**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](http://en.wikipedia.org/wiki/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.



How do I describe the GL_n torsor attached to a smooth morphism of relative dimension n?
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](http://arxiv.org/abs/1410.1716), 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.


