Can not tell colimits from limitsProjectivity of free O_X modules with respect to the sheafy hom?A left adjoint for the evaluation functor Gamma(, U)Cov. right-exact additive functors that don't commute with direct sums?Sheaf Hom and the functor HomColimits of quasi-coherent sheaves on a ringed spaceFlat and injective quasi-coherent sheavesWhat kind of ringed space $X$ has the property that a locally free sheaf is projective in Qcoh$(X)$?locally noetherian categories and the category of quasi-coherent sheaves over a noetherian schemeLifting a local section to a global section along a homomorphism of quasi-coherent sheavesBasic questions about formal schemes
Can not tell colimits from limits
Projectivity of free O_X modules with respect to the sheafy hom?A left adjoint for the evaluation functor Gamma(, U)Cov. right-exact additive functors that don't commute with direct sums?Sheaf Hom and the functor HomColimits of quasi-coherent sheaves on a ringed spaceFlat and injective quasi-coherent sheavesWhat kind of ringed space $X$ has the property that a locally free sheaf is projective in Qcoh$(X)$?locally noetherian categories and the category of quasi-coherent sheaves over a noetherian schemeLifting a local section to a global section along a homomorphism of quasi-coherent sheavesBasic questions about formal schemes
$begingroup$
Proposition 71 here reads:
Let $X$ be a concentrated scheme and $F$ a quasi-coherent sheaf of modules. The
following are equivalent:
(a) The functor $mathrmHom(F, −):Qco(X)rightarrow Ab$ preserves direct limits. In other words, $F$ is a
finitely presented object of $Qco(X)$.
(b) The functor $mathitHom(F, −) : Qco(X)rightarrow Mod(X)$ preserves direct limits.
The question is: is there a typo? My understanding is that the covariant $mathrmHom$ (not the internal one) preserves limits for any category. Should it be "colimits" instead of "limits" in the point (a)?
homological-algebra sheaf-theory schemes
asked 1 hour ago
gelfand_dominatesgelfand_dominates
111
New contributor
gelfand_dominates is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$
add a comment |
$begingroup$
Proposition 71 here reads:
Let $X$ be a concentrated scheme and $F$ a quasi-coherent sheaf of modules. The
following are equivalent:
(a) The functor $mathrmHom(F, −):Qco(X)rightarrow Ab$ preserves direct limits. In other words, $F$ is a
finitely presented object of $Qco(X)$.
(b) The functor $mathitHom(F, −) : Qco(X)rightarrow Mod(X)$ preserves direct limits.
The question is: is there a typo? My understanding is that the covariant $mathrmHom$ (not the internal one) preserves limits for any category. Should it be "colimits" instead of "limits" in the point (a)?
homological-algebra sheaf-theory schemes
asked 1 hour ago
gelfand_dominatesgelfand_dominates
111
New contributor
gelfand_dominates is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Proposition 71 here reads:
Let $X$ be a concentrated scheme and $F$ a quasi-coherent sheaf of modules. The
following are equivalent:
(a) The functor $mathrmHom(F, −):Qco(X)rightarrow Ab$ preserves direct limits. In other words, $F$ is a
finitely presented object of $Qco(X)$.
(b) The functor $mathitHom(F, −) : Qco(X)rightarrow Mod(X)$ preserves direct limits.
The question is: is there a typo? My understanding is that the covariant $mathrmHom$ (not the internal one) preserves limits for any category. Should it be "colimits" instead of "limits" in the point (a)?
homological-algebra sheaf-theory schemes
asked 1 hour ago
gelfand_dominatesgelfand_dominates
111
New contributor
gelfand_dominates is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Proposition 71 here reads:
Let $X$ be a concentrated scheme and $F$ a quasi-coherent sheaf of modules. The
following are equivalent:
(a) The functor $mathrmHom(F, −):Qco(X)rightarrow Ab$ preserves direct limits. In other words, $F$ is a
finitely presented object of $Qco(X)$.
(b) The functor $mathitHom(F, −) : Qco(X)rightarrow Mod(X)$ preserves direct limits.
