemes

A scheme X⊂Pⁿ of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the t×t minors of a homogeneous t×(t+c−1) matrix (f_ij). Given integers a₀≤a₁≤?≤a_t+c−2 and b₁≤?≤b_t, we denote by the stratum of standard determinantal schemes where f_ij are homogeneous polynomials of degrees a_j−b_i and is the Hilbert scheme (if n−c>0, resp. the postulation Hilbert scheme if n−c=0).Focusing mainly on zero and one dimensional determinantal schemes we determine the codimension of in and we show that is generically smooth along under certain conditions. For zero dimensional schemes (only) we find a counterexample to the conjectured value of appearing in Kleppe and Miró-Roig (2005) [25].     

A nondensity property of preperiodic points on Chebyshev dynamical systems

Let k be a number field with algebraic closure , and let S be a finite set of primes of k, containing all the infinite ones. Consider a Chebyshev dynamical system on P². Fix the effective divisor D of P² that is equal to a line nondegenerate on²[−2,2]. Then we will prove that the set of preperiodic points on which are S-integral relative to D is not Zariski dense in P².     

Geometric degree of finite extensions of mappings from a smooth variety

Let f:V→W be a finite polynomial mapping of algebraic subsets V,W of and , respectively, with n≤m. It is known that f can be extended to a finite polynomial mapping . Moreover, it is known that, if V,W are smooth of dimension k,4k+2≤n=m, and f is dominated on every component (without vertical components) then there exists a finite polynomial extension such that , where means the number of points in the generic fiber of h. In this note we improve this result. Namely we show that there exists a finite polynomial extension such that .     

The coarse shape groups

The (pointed) coarse shape category Sh^* (), having (pointed) topological spaces as objects and having the (pointed) shape category as a subcategory, was recently constructed. Its isomorphisms classify (pointed) topological spaces strictly coarser than the (pointed) shape type classification. In this paper we introduce a new algebraic coarse shape invariant which is an invariant of shape and homotopy, as well. For every pointed space (X,?) and for every k∈N₀, the coarse shape group , having the standard shape group for its subgroup, is defined. Furthermore, a functor is constructed. The coarse shape and shape groups already differ on the class of polyhedra. An explicit formula for computing coarse shape groups of polyhedra is given. The coarse shape groups give us more information than the shape groups. Generally, does not imply (e.g. for solenoids), but from pro-πk(X,?)=0 follows . Moreover, for pointed metric compacta (X,?), the n-shape connectedness is characterized by , for every k?n.     

Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

We define and investigate extension groups in the context of Arakelov geometry. The “arithmetic extension groups” we introduce are extensions by groups of analytic types of the usual extension groups attached to OX-modules F and G over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension group - the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over X - and we especially consider the case when X is an “arithmetic curve”, namely the spectrum SpecOK of the ring of integers in some number field K. Then the study of arithmetic extensions over X is related to old and new problems concerning lattices and the geometry of numbers.Namely, for any two hermitian vector bundles and over X:=SpecOK, we attach a logarithmic size to any element α of , and we give an upper bound on in terms of slope invariants of and . We further illustrate this notion by relating the sizes of restrictions to points in P¹(Z) of the universal extension over to the geometry of PSL₂(Z) acting on Poincaré's upper half-plane, and by deducing some quantitative results in reduction theory from our previous upper bound on sizes. Finally, we investigate the behaviour of size by base change (i.e., under extension of the ground field K to a larger number field K^′): when the base field K is Q, we establish that the size, which cannot increase under base change, is actually invariant when the field K^′ is an abelian extension of K, or when is a direct sum of root lattices and of lattices of Voronoi's first kind.The appendices contain results concerning extensions in categories of sheaves on ringed spaces, and lattices of Voronoi's first kind which might also be of independent interest.     

