Where is the finiteness of product used in this proposition from Hartshorne?
$begingroup$
See this question: Link
I have exactly the same question, but I feel none of the questions explain why the proof fails in the infinite case. I am not looking for a counterexample.
I have two related questions too:
1) Is an arbitrary product(respectively, coproduct) of quasicoherent sheaves on a scheme quasicoherent?
2) Is a finite product of quasicoherent sheaves on a scheme quasicoherent?
The products and coproducts are in the category of $mathcal{O}_X$ modules, NOT that of quasicoherent sheaves. I think I have proven 2), but I'm not sure.
algebraic-geometry sheaf-theory quasicoherent-sheaves
asked Jan 8 at 4:26
Jehu314Jehu314
1549
$endgroup$
add a comment |
$begingroup$
See this question: Link
I have exactly the same question, but I feel none of the questions explain why the proof fails in the infinite case. I am not looking for a counterexample.
I have two related questions too:
1) Is an arbitrary product(respectively, coproduct) of quasicoherent sheaves on a scheme quasicoherent?
2) Is a finite product of quasicoherent sheaves on a scheme quasicoherent?
The products and coproducts are in the category of $mathcal{O}_X$ modules, NOT that of quasicoherent sheaves. I think I have proven 2), but I'm not sure.
algebraic-geometry sheaf-theory quasicoherent-sheaves
asked Jan 8 at 4:26
Jehu314Jehu314
1549
$endgroup$
add a comment |
$begingroup$
See this question: Link
I have exactly the same question, but I feel none of the questions explain why the proof fails in the infinite case. I am not looking for a counterexample.
I have two related questions too:
1) Is an arbitrary product(respectively, coproduct) of quasicoherent sheaves on a scheme quasicoherent?
2) Is a finite product of quasicoherent sheaves on a scheme quasicoherent?
The products and coproducts are in the category of $mathcal{O}_X$ modules, NOT that of quasicoherent sheaves. I think I have proven 2), but I'm not sure.
algebraic-geometry sheaf-theory quasicoherent-sheaves
asked Jan 8 at 4:26
Jehu314Jehu314
1549
$endgroup$
See this question: Link
I have exactly the same question, but I feel none of the questions explain why the proof fails in the infinite case. I am not looking for a counterexample.
I have two related questions too:
1) Is an arbitrary product(respectively, coproduct) of quasicoherent sheaves on a scheme quasicoherent?
2) Is a finite product of quasicoherent sheaves on a scheme quasicoherent?
The products and coproducts are in the category of $mathcal{O}_X$ modules, NOT that of quasicoherent sheaves. I think I have proven 2), but I'm not sure.
algebraic-geometry sheaf-theory quasicoherent-sheaves
algebraic-geometry sheaf-theory quasicoherent-sheaves
asked Jan 8 at 4:26
Jehu314Jehu314
1549
asked Jan 8 at 4:26
Jehu314Jehu314
1549
asked Jan 8 at 4:26
Jehu314Jehu314
1549
asked Jan 8 at 4:26
Jehu314Jehu314
1549
asked Jan 8 at 4:26
Jehu314Jehu314
1549
1549
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
For the questions listed in the body of your post:
In the infinite case: for infinite coproducts over reasonably nice spaces, this is easy. Over a noetherian scheme (or more generally a noetherian topological space) one may check that the direct sum of the local presentations for each quasi-coherent sheaf in the coproduct add together to form local presentations for the direct sum. (Over a non-noetherian topological space, one must take the presheaf given by $Umapsto bigoplus_{iin I}mathcal{F}_i(U)$ and sheafify.) For products, things are more complicated. Quasi-coherent sheaves on a scheme is a Grothendieck Abelian Category, and in particular has arbitrary products. See here for references to Gabber's proof of this fact.
For the finite case: Yes, and this may be deduced from the fact that finite products and coproducts are equivalent in abelian categories like the category of quasi-coherent sheaves on a scheme (and then one applies the argument involving coproducts from the first paragraph).
