Jets

Stream: maths

[Yury G. Kudryashov]

What is the right way to define r-jets over any field? ∀ k : Fin n, E [×k]→L[K] F + quotient over "the restrictions to the diagonal are equal"? Something else?

[Eric Wieser]

Isn't that basically docs3#formal_multilinear_series?

[Eric Wieser]

Oh, plus a quotient

[Eric Wieser]

Is "restrictions to the diagonal are equal" the same as "swapping any two arguments leaves the input unchanged"? That could then be a symmetric_multilinear_map in contrast to alternating_map

[Yury G. Kudryashov]

We use docs#FormalMultilinearSeries in two incompatible ways: for analytic functions and for iterated derivatives. E.g., multiplication should be different in these two use cases.
Also, docs#FormalMultilinearSeries is about a whole series, not its initial segment.

[Yury G. Kudryashov]

As for symmetric maps, as @Sebastien Gouezel explained me some time ago, you can't represent $F(x, y)=x_1y_2$, $F\colon \mathbb k^2\to \mathbb k^2\to \mathbb k$ as a symmetric form if $\mathbb k$ has characteristic two.

[Eric Wieser]

Don't we have the same problem for docs#MultilinearMap.alternatization ?

[Yury G. Kudryashov]

We do but I don't care about it (as is going to be used in de Rham cohomologies) over fields of positive characteristic.

[Yury G. Kudryashov]

It seems that for applications I have in mind, a better approach is docs#HasFTaylorSeriesUpToOn - like

[Yury G. Kudryashov]

With jets representing a sequence of derivatives.

[Antoine Chambert-Loir]

This is a classic issue in algebraic geometry. There is an adequate (slightly abstract) definition in Grothendieck's EGA but in the end, it's just about $f(x+\varepsilon)$ with $\varepsilon ^{k+1} = 0$.

[Antoine Chambert-Loir]

The other issue (symmetric algebra vs symmetric tensors) is also classical.

[Yury G. Kudryashov]

And what is/are the answer(s) to this classical question?

[Antoine Chambert-Loir]

That the symmetric algebra is a quotient of the tensor algebra, not the algebra of symmetric tensors.

[Antoine Chambert-Loir]

It is also related to (what shouldcould be) a general definition of quadratic forms over an arbitrary ring, which takes its values in a different module than the bilinear form, both being related by two maps whose composition is multiplication by 2.

[Eric Wieser]

Antoine Chambert-Loir said:

It is also related to (what shouldcould be) a general definition of quadratic forms over an arbitrary ring, which takes its values in a different module than the bilinear form, both being related by two maps whose composition is multiplication by 2.

Is this related to my comment in the other thread that quadratic forms could be defined as a quotient of bilinear forms?

[Kevin Buzzard]

My understanding of what Antoine is saying is that one way (and some people think it's the correct way) of doing quadratic forms for R-modules (R a commutative ring) is something which looks a bit like this (but may not be exactly this because I'm just extrapolating from what he's said): a quadratic form on an R-module M is another R-module N, and a surjective R-linear map N -> M and another map M -> N whose composite is multiplication by 2 on N, and a bilinear form B on N with the property that if x is in the kernel of N -> M then B(x,y)=0 for all y and B(y,x)=0 for all y (this might not be exactly right). Now you define Q on M by Q(m)=B(n,n) for n any element of N which maps to m, and hopefully I've made this well-defined. I've probably not got the details right but this is the sort of thing that he seems to be alluding to.

[Eric Wieser]

Antoine Chambert-Loir said:

That the symmetric algebra is a quotient of the tensor algebra, not the algebra of symmetric tensors.

I stumbled across this page earlier today, which makes the same comparison in the context of quadratic forms.

[Kevin Buzzard]

This seems to stick to the finite-dimensional case