I recently came across this elegant translation of etale $\mathbb{G}_m$-torsors into line bundles:

Let $\text{Spec }R$ be a $\mathbb{G}_m$-torsor over $\text{Spec }A$ for the etale topology, where $\mathbb{G}_m = \text{Spec }A[x,x^{-1}]$, then the action of $\mathbb{G}_m$ on $\text{Spec }R$ is given by a homomorphism of rings (the coaction): $$\mu : R\rightarrow A[x,x^{-1}]\otimes_A R$$ Somehow, this induces a decomposition $R = \bigoplus_{n\in\mathbb{Z}} R_1^{\otimes n}$ making $R$ into a graded $A$-algebra, where $R_1$ is a projective $A$-module.

Unfortunately, I don't understand how to prove this decomposition, and why $R_1$ is projective. I can imagine that the etale local triviality of $R$ would somehow result in $R$ being etale-locally free, but I don't see why this would imply Zariski-freeness (ie, projectivity).

  • 1
    $\begingroup$ The fact that étale locally free modules are Zariski locally free is sometimes referred to as Hilbert's theorem 90 (after the analogous statement in Galois cohomology). See for example Tag 03P7 in the Stacks project. $\endgroup$ Nov 7 '16 at 0:57
  • $\begingroup$ Let $R_n$ be the kernel of the map $\mu - x^n \otimes$. Now use coassociativity to show that $R_n = R_1^{\otimes n}$. $\endgroup$ Nov 7 '16 at 1:57
  • $\begingroup$ You can also define a line bundle geometrically as $G \times_{G_m} A^1$, where $G$ is a torsor. $\endgroup$
    – Sasha
    Nov 7 '16 at 7:26
  • $\begingroup$ The first question is a general fact about diagonalizable groups and is extremely well documented. For the second point, being locally free of finite rank is even fpqc local, by descent theory, see mathoverflow.net/questions/155224/… $\endgroup$
    – Niels
    Nov 7 '16 at 7:43
  • $\begingroup$ I think you can show this via descent. It is discussed in slightly more general terms in this lecture youtube.com/watch?v=dY6-TenYUEo. $\endgroup$
    – JJJ
    Nov 8 '16 at 0:32

If $G$ is a commutative monoid (for your question we will want $G = \mathbb{Z}$), then $A[G]$-comodules identify with $G$-graded $A$-modules. A reference is Demazure, Gabriel, Introduction to Algebraic Geometry and Algebraic Groups, II, §2, no 2, Example 1.

Indeed, if $M$ is an $A$-module with an $A[G]$-module structure given by the coaction $\mu: M \to M\otimes_A A[G]$, the corresponding grading $M = \bigoplus_{g \in G} M_g$ is given by $M_g = \{m \in M : \mu(m)=m\otimes g\}$.

Conversely, given a $G$-grading $M = \bigoplus_{g\in G}M_g$, for $m\in M$, let $m_g$ be the projection of $m$ onto $M_g$, then the map $$\mu : M\longrightarrow M\otimes_A A[G],\quad m\mapsto m_g\otimes g$$ is an $A[G]$-coaction on $M$.

These functors are mutually inverse, and define an equivalence of symmetric monoidal categories. Hence, $A[G]$-comodule commutative algebras identify with $G$-graded commutative $A$-algebras.

Now assume that $R \to \underbrace{R\otimes_A A[\mathbb{Z}]}_{R[x,x^{-1}]} = R\otimes_A \underbrace{A[x,x^{-1}]}_{(\mathbb{G}_m)_{/A}}$ is a comodule commutative algebra. The corresponding $\mathbb{G}_m$-scheme $\mathrm{Spec}(R)$ is a torsor if and only if

  1. $R$ is faithfully flat over $A$,
  2. The natural morphism $\mathrm{Spec}(R)/\mathbb{G}_m \to \mathrm{Spec}(A)$ is an isomorphism, i.e. the natural morphism $A \to R_0$ is an isomorphism,
  3. The natural morphism $\mathbb{G}_m \times_{\mathrm{Spec}(A)} \mathrm{Spec}(R) \to \mathrm{Spec}(R) \times_{\mathrm{Spec}(A)} \mathrm{Spec}(R)$ is an isomorphism, i.e. the natural morphism $R \otimes_A R \to R[x,x^{-1}]$ is an isomorphism.

For $n,m \in \mathbb{Z}$ the natural morphism $R_n \otimes_A R_m \to R_{n+m}$ is an isomorphism since this is so when we tensor with $R$ over $A$ (by faithful flatness). It follows that $R_1$ is invertible and that $R_n \cong R_1^{\otimes n}$ for $n \in \mathbb{Z}$.

  • $\begingroup$ Am I right in saying that property (3) implies the others? $\endgroup$ Nov 7 '16 at 21:54
  • $\begingroup$ @rtz: No, consider $R=0$ for instance. But I think that 2. is implied by 1. and 3. $\endgroup$
    – HeinrichD
    Nov 8 '16 at 7:20
  • $\begingroup$ In the book by Demazure, he sometimes writes functors with underlines, and sometimes not. For example in II, $\S 1$, no 2, 2.8 ("Diagonalizable Groups, end of page 190 and beginning of p191), he writes $\underline{D}(\Gamma)$, and later at the top of p191 he writes $D(\Gamma)$. They both seem like just functors from Rings to Sets. Is there a distinction? $\endgroup$ Nov 8 '16 at 17:09
  • $\begingroup$ Also, what is an "affine algebraic scheme" (p183, near bottom, p184,185...)? Is it just an affine scheme? I can't seem to find the definition anywhere... I apologize for these questions. It's just difficult navigating a book using an unfamiliar language for the first time, and since you seem to be familiar with it, I thought I'd just ask you... $\endgroup$ Nov 8 '16 at 17:24
  • $\begingroup$ About the underlines - perhaps it's most striking in the statement of the two Propositions on p197. $\endgroup$ Nov 8 '16 at 18:00

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