To address the issue that comes up in the linked proof, the problem is that the listed exact sequence with direct sums should really be a sequence of direct products, which are not necessarily equivalent to direct sums except when the index set is finite.
answered Jan 8 at 4:58
KReiserKReiser
10.1k21435
$endgroup$
$begingroup$
Thanks, but I was asking about products in the category of Sheaves over a scheme, not the category of QUASI-COHERENT sheaves.
$endgroup$
– Jehu314
Jan 8 at 8:05
$begingroup$
Also, I don't quite understand your statement about infinite coproducts. Each point has an open neighbourhood on which the sheaf has a presentation, but if there are infinitely many sheaves in the coproduct, why is there necessarily an open neighbourhood on which ALL the sheaves have a presentation?
$endgroup$
– Jehu314
Jan 8 at 8:15
1
$begingroup$
I had hastily omitted a condition: one needs a noetherian topological space for infinite coproducts to avoid sheafifying. And as long as one is working in sheaves of $mathcal{O}_X$ modules, there is no need to adjust anything here - quasicoherent sheaves are a full subcategory of $mathcal{O}_X$ modules so any categorical construction made in the smaller category is automatically the correct construction in the larger.
$endgroup$
– KReiser
Jan 8 at 8:35
1
$begingroup$
Yes to the first. After doing a bit of reading, it appears that the product side is perhaps more complicated than I originally thought (and one should entirely ignore the last sentence of my previous comment, it is incorrect). You'll want to go and have a look at Thomason and Trobaugh "Higher algebraic K-theory of schemes and of derived categories" in The Grothendieck Festschrift, Vol. III, 247--435, Progr. Math., 88, Birkhäuser, Boston, 1990. (MR11069118) for probably the best reference of figuring this out. Either way, it is surprising to me that you are worrying so much about...
$endgroup$
– KReiser
Jan 8 at 10:16
1
$begingroup$
... this interaction between qc sheaves and arbitrary sheaves this (apparently) early in your algebraic geometry career. The reason qc sheaves of modules are worked with is that they behave reasonably nicely - until you have a specific research problem involving some sheaf which is not qc, it will save you some amount of headaches if you do not worry to hard about them (similar to the way in which one does not worry too much about very large groups until one has to).
$endgroup$
– KReiser
Jan 8 at 10:20
|
show 3 more comments
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$begingroup$
For the questions listed in the body of your post:
In the infinite case: for infinite coproducts over reasonably nice spaces, this is easy. Over a noetherian scheme (or more generally a noetherian topological space) one may check that the direct sum of the local presentations for each quasi-coherent sheaf in the coproduct add together to form local presentations for the direct sum. (Over a non-noetherian topological space, one must take the presheaf given by $Umapsto bigoplus_{iin I}mathcal{F}_i(U)$ and sheafify.) For products, things are more complicated. Quasi-coherent sheaves on a scheme is a Grothendieck Abelian Category, and in particular has arbitrary products. See here for references to Gabber's proof of this fact.
For the finite case: Yes, and this may be deduced from the fact that finite products and coproducts are equivalent in abelian categories like the category of quasi-coherent sheaves on a scheme (and then one applies the argument involving coproducts from the first paragraph).
To address the issue that comes up in the linked proof, the problem is that the listed exact sequence with direct sums should really be a sequence of direct products, which are not necessarily equivalent to direct sums except when the index set is finite.
answered Jan 8 at 4:58
KReiserKReiser
10.1k21435
$endgroup$
$begingroup$
Thanks, but I was asking about products in the category of Sheaves over a scheme, not the category of QUASI-COHERENT sheaves.
$endgroup$
– Jehu314
Jan 8 at 8:05
$begingroup$
Also, I don't quite understand your statement about infinite coproducts. Each point has an open neighbourhood on which the sheaf has a presentation, but if there are infinitely many sheaves in the coproduct, why is there necessarily an open neighbourhood on which ALL the sheaves have a presentation?
