Derived Morita equivalence of associative algebras

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An associative algebra $A$ is said to be Morita equivalent to another one $B$ if there is an equivalence $$mathsfMod_Asimeq mathsfMod_B$$ between its corresponding abelian categories of modules. Moreover, whenever $A$ and $B$ are commutative, they are Morita equivalent iff they are isomorphic. On the other hand, we say that $A$ is derived Morita equivalent to $B$ if there is an equivalence $$D(A)simeq D(B)$$ between its triangulated derived categories. My question is:



Q. If $A$ is commutative and derived Morita equivelent to $B$, is $A$ Morita equivalent to $B$?










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    An associative algebra $A$ is said to be Morita equivalent to another one $B$ if there is an equivalence $$mathsfMod_Asimeq mathsfMod_B$$ between its corresponding abelian categories of modules. Moreover, whenever $A$ and $B$ are commutative, they are Morita equivalent iff they are isomorphic. On the other hand, we say that $A$ is derived Morita equivalent to $B$ if there is an equivalence $$D(A)simeq D(B)$$ between its triangulated derived categories. My question is:



    Q. If $A$ is commutative and derived Morita equivelent to $B$, is $A$ Morita equivalent to $B$?










    share|cite|improve this question























      up vote
      5
      down vote

      favorite









      up vote
      5
      down vote

      favorite











      An associative algebra $A$ is said to be Morita equivalent to another one $B$ if there is an equivalence $$mathsfMod_Asimeq mathsfMod_B$$ between its corresponding abelian categories of modules. Moreover, whenever $A$ and $B$ are commutative, they are Morita equivalent iff they are isomorphic. On the other hand, we say that $A$ is derived Morita equivalent to $B$ if there is an equivalence $$D(A)simeq D(B)$$ between its triangulated derived categories. My question is:



      Q. If $A$ is commutative and derived Morita equivelent to $B$, is $A$ Morita equivalent to $B$?










      share|cite|improve this question













      An associative algebra $A$ is said to be Morita equivalent to another one $B$ if there is an equivalence $$mathsfMod_Asimeq mathsfMod_B$$ between its corresponding abelian categories of modules. Moreover, whenever $A$ and $B$ are commutative, they are Morita equivalent iff they are isomorphic. On the other hand, we say that $A$ is derived Morita equivalent to $B$ if there is an equivalence $$D(A)simeq D(B)$$ between its triangulated derived categories. My question is:



      Q. If $A$ is commutative and derived Morita equivelent to $B$, is $A$ Morita equivalent to $B$?







      homological-algebra derived-categories derived-algebraic-geometry






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      asked Aug 29 at 6:28









      Enrique Becerra

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          This seems to be true by https://arxiv.org/pdf/math/9810134.pdf , theorem 2.7.






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          1 Answer
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          1 Answer
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          active

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          active

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          up vote
          5
          down vote



          accepted










          This seems to be true by https://arxiv.org/pdf/math/9810134.pdf , theorem 2.7.






          share|cite|improve this answer




















          • cambridge.org/core/journals/… for non-pdf version
            – AHusain
            Aug 29 at 6:57














          up vote
          5
          down vote



          accepted










          This seems to be true by https://arxiv.org/pdf/math/9810134.pdf , theorem 2.7.






          share|cite|improve this answer




















          • cambridge.org/core/journals/… for non-pdf version
            – AHusain
            Aug 29 at 6:57












          up vote
          5
          down vote



          accepted







          up vote
          5
          down vote



          accepted






          This seems to be true by https://arxiv.org/pdf/math/9810134.pdf , theorem 2.7.






          share|cite|improve this answer












          This seems to be true by https://arxiv.org/pdf/math/9810134.pdf , theorem 2.7.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 29 at 6:44









          Mare

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          2,4662926











          • cambridge.org/core/journals/… for non-pdf version
            – AHusain
            Aug 29 at 6:57
















          • cambridge.org/core/journals/… for non-pdf version
            – AHusain
            Aug 29 at 6:57















          cambridge.org/core/journals/… for non-pdf version
          – AHusain
          Aug 29 at 6:57




          cambridge.org/core/journals/… for non-pdf version
          – AHusain
          Aug 29 at 6:57

















           

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