Critical points between varieties generated by subspace lattices of vector spaces

We denote by the semilattice of all compact congruences of an algebra A. Given a variety V of algebras, we denote by the class of all semilattices isomorphic to for some A∈V. Given varieties V and W of algebras, the critical point of V under W is defined as . Given a finitely generated variety V of modular lattices, we obtain an integer ?, depending on V, such that for any n≥? and any field F.In a second part, using tools introduced in Gillibert (2009) [5], we prove that:

     

Gorenstein dimension of modules over homomorphisms

Given a homomorphism of commutative noetherian rings R→S and an S-module N, it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals , where E is the injective hull of the residue field of R. This result is analogous to a theorem of André on flat dimension.     

Chen-Ruan cohomology of some moduli spaces of parabolic vector bundles

Let (X,D) be an ?-pointed compact Riemann surface of genus at least two. For each point x∈D, fix parabolic weights such that . Fix a holomorphic line bundle ξ over X of degree one. Let PMξ denote the moduli space of stable parabolic vector bundles, of rank two and determinant ξ, with parabolic structure over D and parabolic weights . The group of order two line bundles over X acts on PMξ by the rule E_∗⊗L?E_∗⊗L. We compute the Chen-Ruan cohomology ring of the corresponding orbifold.     

On certain cohomological invariants of groups

Let R be a left and right ℵ₀-Noetherian ring. We show that if all projective left and all projective right R-modules have finite injective dimension, then all injective left and all injective right R-modules have finite projective dimension. Using this result, we prove that the invariants and , which were introduced by Gedrich and Gruenberg (1987) [15], are equal for any group G. As an application of the latter equality, we show that a group G is finite if and only if , where is the generalized cohomological dimension of groups introduced by Ikenaga (1984) [21].     

Integral bilinear forms, Coxeter transformations and Coxeter polynomials of finite posets

Linear algebra technique in the study of linear representations of finite posets is developed in the paper. A concept of a quadratic wandering on a class of posets I is introduced and finite posets I are studied by means of the four integral bilinear forms (1.1), the associated Coxeter transformations, and the Coxeter polynomials (in connection with bilinear forms of Dynkin diagrams, extended Dynkin diagrams and irreducible root systems are also studied). Bilinear equivalences between some of the forms are established and equivalences with the bilinear forms of Dynkin diagrams and extended Dynkin diagrams are discussed. A homological interpretation of the bilinear forms (1.1) is given and Z-bilinear equivalences between them are discussed. By applying well-known results of Bongartz, Loupias, and Zavadskij-Shkabara, we give several characterisations of posets I, with the Euler form weakly positive (resp. with the reduced Euler form weakly positive), and posets I, with the Tits form weakly positive.     

New applications of Besov-type and Triebel-Lizorkin-type spaces

In this paper, the authors prove that Besov-Morrey spaces are proper subspaces of Besov-type spaces and that Triebel-Lizorkin-Morrey spaces are special cases of Triebel-Lizorkin-type spaces . The authors also establish an equivalent characterization of when τ∈[0,1/p). These Besov-type spaces and Triebel-Lizorkin-type spaces were recently introduced to connect Besov spaces and Triebel-Lizorkin spaces with Q spaces. Moreover, for the spaces and , the authors investigate their trace properties and the boundedness of the pseudo-differential operators with homogeneous symbols in these spaces, which generalize the corresponding classical results of Jawerth and Grafakos-Torres by taking τ=0.     

How to play the Majority game with a liar

The Majority game is played by a questioner () and an answerer (). holds n elements, each of which can be labeled as 0 or 1. is trying to identify some element holds as having the Majority label or, in the case of a tie, claim there is none. To do this asks questions comparing whether two elements have the same or different label. ’s goal is to ask as few questions as possible while ’s goal is to delay as much as possible. Let q^∗ denote the minimal number of questions needed for to identify a Majority element regardless of ’s answers.In this paper we investigate upper and lower bounds for q^∗ in a variation of the Majority game, where is allowed to lie up to t times. We consider two versions of the game, the adaptive (where questions are asked sequentially) and the oblivious (where questions are asked in one batch).