Israel Journal of Mathematics最新文献

Let ⪯ be a preorder on a monoid H with identity 1_H and s be an integer ≥ 2. The ⪯-height of an element x ∈ H is the supremum of the integers k ≥ 1 for which there is a (strictly) ⪯-decreasing sequence x₁, …, x_k of ⪯-non-units of H with x₁ = x, where u ∈ H is a ⪯-unit if u ⪯ 1_H ⪯ u and a ⪯-non-unit otherwise. We say H is ⪯-artinian if there is no infinite ⪯-decreasing sequence of elements of H, and strongly ⪯-artinian if the ⪯-height of each element is finite.

We establish that, if H is ⪯-artinian, then each ⪯-non-unit x ∈ H factors through the ⪯-irreducibles of degree s, where a ⪯-irreducible of degree s is a ⪯-non-unit a ∈ H that cannot be written as a product of s or fewer ⪯-non-units each of which is (strictly) smaller than a with respect to ⪯. In addition, we show that, if H is strongly ⪯-artinian, then x factors through the ⪯-quarks of H, where a ⪯-quark is a ⪯-minimal ⪯-non-unit. In the process, we obtain upper bounds for the length of a shortest factorization of x into ⪯-irreducibles of degree s (resp., ⪯-quarks) in terms of its ⪯-height.

Next, we specialize these results to the case in which (i) H is the multiplicative submonoid of a ring R formed by the zero divisors of R (and the identity 1_R) and (ii) a ⪯ b if and only if the right annihilator of 1_R − b is contained in the right annihilator of 1_R − a. If H is ⪯-artinian (resp., strongly ⪯-artinian), then every zero divisor of R factors as a product of ⪯-irreducibles of degree s (resp., ⪯-quarks); and we prove that, for a variety of right Rickart rings, either the ⪯-quarks or the ⪯-irreducibles of degree 2 or 3 are coprimitive idempotents (an idempotent e ∈ R is coprimitive if 1_R − e is primitive). In the latter case, we also derive sharp upper bounds for the length of a shortest idempotent factorization of a zero divisor x ∈ R in terms of the ⪯-height of x and the uniform dimension of R_R. In particular, we can thus recover and improve on classical theorems of J. A. Erdos (1967), R.J.H. Dawlings (1981), and J. Fountain (1991) on idempotent factorizations in the endomorphism ring of a free module of finite rank over a skew field or a commutative DVD (e.g., we find that every singular n-by-n matrix over a commutative DVD, with n ≥ 2, is a product of 2n − 2 or fewer idempotent matrices of rank n − 1).

Dobrinen, Hathaway and Prikry studied a forcing ℙ_κ consisting of perfect trees of height λ and width κ where κ is a singular λ-strong limit of cofinality λ. They showed that if κ is singular of countable cofinality, then ℙ_κ is minimal for ω-sequences assuming that κ is a supremum of a sequence of measurable cardinals. We obtain this result without the measurability assumption.

Prikry proved that ℙ_κ is (ω, ν)-distributive for all ν < κ given a singular ω-strong limit cardinal κ of countable cofinality, and Dobrinen et al. asked whether this result generalizes if κ has uncountable cofinality. We answer their question in the negative by showing that ℙ_κ is not (λ, 2)-distributive if κ is a λ-strong limit of uncountable cofinality λ and we obtain the same result for a range of similar forcings, including one that Dobrinen et al. consider that consists of pre-perfect trees. We also show that ℙ_κ in particular is not (ω, ·, λ⁺)-distributive under these assumptions.

While developing these ideas, we address natural questions regarding minimality and collapses of cardinals.

Let A and B be graded algebras in the same variety of trace algebras, such that A is a finite-dimensional, central simple power associative algebra (in the ordinary sense). Over a field K of characteristic zero, we study sufficient conditions that ensure B to be a graded subalgebra of A. More precisely, we prove, under additional hypotheses, that there is a graded and trace-preserving embedding from B to A over some associative and commutative K-algebra C if and only if B satisfies all G-trace identities of A over K. As a consequence of these results, we give a geometric interpretation of our main theorem under the context of graded algebras, and we apply them beyond the Cayley–Hamilton algebras presented in [24, 29]. Such results open a wide range of opportunities to study geometry in Jordan and alternative algebras (with trivial grading).

