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

Question

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:

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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.

Answer 1

If $V$ is a vector bundle of rank $n$, the corresponding universal algebra $A$ which makes $V$ trivial (i.e. $V \otimes A \cong A^n$), or equivalently the algebra of the corresponding $\mathrm{GL}_n$-torsor, is given by $$A = \mathrm{Sym}(V^n) \otimes_{\mathrm{Sym}(\Lambda^n V)} \mathrm{Sym}^{\mathbb{Z}}(\Lambda^n V).$$ Here, we define $\mathrm{Sym}^{\mathbb{Z}}(\mathcal{L})=\bigoplus_{z \in \mathbb{Z}} \mathcal{L}^{\otimes z}$ for the line bundle $\mathcal{L}=\Lambda^n V$, and the tensor product is taken with respect to the morphism $\delta : \mathcal{L} \to \mathrm{Sym}^n(V^n)$ which maps $v_1 \wedge \dotsc \wedge v_n$ to $\sum_{\sigma \in \Sigma_n} \mathrm{sgn}(\sigma) \prod_{i=1}^{n} \iota_i(v_{\sigma(i)})$. This description is global in nature and actually generalizes to arbitrary cocomplete linear tensor categories. Some details can be found in my thesis, Section 4.9.

The idea of the construction of $A$ is the following: $\mathrm{Sym}(V^n)$ is the universal algebra $B$ with a morphism of $B$-modules $V \otimes B \to B^n$. Then we construct $B \to A$ so that the determinant of this morphism becomes invertible over $A$, so that $V \otimes A \cong A^n$.

We could also construct $A$ as a quotient of $\mathrm{Sym}(V^n) \otimes \mathrm{Sym}((V^*)^n)$, which introduces morphisms $V \otimes A \to V^n$ and $A^n \to V \otimes A$, and the quotient should be made in such a way that these morphisms become inverse to each other.

Answer 2

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

Answer 3

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.