The question is: is there a typo? My understanding is that the covariant $mathrmHom$ (not the internal one) preserves limits for any category. Should it be "colimits" instead of "limits" in the point (a)?
homological-algebra sheaf-theory schemes
homological-algebra sheaf-theory schemes
asked 1 hour ago
gelfand_dominatesgelfand_dominates
111
New contributor
gelfand_dominates is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
gelfand_dominatesgelfand_dominates
111
New contributor
gelfand_dominates is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
gelfand_dominatesgelfand_dominates
111
New contributor
gelfand_dominates is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
gelfand_dominatesgelfand_dominates
111
asked 1 hour ago
gelfand_dominatesgelfand_dominates
111
111
New contributor
gelfand_dominates is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
gelfand_dominates is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
EDIT: The main body of my answer is there because I didn't understand the question, so let me answer it in this edit (I'll leave the main body of the answer there, just in case)
No, there is no typo : direct limits are a special kind of colimits. The terminology for them was presumably coined before category theory defined the general notions of "limits" and "colimits". In particular, "direct limit" means "colimit over a directed system" and "inverse" or "projective limit" means "limit over a (directed system)$^op$"; thankfully, in general there isn't too much confusion about what one means in a given situation.
The rest does not adress the question, because I had misunderstood it. I'll leave it there regardless, hoping it's not too much of a problem.
No there is no typo : indeed the covariant hom always preserves limits, but when $F$ is "small enough" (here : finitely presented) it also preserves direct limits.
Here's a sketch of proof where I replace sheaves by mere modules : let $F$ be an $R$-module; then $F$ is finitely presented iff $hom(F,-)$ preserves direct limits.
Indeed if you have a directed system $(M_i)_iin I$ and a map $Fto varinjlim M_i$, the generators land in some $M_i_0$ (there's finitely many of them) and every relation is satisfied in some $M_j_0, j_0geq i_0$ because the relations are satisfied in the colimit and there's only finitely many of them too.
For the converse you may want to take the directed system of finitely generated submodules of $F$ to get "finitely generated" and then you work in a similar fashion to get finitely presented (taking quotients of the free module on the generators by finitely generated submodules of the relations for instance)
This works for modules and actually all algebraic structures so that "$hom(F,-)$ preserves direct limits" is a good generalization of "$F$ is finitely presented"
answered 1 hour ago
MaxMax
6791619
$endgroup$
2
$begingroup$
I think the OP maybe confused because the terminology kind of sucks. "Direct limits" are colimits, right?
$endgroup$
– Aknazar Kazhymurat
1 hour ago
$begingroup$
@AknazarKazhymurat yes, I'll add a word about that !
$endgroup$
– Max
1 hour ago
$begingroup$
@AknazarKazhymurat oh wait, sorry, I had completely misunderstood your point and the point of the OP !
$endgroup$
– Max
1 hour ago
add a comment |
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1 Answer
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1 Answer
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votes
$begingroup$
EDIT: The main body of my answer is there because I didn't understand the question, so let me answer it in this edit (I'll leave the main body of the answer there, just in case)
No, there is no typo : direct limits are a special kind of colimits. The terminology for them was presumably coined before category theory defined the general notions of "limits" and "colimits". In particular, "direct limit" means "colimit over a directed system" and "inverse" or "projective limit" means "limit over a (directed system)$^op$"; thankfully, in general there isn't too much confusion about what one means in a given situation.
The rest does not adress the question, because I had misunderstood it. I'll leave it there regardless, hoping it's not too much of a problem.
No there is no typo : indeed the covariant hom always preserves limits, but when $F$ is "small enough" (here : finitely presented) it also preserves direct limits.
Here's a sketch of proof where I replace sheaves by mere modules : let $F$ be an $R$-module; then $F$ is finitely presented iff $hom(F,-)$ preserves direct limits.
Indeed if you have a directed system $(M_i)_iin I$ and a map $Fto varinjlim M_i$, the generators land in some $M_i_0$ (there's finitely many of them) and every relation is satisfied in some $M_j_0, j_0geq i_0$ because the relations are satisfied in the colimit and there's only finitely many of them too.