$endgroup$
– Jehu314
Jan 8 at 8:15
1
$begingroup$
I had hastily omitted a condition: one needs a noetherian topological space for infinite coproducts to avoid sheafifying. And as long as one is working in sheaves of $mathcal{O}_X$ modules, there is no need to adjust anything here - quasicoherent sheaves are a full subcategory of $mathcal{O}_X$ modules so any categorical construction made in the smaller category is automatically the correct construction in the larger.
$endgroup$
– KReiser
Jan 8 at 8:35
1
$begingroup$
Yes to the first. After doing a bit of reading, it appears that the product side is perhaps more complicated than I originally thought (and one should entirely ignore the last sentence of my previous comment, it is incorrect). You'll want to go and have a look at Thomason and Trobaugh "Higher algebraic K-theory of schemes and of derived categories" in The Grothendieck Festschrift, Vol. III, 247--435, Progr. Math., 88, Birkhäuser, Boston, 1990. (MR11069118) for probably the best reference of figuring this out. Either way, it is surprising to me that you are worrying so much about...
$endgroup$
– KReiser
Jan 8 at 10:16
1
$begingroup$
... this interaction between qc sheaves and arbitrary sheaves this (apparently) early in your algebraic geometry career. The reason qc sheaves of modules are worked with is that they behave reasonably nicely - until you have a specific research problem involving some sheaf which is not qc, it will save you some amount of headaches if you do not worry to hard about them (similar to the way in which one does not worry too much about very large groups until one has to).
$endgroup$
– KReiser
Jan 8 at 10:20
|
show 3 more comments
$begingroup$
For the questions listed in the body of your post:
In the infinite case: for infinite coproducts over reasonably nice spaces, this is easy. Over a noetherian scheme (or more generally a noetherian topological space) one may check that the direct sum of the local presentations for each quasi-coherent sheaf in the coproduct add together to form local presentations for the direct sum. (Over a non-noetherian topological space, one must take the presheaf given by $Umapsto bigoplus_{iin I}mathcal{F}_i(U)$ and sheafify.) For products, things are more complicated. Quasi-coherent sheaves on a scheme is a Grothendieck Abelian Category, and in particular has arbitrary products. See here for references to Gabber's proof of this fact.
For the finite case: Yes, and this may be deduced from the fact that finite products and coproducts are equivalent in abelian categories like the category of quasi-coherent sheaves on a scheme (and then one applies the argument involving coproducts from the first paragraph).
To address the issue that comes up in the linked proof, the problem is that the listed exact sequence with direct sums should really be a sequence of direct products, which are not necessarily equivalent to direct sums except when the index set is finite.
answered Jan 8 at 4:58
KReiserKReiser
10.1k21435
$endgroup$
$begingroup$
Thanks, but I was asking about products in the category of Sheaves over a scheme, not the category of QUASI-COHERENT sheaves.
$endgroup$
– Jehu314
Jan 8 at 8:05
$begingroup$
Also, I don't quite understand your statement about infinite coproducts. Each point has an open neighbourhood on which the sheaf has a presentation, but if there are infinitely many sheaves in the coproduct, why is there necessarily an open neighbourhood on which ALL the sheaves have a presentation?
$endgroup$
– Jehu314
Jan 8 at 8:15
1
$begingroup$
I had hastily omitted a condition: one needs a noetherian topological space for infinite coproducts to avoid sheafifying. And as long as one is working in sheaves of $mathcal{O}_X$ modules, there is no need to adjust anything here - quasicoherent sheaves are a full subcategory of $mathcal{O}_X$ modules so any categorical construction made in the smaller category is automatically the correct construction in the larger.
