Beitrage zur Algebra und Geometrie

Sera Gunn

Philipp Jell

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry

Construction of fully faithful tropicalizations for curves in ambient dimension 3

三维空间中曲线的完全忠实热带化构造

Andrés Chacón

María Camila Ramírez

Armando Reyes

Maps between schematic semi-graded rings

图式半分级环之间的映射

Abstract A ring $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> has stably free cancellation when every stably free $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> -module is free. Let $$ G \; = \; C_p \rtimes C_q $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mspace/> <mml:mo>=</mml:mo> <mml:mspace/> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>⋊</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> </mml:math> be a finite metacyclic group where p is an odd prime and q is a positive integral divisor of $$p-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . We show that the group ring $$\mathcal{R}[G]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>[</mml:mo> <mml:mi>G</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> has stably free cancellation when $$\;\mathcal{R} \; = \; {\mathbb {Z}}[t_1, t_1^{-1}, \dots t_m, t_m^{-1}, x_1, \dots x_n] \;$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mi>R</mml:mi> <mml:mspace/> <mml:mo>=</mml:mo> <mml:mspace/> <mml:mi>Z</mml:mi> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mi>t</mml:mi> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mi>t</mml:mi> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>]</mml:mo> <mml:mspace/> </mml:mrow> </mml:math> is a ring of mixed polynomials and Laurent polynomials over the integers. As a consequence, when $$C_\infty ^{(m)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>∞</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> is the free abelian group of rank m then the integral group ring $${\mathbb {Z}}[G(p,q) \times C_\infty ^{(m)}]\; $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>[</mml:mo> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>∞</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>]</mml:mo> <mml:mspace/> </mml:mrow> </mml:math> has stably free cancellation.

Abbreviation	Beitr. Algebra Geom.
Journal Impact	0.29
Quartiles(Global)	MATHEMATICS(Q4)

ISSN	0138-4821, 2191-0383
h-index	22

Publication Information	Publisher: Springer Nature，Publishing cycle: 4 issues per year，Journal Type: journal，Open Access Journals: No
Basic data	Year of publication: 2004，Proportion of original research papers: 100.00%，Self Citation Rate:0.00%， Gold OA Rate: 25.44%

Beitrage zur Algebra und Geometrie

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Construction of fully faithful tropicalizations for curves in ambient dimension 3

Maps between schematic semi-graded rings

A cancellation theorem for metacyclic group rings

Correction to: H-stability of syzygy bundles on some regular algebraic surfaces

A special issue: Computer algebra

Maps between schematic semi-graded rings

Construction of fully faithful tropicalizations for curves in ambient dimension 3

A cancellation theorem for metacyclic group rings

Correction to: H-stability of syzygy bundles on some regular algebraic surfaces

A special issue: Computer algebra