Two questions about the functor $DHom_Gamma(-,I)$

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Let $Gamma$ be an Artin algebra with standard duality $D$. $I$ is the injective envelop of $Gamma$ as $Gamma$-module. Let $S$ be a set consisting of $Gamma$-modules such that $X in S$ iff there is an exact sequence $X rightarrow B_0 rightarrow B_1$ with $B_0,B_1 in add (I)$. Then



$(1)$ How to show that $DHom_Gamma(-,I): S rightarrow mod End_Gamma(I)$ is an equivalence? (I can not get the inverse)



$(2)$ If $Gamma in S$, how to get that $DHom_Gamma(Gamma,I)cong DI$ is a generator and cogenerator of $modEnd_Gamma(I)?$










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    Let $Gamma$ be an Artin algebra with standard duality $D$. $I$ is the injective envelop of $Gamma$ as $Gamma$-module. Let $S$ be a set consisting of $Gamma$-modules such that $X in S$ iff there is an exact sequence $X rightarrow B_0 rightarrow B_1$ with $B_0,B_1 in add (I)$. Then



    $(1)$ How to show that $DHom_Gamma(-,I): S rightarrow mod End_Gamma(I)$ is an equivalence? (I can not get the inverse)



    $(2)$ If $Gamma in S$, how to get that $DHom_Gamma(Gamma,I)cong DI$ is a generator and cogenerator of $modEnd_Gamma(I)?$










    share|cite|improve this question

























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Let $Gamma$ be an Artin algebra with standard duality $D$. $I$ is the injective envelop of $Gamma$ as $Gamma$-module. Let $S$ be a set consisting of $Gamma$-modules such that $X in S$ iff there is an exact sequence $X rightarrow B_0 rightarrow B_1$ with $B_0,B_1 in add (I)$. Then



      $(1)$ How to show that $DHom_Gamma(-,I): S rightarrow mod End_Gamma(I)$ is an equivalence? (I can not get the inverse)



      $(2)$ If $Gamma in S$, how to get that $DHom_Gamma(Gamma,I)cong DI$ is a generator and cogenerator of $modEnd_Gamma(I)?$










      share|cite|improve this question















      Let $Gamma$ be an Artin algebra with standard duality $D$. $I$ is the injective envelop of $Gamma$ as $Gamma$-module. Let $S$ be a set consisting of $Gamma$-modules such that $X in S$ iff there is an exact sequence $X rightarrow B_0 rightarrow B_1$ with $B_0,B_1 in add (I)$. Then



      $(1)$ How to show that $DHom_Gamma(-,I): S rightarrow mod End_Gamma(I)$ is an equivalence? (I can not get the inverse)



      $(2)$ If $Gamma in S$, how to get that $DHom_Gamma(Gamma,I)cong DI$ is a generator and cogenerator of $modEnd_Gamma(I)?$







      modules representation-theory homological-algebra






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      edited Sep 4 at 8:41









      Bernard

      112k635104




      112k635104










      asked Sep 4 at 8:17









      Xiaosong Peng

      651414




      651414

























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