Exactness of direct image functor of presheaves
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Suppose $f:Xlongrightarrow Y$ is a morphism of schemes. Take the categories $mathbfX_et,,mathbfY_et$ of étale morphisms over $X$ and $Y$. Then is the direct image functor:
$f_*:mathbfPSh(mathbfX_et)longrightarrowmathbfPSh(mathbfY_et)$
on category of presheaves exact? I know it's left-exact if restricted to the category of sheaves on étale site, but what does exactness mean in the category of presheaves on a site?
algebraic-geometry category-theory etale-cohomology grothendieck-topologies
edited Mar 18 '17 at 13:22
Armando j18eos
2,43611226
asked Mar 18 '17 at 9:58
Pavle Papunashvili
4616
add a comment |Â
up vote
1
down vote
favorite
Suppose $f:Xlongrightarrow Y$ is a morphism of schemes. Take the categories $mathbfX_et,,mathbfY_et$ of étale morphisms over $X$ and $Y$. Then is the direct image functor:
$f_*:mathbfPSh(mathbfX_et)longrightarrowmathbfPSh(mathbfY_et)$
on category of presheaves exact? I know it's left-exact if restricted to the category of sheaves on étale site, but what does exactness mean in the category of presheaves on a site?
algebraic-geometry category-theory etale-cohomology grothendieck-topologies
edited Mar 18 '17 at 13:22
Armando j18eos
2,43611226
asked Mar 18 '17 at 9:58
Pavle Papunashvili
4616
1
Exactness means what it usually does, i.e. preservation of short exact sequences. A short exact sequence of presheaves is just a short exact sequence at every value.
– Kevin Carlson
Mar 18 '17 at 17:09
Yes but in case of sheaves because the etale site is small, we have a good characterisation of epimorphism: ncatlab.org/nlab/show/category+of+sheaves#EpiMonoIsomorphisms
– Pavle Papunashvili
Mar 18 '17 at 20:19
1
Ah, but an epimorphism of presheaves is just a map which is an epimorphism at every level-exactly dual to a monomorphism.
– Kevin Carlson
Mar 19 '17 at 1:38
Ah, thanks. But is the functor exact? It came up in the proof that $Rf_*G(U)$ is the sheaf associated to $H_et^q(U,G))$ and without exactness of $f_*$ the proof falls apart.
– Pavle Papunashvili
Mar 19 '17 at 7:20
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose $f:Xlongrightarrow Y$ is a morphism of schemes. Take the categories $mathbfX_et,,mathbfY_et$ of étale morphisms over $X$ and $Y$. Then is the direct image functor:
$f_*:mathbfPSh(mathbfX_et)longrightarrowmathbfPSh(mathbfY_et)$
on category of presheaves exact? I know it's left-exact if restricted to the category of sheaves on étale site, but what does exactness mean in the category of presheaves on a site?
algebraic-geometry category-theory etale-cohomology grothendieck-topologies
edited Mar 18 '17 at 13:22
Armando j18eos
2,43611226
asked Mar 18 '17 at 9:58
Pavle Papunashvili
4616
Suppose $f:Xlongrightarrow Y$ is a morphism of schemes. Take the categories $mathbfX_et,,mathbfY_et$ of étale morphisms over $X$ and $Y$. Then is the direct image functor:
$f_*:mathbfPSh(mathbfX_et)longrightarrowmathbfPSh(mathbfY_et)$
on category of presheaves exact? I know it's left-exact if restricted to the category of sheaves on étale site, but what does exactness mean in the category of presheaves on a site?
algebraic-geometry category-theory etale-cohomology grothendieck-topologies
edited Mar 18 '17 at 13:22
Armando j18eos
2,43611226
asked Mar 18 '17 at 9:58
Pavle Papunashvili
4616
edited Mar 18 '17 at 13:22
Armando j18eos
2,43611226
edited Mar 18 '17 at 13:22
Armando j18eos
2,43611226
edited Mar 18 '17 at 13:22
Armando j18eos
2,43611226
2,43611226
asked Mar 18 '17 at 9:58
Pavle Papunashvili
4616
asked Mar 18 '17 at 9:58
Pavle Papunashvili
4616
asked Mar 18 '17 at 9:58
Pavle Papunashvili
4616
4616
1
Exactness means what it usually does, i.e. preservation of short exact sequences. A short exact sequence of presheaves is just a short exact sequence at every value.
– Kevin Carlson
Mar 18 '17 at 17:09
Yes but in case of sheaves because the etale site is small, we have a good characterisation of epimorphism: ncatlab.org/nlab/show/category+of+sheaves#EpiMonoIsomorphisms
– Pavle Papunashvili
Mar 18 '17 at 20:19
1
Ah, but an epimorphism of presheaves is just a map which is an epimorphism at every level-exactly dual to a monomorphism.
– Kevin Carlson
Mar 19 '17 at 1:38
Ah, thanks. But is the functor exact? It came up in the proof that $Rf_*G(U)$ is the sheaf associated to $H_et^q(U,G))$ and without exactness of $f_*$ the proof falls apart.
– Pavle Papunashvili
Mar 19 '17 at 7:20
add a comment |Â
1
Exactness means what it usually does, i.e. preservation of short exact sequences. A short exact sequence of presheaves is just a short exact sequence at every value.
– Kevin Carlson
Mar 18 '17 at 17:09
Yes but in case of sheaves because the etale site is small, we have a good characterisation of epimorphism: ncatlab.org/nlab/show/category+of+sheaves#EpiMonoIsomorphisms
– Pavle Papunashvili
Mar 18 '17 at 20:19
1
Ah, but an epimorphism of presheaves is just a map which is an epimorphism at every level-exactly dual to a monomorphism.