$endgroup$
– KReiser
Jan 8 at 8:35
1
$begingroup$
Yes to the first. After doing a bit of reading, it appears that the product side is perhaps more complicated than I originally thought (and one should entirely ignore the last sentence of my previous comment, it is incorrect). You'll want to go and have a look at Thomason and Trobaugh "Higher algebraic K-theory of schemes and of derived categories" in The Grothendieck Festschrift, Vol. III, 247--435, Progr. Math., 88, Birkhäuser, Boston, 1990. (MR11069118) for probably the best reference of figuring this out. Either way, it is surprising to me that you are worrying so much about...
$endgroup$
– KReiser
Jan 8 at 10:16
1
$begingroup$
... this interaction between qc sheaves and arbitrary sheaves this (apparently) early in your algebraic geometry career. The reason qc sheaves of modules are worked with is that they behave reasonably nicely - until you have a specific research problem involving some sheaf which is not qc, it will save you some amount of headaches if you do not worry to hard about them (similar to the way in which one does not worry too much about very large groups until one has to).
$endgroup$
– KReiser
Jan 8 at 10:20
|
show 3 more comments
$begingroup$
For the questions listed in the body of your post:
In the infinite case: for infinite coproducts over reasonably nice spaces, this is easy. Over a noetherian scheme (or more generally a noetherian topological space) one may check that the direct sum of the local presentations for each quasi-coherent sheaf in the coproduct add together to form local presentations for the direct sum. (Over a non-noetherian topological space, one must take the presheaf given by $Umapsto bigoplus_{iin I}mathcal{F}_i(U)$ and sheafify.) For products, things are more complicated. Quasi-coherent sheaves on a scheme is a Grothendieck Abelian Category, and in particular has arbitrary products. See here for references to Gabber's proof of this fact.
For the finite case: Yes, and this may be deduced from the fact that finite products and coproducts are equivalent in abelian categories like the category of quasi-coherent sheaves on a scheme (and then one applies the argument involving coproducts from the first paragraph).
To address the issue that comes up in the linked proof, the problem is that the listed exact sequence with direct sums should really be a sequence of direct products, which are not necessarily equivalent to direct sums except when the index set is finite.
answered Jan 8 at 4:58
KReiserKReiser
10.1k21435
$endgroup$
For the questions listed in the body of your post:
In the infinite case: for infinite coproducts over reasonably nice spaces, this is easy. Over a noetherian scheme (or more generally a noetherian topological space) one may check that the direct sum of the local presentations for each quasi-coherent sheaf in the coproduct add together to form local presentations for the direct sum. (Over a non-noetherian topological space, one must take the presheaf given by $Umapsto bigoplus_{iin I}mathcal{F}_i(U)$ and sheafify.) For products, things are more complicated. Quasi-coherent sheaves on a scheme is a Grothendieck Abelian Category, and in particular has arbitrary products. See here for references to Gabber's proof of this fact.
For the finite case: Yes, and this may be deduced from the fact that finite products and coproducts are equivalent in abelian categories like the category of quasi-coherent sheaves on a scheme (and then one applies the argument involving coproducts from the first paragraph).
To address the issue that comes up in the linked proof, the problem is that the listed exact sequence with direct sums should really be a sequence of direct products, which are not necessarily equivalent to direct sums except when the index set is finite.
answered Jan 8 at 4:58
KReiserKReiser
10.1k21435
edited Jan 8 at 8:29
answered Jan 8 at 4:58
KReiserKReiser
10.1k21435
answered Jan 8 at 4:58
KReiserKReiser
10.1k21435
answered Jan 8 at 4:58
KReiserKReiser
10.1k21435
10.1k21435
$begingroup$
Thanks, but I was asking about products in the category of Sheaves over a scheme, not the category of QUASI-COHERENT sheaves.
$endgroup$
– Jehu314
Jan 8 at 8:05
$begingroup$
Also, I don't quite understand your statement about infinite coproducts. Each point has an open neighbourhood on which the sheaf has a presentation, but if there are infinitely many sheaves in the coproduct, why is there necessarily an open neighbourhood on which ALL the sheaves have a presentation?