We construct a model in which there exists a refining matrix of regular height λ larger than (mathfrak{h}); both (lambda = mathfrak{c}) and (lambda < mathfrak{c}) are possible. A refining matrix is a refining system of mad families without common refinement. Of particular interest in our proof is the preservation of ({cal B})-Canjarness.

We show that compactly generated t-structures in the derived category of a commutative ring R are in a bijection with certain families of compactly generated t-structures over the local rings (R_{frak{m}}) where (frak{m}) runs through the maximal ideals in the Zariski spectrum Spec(R). The families are precisely those satisfying a gluing condition for the associated sequence of Thomason subsets of Spec(R). As one application, we show that the compact generation of a homotopically smashing t-structure can be checked locally over localizations at maximal ideals. In combination with a result due to Balmer and Favi, we conclude that the ⊗-Telescope Conjecture for a quasi-coherent and quasi-separated scheme is a stalk-local property. Furthermore, we generalize the results of Trlifaj and Şahinkaya and establish an explicit bijection between cosilting objects of cofinite type over R and compatible families of cosilting objects of cofinite type over all localizations (R_{frak{m}}) at maximal primes.

We develop algebraic and geometrical approaches toward canonical bases for affine q-Schur algebras of arbitrary type introduced in this paper. A duality between an affine q-Schur algebra and a corresponding affine Hecke algebra is established. We introduce an inner product on the affine q-Schur algebra, with respect to which the canonical basis is shown to be positive and almost orthonormal. We then formulate the cells and asymptotic forms for affine q-Schur algebras, and develop their basic properties analogous to the cells and asymptotic forms for affine Hecke algebras established by Lusztig. The results on cells and asymptotic algebras are also valid for q-Schur algebras of arbitrary finite type.

We study the étale fundamental group of a singular reduced connected curve defined over an algebraically closed field of an arbitrary prime characteristic. It is shown that when the curve is projective, the étale fundamental group is a free product of the étale fundamental group of its normalization with a free finitely generated profinite group whose rank is well determined. A similar result is established for the tame fundamental groups of seminormal affine curves. In the affine case, we provide an Abhyankar-type complete group theoretic classification on which finite groups occur as the Galois groups for Galois étale connected covers over (singular) affine curves. An analogue of the Inertia Conjecture is also posed for certain singular curves.

In his famous paper [11], J. Franke has defined a certain finite filtration of the space of automorphic forms of a general reductive group, which captures most of its internal representation theory. The purpose of this paper is to provide several concrete examples of yet unexpected phenomena, which occur in the Franke filtration for the general linear group. More precisely, we show that the degenerate Eisenstein series arising from the parabolic subgroups of the same rank are not necessarily contributing to the same quotient of the filtration, and that, even more, the Eisenstein series arising from the parabolic subgroups of higher relative rank may contribute to a deeper quotient of the filtration. These are the first structural counterexamples to an expectation, mentioned in [11].

Abstract

We show that the n × n matrix differential equation δ(Y) = AY with n² general coefficients cannot be simplified to an equation in less than n parameters by using gauge transformations whose coefficients are rational functions in the matrix entries of A and their derivatives. Our proof uses differential Galois theory and a differential analogue of essential dimension. We also bound the minimum number of parameters needed to describe some generic Picard–Vessiot extensions.

We explore the diversity of subsymmetric basic sequences in spaces with a subsymmetric basis. We prove that the subsymmetrization Su(T*) of Tsirelson’s original Banach space provides the first known example of a space with a unique subsymmetric basic sequence that is additionally non-symmetric. Contrastingly, we provide a criterion for a space with a sub-symmetric basis to contain a continuum of nonequivalent subsymmetric basic sequences and apply it to Su(T*)*. Finally, we provide a criterion for a subsymmetric sequence to be equivalent to the unit vector basis of some ({ell _p}) or c₀.