For the converse you may want to take the directed system of finitely generated submodules of $F$ to get "finitely generated" and then you work in a similar fashion to get finitely presented (taking quotients of the free module on the generators by finitely generated submodules of the relations for instance)
This works for modules and actually all algebraic structures so that "$hom(F,-)$ preserves direct limits" is a good generalization of "$F$ is finitely presented"
answered 1 hour ago
MaxMax
6791619
$endgroup$
2
$begingroup$
I think the OP maybe confused because the terminology kind of sucks. "Direct limits" are colimits, right?
$endgroup$
– Aknazar Kazhymurat
1 hour ago
$begingroup$
@AknazarKazhymurat yes, I'll add a word about that !
$endgroup$
– Max
1 hour ago
$begingroup$
@AknazarKazhymurat oh wait, sorry, I had completely misunderstood your point and the point of the OP !
$endgroup$
– Max
1 hour ago
add a comment |
$begingroup$
EDIT: The main body of my answer is there because I didn't understand the question, so let me answer it in this edit (I'll leave the main body of the answer there, just in case)
No, there is no typo : direct limits are a special kind of colimits. The terminology for them was presumably coined before category theory defined the general notions of "limits" and "colimits". In particular, "direct limit" means "colimit over a directed system" and "inverse" or "projective limit" means "limit over a (directed system)$^op$"; thankfully, in general there isn't too much confusion about what one means in a given situation.
The rest does not adress the question, because I had misunderstood it. I'll leave it there regardless, hoping it's not too much of a problem.
No there is no typo : indeed the covariant hom always preserves limits, but when $F$ is "small enough" (here : finitely presented) it also preserves direct limits.
Here's a sketch of proof where I replace sheaves by mere modules : let $F$ be an $R$-module; then $F$ is finitely presented iff $hom(F,-)$ preserves direct limits.
Indeed if you have a directed system $(M_i)_iin I$ and a map $Fto varinjlim M_i$, the generators land in some $M_i_0$ (there's finitely many of them) and every relation is satisfied in some $M_j_0, j_0geq i_0$ because the relations are satisfied in the colimit and there's only finitely many of them too.
For the converse you may want to take the directed system of finitely generated submodules of $F$ to get "finitely generated" and then you work in a similar fashion to get finitely presented (taking quotients of the free module on the generators by finitely generated submodules of the relations for instance)
This works for modules and actually all algebraic structures so that "$hom(F,-)$ preserves direct limits" is a good generalization of "$F$ is finitely presented"
answered 1 hour ago
MaxMax
6791619
$endgroup$
2
$begingroup$
I think the OP maybe confused because the terminology kind of sucks. "Direct limits" are colimits, right?
$endgroup$
– Aknazar Kazhymurat
1 hour ago
$begingroup$
@AknazarKazhymurat yes, I'll add a word about that !
$endgroup$
– Max
1 hour ago
$begingroup$
@AknazarKazhymurat oh wait, sorry, I had completely misunderstood your point and the point of the OP !
$endgroup$
– Max
1 hour ago
add a comment |
$begingroup$
EDIT: The main body of my answer is there because I didn't understand the question, so let me answer it in this edit (I'll leave the main body of the answer there, just in case)
No, there is no typo : direct limits are a special kind of colimits. The terminology for them was presumably coined before category theory defined the general notions of "limits" and "colimits". In particular, "direct limit" means "colimit over a directed system" and "inverse" or "projective limit" means "limit over a (directed system)$^op$"; thankfully, in general there isn't too much confusion about what one means in a given situation.
The rest does not adress the question, because I had misunderstood it. I'll leave it there regardless, hoping it's not too much of a problem.
No there is no typo : indeed the covariant hom always preserves limits, but when $F$ is "small enough" (here : finitely presented) it also preserves direct limits.
Here's a sketch of proof where I replace sheaves by mere modules : let $F$ be an $R$-module; then $F$ is finitely presented iff $hom(F,-)$ preserves direct limits.
Indeed if you have a directed system $(M_i)_iin I$ and a map $Fto varinjlim M_i$, the generators land in some $M_i_0$ (there's finitely many of them) and every relation is satisfied in some $M_j_0, j_0geq i_0$ because the relations are satisfied in the colimit and there's only finitely many of them too.
