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Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">T</mi></semantics></math></inline-formula> be a compactly generated triangulated category. In this paper, the width and local homology dimension of an object <i>X</i> with respect to a homogeneous ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">a</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>width</mi><mi>R</mi></msub><mrow><mo>(</mo><mi mathvariant="fraktur">a</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>hd</mi><mo>(</mo><mi mathvariant="fraktur">a</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula>, respectively, are introduced. The local nature and some basic properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>width</mi><mi>R</mi></msub><mrow><mo>(</mo><mi mathvariant="fraktur">a</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>hd</mi><mo>(</mo><mi mathvariant="fraktur">a</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> are provided. In addition, we give an upper bound and lower bound of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>width</mi><mi>R</mi></msub><mrow><mo>(</mo><mi mathvariant="fraktur">a</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. What is more, we give the relationship between the local homology dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>hd</mi><mo>(</mo><mi mathvariant="fraktur">a</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> and the arithmetic rank of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">a</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>dim</mi><mi>R</mi></mrow></semantics></math></inline-formula>.