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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 6th 2020
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded a third definition of fine sheaves due to Godement. <a href="https://ncatlab.org/nlab/revision/diff/fine+sheaf/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/fine+sheaf/11">v11</a>, <a href="https://ncatlab.org/nlab/show/fine+sheaf">current</a>

   Added a third definition of fine sheaves due to Godement.

   diff, v11, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 6th 2020
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexI am moving here an old discussion: ## Discussion ## David Speyer asks: Voisin, in _Hodge Theory and Complex Algebraic Geometry I_, definition 4.35 makes a different definition of fine sheaf. I can see that they are related, but I can't see precisely what the relation is. According to Voisin: :A fine sheaf $\mathcal{F}$ over $X$ is a sheaf of $\mathcal{A}$-modules, where $\mathcal{A}$ is a sheaf of rings such that, for every open cover $U_i$ of $X$, there is a partition of unity $1 = \sum f_i$ (where the sum is locally finite) subordinate to this covering. A technical point: I infer from context that, for Voisin, being subordinate to $U_i$ means that, for each $U_i$, there is an open set $V_i$ such that $X = U_i \cup V_i$ and $f|_{V_i}=0$. This is slightly stronger than requiring that $f|_{X \setminus U_i} =0$. When working on a regular (T3) space, I believe that, if partitions of unity exist in the weaker sense, than they also exist in the stronger sense. Zoran: paracompact Hausdorff space is automatically normal (Dieudonne's theorem) so *a fortiori* $T_3$. A partition of unity subordinate to the covering means as usual that for each $i$ there is $j$ such that $supp f_i \subset U_j$. Thanks for the other correction.

   I am moving here an old discussion:

   Discussion

   David Speyer asks: Voisin, in Hodge Theory and Complex Algebraic Geometry I, definition 4.35 makes a different definition of fine sheaf. I can see that they are related, but I can’t see precisely what the relation is.

   According to Voisin:

   :A fine sheaf $\mathcal{F}$ over $X$ is a sheaf of $\mathcal{A}$-modules, where $\mathcal{A}$ is a sheaf of rings such that, for every open cover $U_i$ of $X$, there is a partition of unity $1 = \sum f_i$ (where the sum is locally finite) subordinate to this covering.

   A technical point: I infer from context that, for Voisin, being subordinate to $U_i$ means that, for each $U_i$, there is an open set $V_i$ such that $X = U_i \cup V_i$ and $f|_{V_i}=0$. This is slightly stronger than requiring that $f|_{X \setminus U_i} =0$. When working on a regular (T3) space, I believe that, if partitions of unity exist in the weaker sense, than they also exist in the stronger sense.

   Zoran: paracompact Hausdorff space is automatically normal (Dieudonne’s theorem) so a fortiori $T_3$. A partition of unity subordinate to the covering means as usual that for each $i$ there is $j$ such that $supp f_i \subset U_j$. Thanks for the other correction.

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 6th 2020
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexRemoved an old discussion that was already reflected in the text. <a href="https://ncatlab.org/nlab/revision/diff/fine+sheaf/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/fine+sheaf/11">v11</a>, <a href="https://ncatlab.org/nlab/show/fine+sheaf">current</a>

   Removed an old discussion that was already reflected in the text.

   diff, v11, current

4. • CommentRowNumber4.
   • CommentAuthorperezl.alonso
   • CommentTimeOct 17th 2023
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexcouple properties related to [[sheaf cohomology]]. <a href="https://ncatlab.org/nlab/revision/diff/fine+sheaf/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/fine+sheaf/12">v12</a>, <a href="https://ncatlab.org/nlab/show/fine+sheaf">current</a>

   couple properties related to sheaf cohomology.

   diff, v12, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 18th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the bibitems for Godement and for Voisin that are referred to elsewhere in the text <a href="https://ncatlab.org/nlab/revision/diff/fine+sheaf/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/fine+sheaf/14">v14</a>, <a href="https://ncatlab.org/nlab/show/fine+sheaf">current</a>

   added the bibitems for Godement and for Voisin that are referred to elsewhere in the text

   diff, v14, current