For the converse you may want to take the directed system of finitely generated submodules of $F$ to get "finitely generated" and then you work in a similar fashion to get finitely presented (taking quotients of the free module on the generators by finitely generated submodules of the relations for instance)
This works for modules and actually all algebraic structures so that "$hom(F,-)$ preserves direct limits" is a good generalization of "$F$ is finitely presented"
answered 1 hour ago
MaxMax
6791619
$endgroup$
EDIT: The main body of my answer is there because I didn't understand the question, so let me answer it in this edit (I'll leave the main body of the answer there, just in case)
No, there is no typo : direct limits are a special kind of colimits. The terminology for them was presumably coined before category theory defined the general notions of "limits" and "colimits". In particular, "direct limit" means "colimit over a directed system" and "inverse" or "projective limit" means "limit over a (directed system)$^op$"; thankfully, in general there isn't too much confusion about what one means in a given situation.
The rest does not adress the question, because I had misunderstood it. I'll leave it there regardless, hoping it's not too much of a problem.
No there is no typo : indeed the covariant hom always preserves limits, but when $F$ is "small enough" (here : finitely presented) it also preserves direct limits.
Here's a sketch of proof where I replace sheaves by mere modules : let $F$ be an $R$-module; then $F$ is finitely presented iff $hom(F,-)$ preserves direct limits.
Indeed if you have a directed system $(M_i)_iin I$ and a map $Fto varinjlim M_i$, the generators land in some $M_i_0$ (there's finitely many of them) and every relation is satisfied in some $M_j_0, j_0geq i_0$ because the relations are satisfied in the colimit and there's only finitely many of them too.
For the converse you may want to take the directed system of finitely generated submodules of $F$ to get "finitely generated" and then you work in a similar fashion to get finitely presented (taking quotients of the free module on the generators by finitely generated submodules of the relations for instance)
This works for modules and actually all algebraic structures so that "$hom(F,-)$ preserves direct limits" is a good generalization of "$F$ is finitely presented"
answered 1 hour ago
MaxMax
6791619
edited 1 hour ago
answered 1 hour ago
MaxMax
6791619
answered 1 hour ago
MaxMax
6791619
answered 1 hour ago
MaxMax
6791619
6791619
2
$begingroup$
I think the OP maybe confused because the terminology kind of sucks. "Direct limits" are colimits, right?
$endgroup$
– Aknazar Kazhymurat
1 hour ago
$begingroup$
@AknazarKazhymurat yes, I'll add a word about that !
$endgroup$
– Max
1 hour ago
$begingroup$
@AknazarKazhymurat oh wait, sorry, I had completely misunderstood your point and the point of the OP !
$endgroup$
– Max
1 hour ago
add a comment |
2
$begingroup$
I think the OP maybe confused because the terminology kind of sucks. "Direct limits" are colimits, right?
$endgroup$
– Aknazar Kazhymurat
1 hour ago
$begingroup$
@AknazarKazhymurat yes, I'll add a word about that !
$endgroup$
– Max
1 hour ago
$begingroup$
@AknazarKazhymurat oh wait, sorry, I had completely misunderstood your point and the point of the OP !
$endgroup$
– Max
1 hour ago
2
2
$begingroup$
I think the OP maybe confused because the terminology kind of sucks. "Direct limits" are colimits, right?
$endgroup$
– Aknazar Kazhymurat
1 hour ago
$begingroup$
I think the OP maybe confused because the terminology kind of sucks. "Direct limits" are colimits, right?
$endgroup$
– Aknazar Kazhymurat
1 hour ago
$begingroup$
@AknazarKazhymurat yes, I'll add a word about that !
$endgroup$
– Max
1 hour ago
$begingroup$
@AknazarKazhymurat yes, I'll add a word about that !
$endgroup$
– Max
1 hour ago
$begingroup$
@AknazarKazhymurat oh wait, sorry, I had completely misunderstood your point and the point of the OP !
$endgroup$
– Max
1 hour ago
$begingroup$
@AknazarKazhymurat oh wait, sorry, I had completely misunderstood your point and the point of the OP !
$endgroup$
– Max
1 hour ago
add a comment |
gelfand_dominates is a new contributor. Be nice, and check out our Code of Conduct.
gelfand_dominates is a new contributor. Be nice, and check out our Code of Conduct.
gelfand_dominates is a new contributor. Be nice, and check out our Code of Conduct.
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