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• W2014648567 abstract "In this note we study the behavior of the dimension of the perfect derived category <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P e r f left-parenthesis upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>Perf</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {Perf}(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a dg-algebra <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over a field <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under a base field extension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K slash k> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>K/k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, we show that the dimension of a perfect derived category is invariant under a separable algebraic extension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K slash k> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>K/k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As an application we prove the following statement: Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a self-injective algebra over a perfect field <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If the dimension of the stable category <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=ModifyingBelow mod With bar upper A> <mml:semantics> <mml:mrow> <mml:munder> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>mod</mml:mtext> </mml:mrow> <mml:mo>_<!-- _ --></mml:mo> </mml:munder> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>underline {textrm {mod}}A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding=application/x-tex>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is of finite representation type. This theorem is proved by M. Yoshiwaki in the case when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an algebraically closed field. Our proof depends on his result." @default.
• W2014648567 created "2016-06-24" @default.
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• W2014648567 date "2013-09-06" @default.
• W2014648567 modified "2023-09-26" @default.
• W2014648567 title "A note on dimension of triangulated categories" @default.
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• W2014648567 doi "https://doi.org/10.1090/s0002-9939-2013-11723-5" @default.
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