– Kevin Carlson
Mar 19 '17 at 1:38
Ah, thanks. But is the functor exact? It came up in the proof that $Rf_*G(U)$ is the sheaf associated to $H_et^q(U,G))$ and without exactness of $f_*$ the proof falls apart.
– Pavle Papunashvili
Mar 19 '17 at 7:20
1
1
Exactness means what it usually does, i.e. preservation of short exact sequences. A short exact sequence of presheaves is just a short exact sequence at every value.
– Kevin Carlson
Mar 18 '17 at 17:09
Exactness means what it usually does, i.e. preservation of short exact sequences. A short exact sequence of presheaves is just a short exact sequence at every value.
– Kevin Carlson
Mar 18 '17 at 17:09
Yes but in case of sheaves because the etale site is small, we have a good characterisation of epimorphism: ncatlab.org/nlab/show/category+of+sheaves#EpiMonoIsomorphisms
– Pavle Papunashvili
Mar 18 '17 at 20:19
Yes but in case of sheaves because the etale site is small, we have a good characterisation of epimorphism: ncatlab.org/nlab/show/category+of+sheaves#EpiMonoIsomorphisms
– Pavle Papunashvili
Mar 18 '17 at 20:19
1
1
Ah, but an epimorphism of presheaves is just a map which is an epimorphism at every level-exactly dual to a monomorphism.
– Kevin Carlson
Mar 19 '17 at 1:38
Ah, but an epimorphism of presheaves is just a map which is an epimorphism at every level-exactly dual to a monomorphism.
– Kevin Carlson
Mar 19 '17 at 1:38
Ah, thanks. But is the functor exact? It came up in the proof that $Rf_*G(U)$ is the sheaf associated to $H_et^q(U,G))$ and without exactness of $f_*$ the proof falls apart.
– Pavle Papunashvili
Mar 19 '17 at 7:20
Ah, thanks. But is the functor exact? It came up in the proof that $Rf_*G(U)$ is the sheaf associated to $H_et^q(U,G))$ and without exactness of $f_*$ the proof falls apart.
– Pavle Papunashvili
Mar 19 '17 at 7:20
add a comment |Â
1 Answer
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The answer to your question is yes.
Exactness for presheaves on a site means that for every object $U$ in the topology, applying the functor $Gamma(U,-|_U)$ is exact. The direct image of a presheaf $mathscr F$ is the presheaf $UmapstoGamma(U_X,-|_U_X)$, where $U_X=Utimes_Y X$. It's now tautological that direct image is an exact functor on presheaves.
answered Aug 10 at 19:12
Owen Barrett
982814
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The answer to your question is yes.
Exactness for presheaves on a site means that for every object $U$ in the topology, applying the functor $Gamma(U,-|_U)$ is exact. The direct image of a presheaf $mathscr F$ is the presheaf $UmapstoGamma(U_X,-|_U_X)$, where $U_X=Utimes_Y X$. It's now tautological that direct image is an exact functor on presheaves.
answered Aug 10 at 19:12
Owen Barrett
982814
add a comment |Â
up vote
0
down vote
The answer to your question is yes.
Exactness for presheaves on a site means that for every object $U$ in the topology, applying the functor $Gamma(U,-|_U)$ is exact. The direct image of a presheaf $mathscr F$ is the presheaf $UmapstoGamma(U_X,-|_U_X)$, where $U_X=Utimes_Y X$. It's now tautological that direct image is an exact functor on presheaves.
answered Aug 10 at 19:12
Owen Barrett
982814
add a comment |Â
up vote
0
down vote
up vote
0
down vote
The answer to your question is yes.
Exactness for presheaves on a site means that for every object $U$ in the topology, applying the functor $Gamma(U,-|_U)$ is exact. The direct image of a presheaf $mathscr F$ is the presheaf $UmapstoGamma(U_X,-|_U_X)$, where $U_X=Utimes_Y X$. It's now tautological that direct image is an exact functor on presheaves.
answered Aug 10 at 19:12
Owen Barrett
982814
The answer to your question is yes.
Exactness for presheaves on a site means that for every object $U$ in the topology, applying the functor $Gamma(U,-|_U)$ is exact. The direct image of a presheaf $mathscr F$ is the presheaf $UmapstoGamma(U_X,-|_U_X)$, where $U_X=Utimes_Y X$. It's now tautological that direct image is an exact functor on presheaves.
answered Aug 10 at 19:12
Owen Barrett
982814
answered Aug 10 at 19:12
Owen Barrett
982814
answered Aug 10 at 19:12
Owen Barrett
982814
answered Aug 10 at 19:12
Owen Barrett
982814
982814
add a comment |Â
add a comment |Â
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1
Exactness means what it usually does, i.e. preservation of short exact sequences. A short exact sequence of presheaves is just a short exact sequence at every value.
– Kevin Carlson
Mar 18 '17 at 17:09
Yes but in case of sheaves because the etale site is small, we have a good characterisation of epimorphism: ncatlab.org/nlab/show/category+of+sheaves#EpiMonoIsomorphisms
– Pavle Papunashvili
Mar 18 '17 at 20:19
1
Ah, but an epimorphism of presheaves is just a map which is an epimorphism at every level-exactly dual to a monomorphism.
– Kevin Carlson
Mar 19 '17 at 1:38
Ah, thanks. But is the functor exact? It came up in the proof that $Rf_*G(U)$ is the sheaf associated to $H_et^q(U,G))$ and without exactness of $f_*$ the proof falls apart.
– Pavle Papunashvili
Mar 19 '17 at 7:20