Relative Gorenstein Dimensions over Triangular Matrix Rings
Let <i>A</i> and <i>B</i> be rings, <i>U</i> a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><...
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| Autores principales: | , , , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/93474e7f450d4fe8815f85062b920e59 |
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| Sumario: | Let <i>A</i> and <i>B</i> be rings, <i>U</i> a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-bimodule, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>=</mo><mfenced open="(" close=")"><mtable><mtr><mtd><mi>A</mi></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mi>U</mi></mtd><mtd><mi>B</mi></mtd></mtr></mtable></mfenced></mrow></semantics></math></inline-formula> the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over <i>T</i> using the corresponding ones over <i>A</i> and <i>B</i>. We show that when <i>U</i> is relative (weakly) compatible, we are able to describe the structure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mi>C</mi></msub></semantics></math></inline-formula>-projective modules over <i>T</i>. As an application, we study when a morphism in <i>T</i>-Mod is a special <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>C</mi></msub><mi>P</mi><mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>-precover and when the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>C</mi></msub><mi>P</mi><mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> is a special precovering class. In addition, we study the relative global dimension of <i>T</i>. In some cases, we show that it can be computed from the relative global dimensions of <i>A</i> and <i>B</i>. We end the paper with a counterexample to a result that characterizes when a <i>T</i>-module has a finite projective dimension. |
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