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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 6th 2020

    Added a third definition of fine sheaves due to Godement.

    diff, v11, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 6th 2020

    I am moving here an old 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 XX is a sheaf of 𝒜\mathcal{A}-modules, where 𝒜\mathcal{A} is a sheaf of rings such that, for every open cover U iU_i of XX, there is a partition of unity 1=f i1 = \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 iU_i means that, for each U iU_i, there is an open set V iV_i such that X=U iV iX = U_i \cup V_i and f| V i=0f|_{V_i}=0. This is slightly stronger than requiring that f| XU i=0f|_{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 3T_3. A partition of unity subordinate to the covering means as usual that for each ii there is jj such that suppf iU jsupp f_i \subset U_j. Thanks for the other correction.

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 6th 2020

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

    diff, v11, current