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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):

  1. 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 $\mathcal{O}_S^{\oplus n} \to g^*\Omega_{Y/X}$?
  2. 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)?
  3. I would be interested in seeing how the torsor can be cut out of the rank $n^2$ bundle of endomorphisms by inverting determinants.
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$V$ is the space of linear functions on $V^*$, and Sym($V$) is the space of polynomial functions on $V^*$, so that Spec(Sym($V$)) is naturally identified with the dual vector space (or dual vector bundle) to $V$. This explains your tangent/cotangent concern. After replacing $\Omega_{Y/X}$ by its dual, the answer to your auxiliary question 1. is "yes," and to answer question 2. using this description note that the S-points of GL_n,Y are the S-automorphisms of O_S^{\oplus n}. – David Treumann Jun 8 2010 at 23:03
Thank you, David. This is very helpful. – S. Carnahan♦ Jun 9 2010 at 1:18
About 3, pay attention that for a vector bundle $E$ on $Y$, the principal bundle $Frame_Y(E)$ is not the same as the (non principal) $GL_n$-bundle $\operatorname{Aut}_Y(E)$. – Qfwfq Jun 9 2010 at 16:04
Thank you, unknown(google). I had not seen the notation $Frame_Y(E)$ before, but it makes a lot of sense. I realized that there was some kind of difference after reading Michael Thaddeus's answer. Perhaps I should edit the question to make it clear where my misconceptions lay. – S. Carnahan♦ Jun 9 2010 at 17:52

2 Answers

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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 $\mathcal{F}=\underline{Isom}_{\mathcal{O}_Y-\text{alg}}(Sym(\mathcal{E}),\mathcal{O}_Y^n[x_1,..,x_n])$.

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Just out of curiosity, how do you show that the sheaf of isomorphisms is represented by a scheme? – Michael Thaddeus Jun 9 2010 at 8:55
I think it follows from standard theorems of Grothendieck, but in this case you can use descent for affine morphisms. More precisely, you have a zariski cover, say $U_i$, of $Y$ on which $Z$ is isomorphic to $\mathbb{A}^n$. Over each of these, $\underline{Isom}_Y(\mathbb{A}^n_Y,Z)$ will be isomorphic to $GL_{n,U_i}$, which is affine over $U_i$. Using descent for affine morphisms you get an affine scheme that represents the sheaf $\underline{Isom}_Y(\mathbb{A}^n_Y,Z)$. – Mattia Talpo Jun 9 2010 at 9:55
On second thought, since we're working with the zariski topology here, calling for descent isn't really necessary.. You just have to glue the various $GL_{n,U_i}$ along the preimages of the intersections $U_i\cap U_j$. – Mattia Talpo Jun 9 2010 at 10:19
OK, since everything is locally trivial this is very easy, but I meant my question in a broader context: do you know a specific reference where these standard theorems of Grothendieck are found? One could no doubt deduce it by hand from the existence of the Hilbert scheme, but this would be a pain. – Michael Thaddeus Jun 9 2010 at 12:38
Oh, I had forgotten about the Isom construction. Thanks! – S. Carnahan♦ Jun 9 2010 at 15:39
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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.

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Thank you, Michael. Now I think I have mixed up two constructions, and that the torsor P with the associated bundle property I wanted is not the same as the adjoint $GL_n$-bundle of automorphisms. Is that the case? – S. Carnahan♦ Jun 9 2010 at 1:19
My understanding is that the 1-jets lie in an extension 0 --> T* --> J(1) --> O --> 0, which is split by the functions vanishing at the given point and the constant functions. So J(1) = T* + O. On the other hand, the extensions defining higher jets 0 --> Sym^k T* --> J(k) --> J(k+1) --> 0 need not split. Since representations of GL(n) are completely reducible, this means you won't obtain the jet bundles from any associated bundle construction starting from the tangent bundle. – Michael Thaddeus Jun 9 2010 at 7:36
This clears up a lot. I think the jet bundles come from an associated bundle construction using automorphism groups of infinitesimal neighborhoods, so first order uses $GL_n$, but higher order uses nilpotent extensions by $1 + M_n[t]/t^k$. – S. Carnahan♦ Jun 9 2010 at 15:36

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