What's the direct sum of infinite sheaves?
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In the Lemma 5.1.3 of Liu Qing's book on algebraic geometry, he uses $O_{X}^{(I)}$ which the direct sum of $O_{X}$indexed by $I$.
What's the global section of this sheaf? Is it $bigoplus O_{X}(X)$? But I think this is impossible since when $I$ is infinite we can suppose $X=bigsqcup U_{i}$ and glue $1_{U_{i}}$ together to get a global section which can't be an element in the direct sum.
Or we just get a presheaf from $O_{X}^{(I)}$ and we consider its associate sheaf instead? If so, is there a big difference between direct sum of infinite sheaves and product of infinite sheaves?
algebraic-geometry coherent-sheaves
asked Dec 13 '18 at 16:36
RH WangRH Wang
111
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add a comment |
$begingroup$
In the Lemma 5.1.3 of Liu Qing's book on algebraic geometry, he uses $O_{X}^{(I)}$ which the direct sum of $O_{X}$indexed by $I$.
What's the global section of this sheaf? Is it $bigoplus O_{X}(X)$? But I think this is impossible since when $I$ is infinite we can suppose $X=bigsqcup U_{i}$ and glue $1_{U_{i}}$ together to get a global section which can't be an element in the direct sum.
Or we just get a presheaf from $O_{X}^{(I)}$ and we consider its associate sheaf instead? If so, is there a big difference between direct sum of infinite sheaves and product of infinite sheaves?
algebraic-geometry coherent-sheaves
asked Dec 13 '18 at 16:36
RH WangRH Wang
111
$endgroup$
$begingroup$
It's exactly what you think: the sheaf associated to the presheaf $U mapsto bigoplus_i mathcal{O}_X (U)$.
$endgroup$
– Rellek
Dec 13 '18 at 23:14
add a comment |
$begingroup$
In the Lemma 5.1.3 of Liu Qing's book on algebraic geometry, he uses $O_{X}^{(I)}$ which the direct sum of $O_{X}$indexed by $I$.
What's the global section of this sheaf? Is it $bigoplus O_{X}(X)$? But I think this is impossible since when $I$ is infinite we can suppose $X=bigsqcup U_{i}$ and glue $1_{U_{i}}$ together to get a global section which can't be an element in the direct sum.
Or we just get a presheaf from $O_{X}^{(I)}$ and we consider its associate sheaf instead? If so, is there a big difference between direct sum of infinite sheaves and product of infinite sheaves?
algebraic-geometry coherent-sheaves
asked Dec 13 '18 at 16:36
RH WangRH Wang
111
$endgroup$
In the Lemma 5.1.3 of Liu Qing's book on algebraic geometry, he uses $O_{X}^{(I)}$ which the direct sum of $O_{X}$indexed by $I$.
What's the global section of this sheaf? Is it $bigoplus O_{X}(X)$? But I think this is impossible since when $I$ is infinite we can suppose $X=bigsqcup U_{i}$ and glue $1_{U_{i}}$ together to get a global section which can't be an element in the direct sum.
Or we just get a presheaf from $O_{X}^{(I)}$ and we consider its associate sheaf instead? If so, is there a big difference between direct sum of infinite sheaves and product of infinite sheaves?
algebraic-geometry coherent-sheaves
algebraic-geometry coherent-sheaves
asked Dec 13 '18 at 16:36
RH WangRH Wang
111
asked Dec 13 '18 at 16:36
RH WangRH Wang
111
asked Dec 13 '18 at 16:36
RH WangRH Wang
111
asked Dec 13 '18 at 16:36
RH WangRH Wang
111
asked Dec 13 '18 at 16:36
RH WangRH Wang
111
111
$begingroup$
It's exactly what you think: the sheaf associated to the presheaf $U mapsto bigoplus_i mathcal{O}_X (U)$.
$endgroup$
– Rellek
Dec 13 '18 at 23:14
add a comment |
$begingroup$
It's exactly what you think: the sheaf associated to the presheaf $U mapsto bigoplus_i mathcal{O}_X (U)$.
$endgroup$
– Rellek
Dec 13 '18 at 23:14
$begingroup$
It's exactly what you think: the sheaf associated to the presheaf $U mapsto bigoplus_i mathcal{O}_X (U)$.
$endgroup$
– Rellek
Dec 13 '18 at 23:14
$begingroup$
It's exactly what you think: the sheaf associated to the presheaf $U mapsto bigoplus_i mathcal{O}_X (U)$.
$endgroup$
– Rellek
Dec 13 '18 at 23:14
add a comment |
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$begingroup$
It's exactly what you think: the sheaf associated to the presheaf $U mapsto bigoplus_i mathcal{O}_X (U)$.
$endgroup$
– Rellek
Dec 13 '18 at 23:14