A local finiteness theorem for corings over semihereditary rings

Hiroshi Kihara

doi:10.1007/s00013-016-0902-6

Matric generators of coalgebras and bialgebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498819501445 ◽

2019 ◽

Vol 18 (08) ◽

pp. 1950144

Author(s):

Hiroshi Kihara

Keyword(s):

Commutative Ring ◽

Finite Subset ◽

Finiteness Theorem ◽

Finitely Generated ◽

Semihereditary Ring ◽

Local Finiteness

Takeuchi asserted that if a bialgebra [Formula: see text] over a field [Formula: see text] is finitely generated as a [Formula: see text]-algebra, then [Formula: see text] is a matric bialgebra. We introduce the notion of a matric coalgebra over a commutative ring [Formula: see text]. We show that if [Formula: see text] is faithfully projective as a [Formula: see text]-module, then [Formula: see text] is a matric coalgebra. Using this, we also show that if a bialgebra [Formula: see text] over a semihereditary ring [Formula: see text] is projective as a [Formula: see text]-module, then any finite subset of [Formula: see text] is contained in some matric subbialgebra. This result is a generalization of Takeuchi’s assertion and can be regarded as a local finiteness theorem on bialgebras.

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The local structure of the free boundary in the fractional obstacle problem

Advances in Calculus of Variations ◽

10.1515/acv-2019-0081 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Matteo Focardi ◽

Emanuele Spadaro

Keyword(s):

Hausdorff Dimension ◽

Free Boundary ◽

Hausdorff Measure ◽

Local Structure ◽

Obstacle Problem ◽

List Type ◽

Minkowski Content ◽

Local Finiteness ◽

Lower Dimensional

AbstractBuilding upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results:(i)local finiteness of the {(n-1)}-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),(ii){{\mathcal{H}}^{n-1}}-rectifiability of the free boundary,(iii)classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most {(n-2)} in the free boundary.Instead, if {\varphi\in C^{k+1}(\mathbb{R}^{n})}, {k\geq 2}, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than {k+1}.

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$L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02616-9 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Jing Li ◽

Shuxiang Feng ◽

Peibiao Zhao

Keyword(s):

Riemannian Manifold ◽

Riemannian Manifolds ◽

Schrödinger Operators ◽

Schrodinger Operators ◽

Finiteness Theorem ◽

Conformally Flat ◽

Locally Conformally Flat ◽

Flat Riemannian Manifold ◽

Ricci Form

AbstractIn this paper, we establish a finiteness theorem for $L^{p}$ L p harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on $L^{2}$ L 2 harmonic 1-forms.

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Noncommutative G-semihereditary rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498818500147 ◽

2018 ◽

Vol 17 (01) ◽

pp. 1850014 ◽

Cited By ~ 2

Author(s):

Jian Wang ◽

Yunxia Li ◽

Jiangsheng Hu

Keyword(s):

Associative Ring ◽

Sufficient Condition ◽

Projective Modules ◽

The Third ◽

Flat Modules ◽

Gorenstein Projective Modules ◽

Direct Limits ◽

Gorenstein Flat ◽

Finitely Presented ◽

Semihereditary Rings

In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. Some new characterizations of left [Formula: see text]-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right [Formula: see text]-module is Gorenstein flat if and only if it is Gorenstein projective provided that [Formula: see text] is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, [Formula: see text]-[Formula: see text] rings and [Formula: see text] rings.

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