How do I describe the GL_n torsor attached to a smooth morphism of relative dimension n? - MathOverflow

How do I describe the GL_n torsor attached to a smooth morphism of relative dimension n?

2010-06-08T22:12:31Z

Edit: It seems I had two different constructions mixed up in my head, namely the frame torsor and the automorphism bundle of a vector bundle. This made the main question a bit confusing. The first two auxiliary questions were about the frame torsor, and the last one was about the automorphism bundle. If anyone knows a published reference for either construction, I would still be most appreciative.

The original question is below the line:

I feel like I should have learned this in grad school, but I never encountered a construction.

Let $n$ be a positive integer, and let $f: Y \to X$ be a smooth morphism of schemes of relative dimension $n$. The sheaf $\Omega_{Y/X}$ is then a rank $n$ locally free $\mathcal{O}_Y$ -module. Using the symmetric algebra functor, we can form the associated rank $n$ vector bundle $\mathbf{V}(\Omega_{Y/X}) = \operatorname{Spec}_Y \operatorname{Sym}_{\mathcal{O}_Y} \Omega_{Y/X}$ (cf. EGA2 1.7.8). I've heard it called the bundle of 1-jets, which ought to mean tangent bundle, but I'm always confused by this, so maybe it's the cotangent bundle.

Main question: Is there a reference for the construction of the commutative $\mathcal{O}_Y$ -algebra $A$ for which $\operatorname{Spec}_Y(A)$ is the $GL_{n,Y}$ -torsor $P$ of automorphisms of $\mathbf{V}(\Omega_{Y/X})$ ? Specifically, I'd like the torsor to satisfy the property that I can retrieve 1-jets by the associated bundle construction: $\mathbf{V}(\Omega_{Y/X}) \cong P \overset{GL_{n,Y}}{\times} \mathcal{O}_Y^{\oplus n}$

This can be viewed as a question about constructing the automorphism torsor of any bundle, but 1-jets seem to have specific structural features that may make a more specialized construction possible. For example, it should be a quotient of some canonical infinite-dimensional torsor of coordinates coming from the Gelfand-Kazhdan formal geometry theory.

Auxiliary questions (not as important):

Is there a concise description of the functor the torsor represents, e.g., are $S$-points on the torsor equal to $S$-points $g:S \to Y$ equipped with isomorphisms $\mathcal{O}_S^{\oplus n} \to g^*\Omega_{Y/X}$ ?
Is there a nice way to describe the $GL_{n,Y}$ -action (since writing an explicit comodule structure sounds like it could be a mess)?
I would be interested in seeing how the torsor can be cut out of the rank $n^2$ bundle of endomorphisms by inverting determinants.

Answer by Michael Thaddeus for How do I describe the GL_n torsor attached to a smooth morphism of relative dimension n?

2010-06-08T23:16:15Z

It seems there is nothing special about the relative Kahler differentials here. One could take any vector bundle V over Y. Then V = Spec Sym V^* as you say. By the same token, End V = Spec Sym (V ⊗ V^*). The determinant gives a section det of this sheaf of algebras. Now since GL(n) is cut out in gl(n) by the non-vanishing of the determinant, if we let R = det^-1 Sym (V ⊗ V^*) = Sym (V ⊗ V^*) [t]/(t det-1) be the localization of Sym (V ⊗ V^*) at the multiplicatively closed subset generated by det, then Spec R is what you want.

Be careful when calling this a torsor, however; it is not a principal GL(n)- bundle, but rather an adjoint GL(n)-bundle, in which the fibers carry a group structure and there is a canonical identity section.

Answer by Mattia Talpo for How do I describe the GL_n torsor attached to a smooth morphism of relative dimension n?

2010-06-09T08:31:51Z

I don't know about jets and you already got an answer regarding the bundle of automorphisms, anyway if you want a $GL_{n,Y}$-torsor over $Y$ that gives you back your original vector bundle $Z=V(\mathcal{E})=Spec_Y Sym(\mathcal{E})\to Y$ when you apply the associate bundle construction with $\mathcal{O}_Y^n$, you should take the bundle of local frames of $Z$, that is $P=\underline{Isom}_Y(\mathbb{A}^n_Y,Z)\to Y$, where $\underline{Isom}$ is the scheme representing the sheaf of isomorphisms. This is a $GL_{n,Y}$-torsor over $Y$ by the action of $GL_{n,Y}$ on $\mathbb{A}^n_Y$, and if you want a sheaf $\mathcal{F}$ of $\mathcal{O}_Y$-algebras such that $P=Spec_Y(\mathcal{F})$, it seems reasonable (but i didn't really check) that you can take $\mathcal{F}=\underline{Isom}_{\mathcal{O}_Y-\text{alg}}(Sym(\mathcal{E}),\mathcal{O}_Y^n[x_1,..,x_n])$.