Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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:

    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

    • CommentRowNumber4.
    • CommentAuthorperezl.alonso
    • CommentTimeOct 17th 2023

    couple properties related to sheaf cohomology.

    diff, v12, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2023

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

    diff, v14, current