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• 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.

• 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 ?