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 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 nforum 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).

nLab > Latest Changes: distribution on an affine group scheme

Bottom of Page

1 to 7 of 7

  1.  
    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeAug 22nd 2023
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2023

    First line:

    Distribution in a sense of a linear functional on some functional space can be somewhat adapted to schemes.

    How about we pause for a moment and turn that into a sentence and add cross-links:

    The notion of distributions – in the sense of linear functionals on some algebra of functions – can to some extent be adapted from smooth manifolds to schemes.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeAug 22nd 2023
    • (edited Aug 22nd 2023)

    Great idea, but need to be improved a bit, I guess: in general they need still to be (continuous) linear functionals on some functional space as it has topology (the space of test functions for example), and for the notion of a distribution it is not essential that the functional space is an “algebra” of functions.

    Here, in the algebraic setup, the powers of ideals generate a (co)filtration amounting to some “formal topology”. In that sense, the algebra structure is incidentally used only to define the formal topology, as far as I understand.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2023
    • (edited Aug 22nd 2023)

    I wrote:

    How about we pause for a moment and turn that into a sentence

    You replied:

    Great idea,

    Glad that you agree. Please make it a habit to write in complete sentences.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeAug 22nd 2023
    • (edited Aug 22nd 2023)

    Well, it was a grammatical sentence (the verb is “can”):

    Distribution in a sense of a linear functional on some functional space can be somewhat adapted to schemes.

    The thing which is essential and which I praise in your intervention is the idea about inline explaining more precisely and linking to what a distribution actually is (I am however not sure if I fully understood the actual historical idea).

    P.S. Oh, you meant “from smooth manifolds”.

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 22nd 2023
    • (edited Aug 22nd 2023)

    Re #3:

    in general they need still to be (continuous) linear functionals on some functional space as it has topology (the space of test functions for example)

    As explained in distributions are the smooth linear functionals, it suffices to require (instead of contunity) that smooth families of functions map to smooth families of numbers.

    That is to say, we need a morphism of sheaves of vector spaces and not just a morphism of vector spaces.

    This reformulation has an obvious analogue for schemes, where the site of smooth manifolds is replaced by the Zariski site (say).

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeAug 23rd 2023

    But which notion you get if you ask this ?

1 to 7 of 7

  1.