$endgroup$
– Jehu314
Jan 8 at 8:15
1
$begingroup$
I had hastily omitted a condition: one needs a noetherian topological space for infinite coproducts to avoid sheafifying. And as long as one is working in sheaves of $mathcal{O}_X$ modules, there is no need to adjust anything here - quasicoherent sheaves are a full subcategory of $mathcal{O}_X$ modules so any categorical construction made in the smaller category is automatically the correct construction in the larger.
$endgroup$
– KReiser
Jan 8 at 8:35
1
$begingroup$
Yes to the first. After doing a bit of reading, it appears that the product side is perhaps more complicated than I originally thought (and one should entirely ignore the last sentence of my previous comment, it is incorrect). You'll want to go and have a look at Thomason and Trobaugh "Higher algebraic K-theory of schemes and of derived categories" in The Grothendieck Festschrift, Vol. III, 247--435, Progr. Math., 88, Birkhäuser, Boston, 1990. (MR11069118) for probably the best reference of figuring this out. Either way, it is surprising to me that you are worrying so much about...
$endgroup$
– KReiser
Jan 8 at 10:16
1
$begingroup$
... this interaction between qc sheaves and arbitrary sheaves this (apparently) early in your algebraic geometry career. The reason qc sheaves of modules are worked with is that they behave reasonably nicely - until you have a specific research problem involving some sheaf which is not qc, it will save you some amount of headaches if you do not worry to hard about them (similar to the way in which one does not worry too much about very large groups until one has to).
$endgroup$
– KReiser
Jan 8 at 10:20
|
show 3 more comments
$begingroup$
Thanks, but I was asking about products in the category of Sheaves over a scheme, not the category of QUASI-COHERENT sheaves.
$endgroup$
– Jehu314
Jan 8 at 8:05
$begingroup$
Also, I don't quite understand your statement about infinite coproducts. Each point has an open neighbourhood on which the sheaf has a presentation, but if there are infinitely many sheaves in the coproduct, why is there necessarily an open neighbourhood on which ALL the sheaves have a presentation?
$endgroup$
– Jehu314
Jan 8 at 8:15
1
$begingroup$
I had hastily omitted a condition: one needs a noetherian topological space for infinite coproducts to avoid sheafifying. And as long as one is working in sheaves of $mathcal{O}_X$ modules, there is no need to adjust anything here - quasicoherent sheaves are a full subcategory of $mathcal{O}_X$ modules so any categorical construction made in the smaller category is automatically the correct construction in the larger.
$endgroup$
– KReiser
Jan 8 at 8:35
1
$begingroup$
Yes to the first. After doing a bit of reading, it appears that the product side is perhaps more complicated than I originally thought (and one should entirely ignore the last sentence of my previous comment, it is incorrect). You'll want to go and have a look at Thomason and Trobaugh "Higher algebraic K-theory of schemes and of derived categories" in The Grothendieck Festschrift, Vol. III, 247--435, Progr. Math., 88, Birkhäuser, Boston, 1990. (MR11069118) for probably the best reference of figuring this out. Either way, it is surprising to me that you are worrying so much about...
$endgroup$
– KReiser
Jan 8 at 10:16
1
$begingroup$
... this interaction between qc sheaves and arbitrary sheaves this (apparently) early in your algebraic geometry career. The reason qc sheaves of modules are worked with is that they behave reasonably nicely - until you have a specific research problem involving some sheaf which is not qc, it will save you some amount of headaches if you do not worry to hard about them (similar to the way in which one does not worry too much about very large groups until one has to).
$endgroup$
– KReiser
Jan 8 at 10:20
$begingroup$
Thanks, but I was asking about products in the category of Sheaves over a scheme, not the category of QUASI-COHERENT sheaves.
