Two questions about the functor $DHom_Gamma(-,I)$
Clash Royale CLAN TAG#URR8PPP
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 $modEnd_Gamma(I)?$
modules representation-theory homological-algebra
add a comment |Â
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 $modEnd_Gamma(I)?$
modules representation-theory homological-algebra
add a comment |Â
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 $modEnd_Gamma(I)?$
modules representation-theory homological-algebra
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
modules representation-theory homological-algebra
edited Sep 4 at 8:41
Bernard
112k635104
112k635104
asked Sep 4 at 8:17
Xiaosong Peng
651414
651414
add a comment |Â
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2904778%2ftwo-questions-about-the-functor-dhom-gamma-i%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password