$endgroup$
– Jehu314
Jan 8 at 8:05
$begingroup$
Thanks, but I was asking about products in the category of Sheaves over a scheme, not the category of QUASI-COHERENT sheaves.
$endgroup$
– Jehu314
Jan 8 at 8:05
$begingroup$
Also, I don't quite understand your statement about infinite coproducts. Each point has an open neighbourhood on which the sheaf has a presentation, but if there are infinitely many sheaves in the coproduct, why is there necessarily an open neighbourhood on which ALL the sheaves have a presentation?
$endgroup$
– Jehu314
Jan 8 at 8:15
$begingroup$
Also, I don't quite understand your statement about infinite coproducts. Each point has an open neighbourhood on which the sheaf has a presentation, but if there are infinitely many sheaves in the coproduct, why is there necessarily an open neighbourhood on which ALL the sheaves have a presentation?
$endgroup$
– Jehu314
Jan 8 at 8:15
1
1
$begingroup$
I had hastily omitted a condition: one needs a noetherian topological space for infinite coproducts to avoid sheafifying. And as long as one is working in sheaves of $mathcal{O}_X$ modules, there is no need to adjust anything here - quasicoherent sheaves are a full subcategory of $mathcal{O}_X$ modules so any categorical construction made in the smaller category is automatically the correct construction in the larger.
$endgroup$
– KReiser
Jan 8 at 8:35
$begingroup$
I had hastily omitted a condition: one needs a noetherian topological space for infinite coproducts to avoid sheafifying. And as long as one is working in sheaves of $mathcal{O}_X$ modules, there is no need to adjust anything here - quasicoherent sheaves are a full subcategory of $mathcal{O}_X$ modules so any categorical construction made in the smaller category is automatically the correct construction in the larger.
$endgroup$
– KReiser
Jan 8 at 8:35
1
1
$begingroup$
Yes to the first. After doing a bit of reading, it appears that the product side is perhaps more complicated than I originally thought (and one should entirely ignore the last sentence of my previous comment, it is incorrect). You'll want to go and have a look at Thomason and Trobaugh "Higher algebraic K-theory of schemes and of derived categories" in The Grothendieck Festschrift, Vol. III, 247--435, Progr. Math., 88, Birkhäuser, Boston, 1990. (MR11069118) for probably the best reference of figuring this out. Either way, it is surprising to me that you are worrying so much about...
$endgroup$
– KReiser
Jan 8 at 10:16
$begingroup$
Yes to the first. After doing a bit of reading, it appears that the product side is perhaps more complicated than I originally thought (and one should entirely ignore the last sentence of my previous comment, it is incorrect). You'll want to go and have a look at Thomason and Trobaugh "Higher algebraic K-theory of schemes and of derived categories" in The Grothendieck Festschrift, Vol. III, 247--435, Progr. Math., 88, Birkhäuser, Boston, 1990. (MR11069118) for probably the best reference of figuring this out. Either way, it is surprising to me that you are worrying so much about...
$endgroup$
– KReiser
Jan 8 at 10:16
1
1
$begingroup$
... this interaction between qc sheaves and arbitrary sheaves this (apparently) early in your algebraic geometry career. The reason qc sheaves of modules are worked with is that they behave reasonably nicely - until you have a specific research problem involving some sheaf which is not qc, it will save you some amount of headaches if you do not worry to hard about them (similar to the way in which one does not worry too much about very large groups until one has to).
$endgroup$
– KReiser
Jan 8 at 10:20
$begingroup$
... this interaction between qc sheaves and arbitrary sheaves this (apparently) early in your algebraic geometry career. The reason qc sheaves of modules are worked with is that they behave reasonably nicely - until you have a specific research problem involving some sheaf which is not qc, it will save you some amount of headaches if you do not worry to hard about them (similar to the way in which one does not worry too much about very large groups until one has to).
$endgroup$
– KReiser
Jan 8 at 10:20
|
show 3 more comments
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