This is a discussion of the difference between \(\mathcal O_X^\times\)-torsors, \(\mathbb G_m\)-torsors and line bundles. This became necessary because global sections of the ghost sheaf are interchangably used to give \(\mathbb G_m\)-torsors and line bundles.
Torsors of sheaves of groups vs group schemes
First let us show that \(\mathcal O^\times_X\)-torsors and \(\mathbb G_m\)-torsors are the same thing up to changing perspectives from sheaves of groups to group schemes; it’s really whether you want to think of a torsor sheaf-theoretically or scheme-theoretically, similar to the locally free sheaf vs geometric vector bundle thing. The quick argument: \(\mathbb G_m\) is the scheme which represents the functor \(T\mapsto \Gamma(T, \mathcal O_T^\times)\). A \(\mathbb G_m\)-torsor over \(X\) is a flat map \(D\to X\) so that \(D\) globally has a
Here are their definitions, see also the definition of a \(G\)-torsor.
Definition
An \(\mathcal O_X^\times\) -torsor is a sheaf of sets \(\mathcal P\) on the space \(X\), equipped with a right action of the sheaf of groups \(\mathcal O^\times_X\) such that the following conditions hold:
- Simply Transitive Action: The sheaf map \[\mathcal P\times \mathcal O^\times_X \to \mathcal P\times\mathcal P\]defined on local sections by \((p,g)\mapsto (p, p\cdot g)\) is an isomorphism of sheaves, i.e. it is simply transitive.
- Local Triviality: There exists an open cover \(\{U_i\}\) of \(X\) such that \(\mathcal P(U_i)\) is nonempty for all \(i\).
Note: replace \(\mathcal O^\times_X\) with any sheaf of groups over \(X\) and you get the general definition.
Definition
We say that \(X \to S\) is a \(\mathbb G_m\)-torsor over \(S\) if \(\mathbb G_m\) acts on \(X\) and
- The following map is an isomorphism of schemes:
\[\mathbb G_m\times_{S} X \to X\times_{S}X, \quad (g, x)\mapsto (g\cdot x, x)\]
- There is an fpqc cover \(\{S_i\to S\}\) so that \(X_{S_i}\to S_i\) has a section, which is equivalent to asking that \(X_{S_i}\) is equivariantly isomorphic to \((\mathbb G_m)_{S_i}\), i.e. that \(X_{S_i}\) is a trivial \((\mathbb G_m)_{S_i}\) torsor.
Note: replace \(\mathbb G_m\) with any other group scheme \(G\) over \(S\) and you get the general definition of a \(G\)-torsor.
The following lemma is useful for thinking about \(G\)-torsors:
Lemma
- A scheme \(X\) is a \(\mathbb G_m\)-torsor if and only if for every scheme \(T\) over \(S\) the set \(X(T)\) is either empty or the action of \(G(T)\) on \(X(T)\) is simply transitive.
- A \(\mathbb G_m\)-torsor is trivial (meaning there is a \(\mathbb G_m\)-equivariant isomorphism \(\mathbb G_m\to X\) where \(\mathbb G_m\) acts on itself on the left by multiplication) if and only if \(X\to S\) admits a section.
Equivalence: Going from scheme to sheaf. Suppose \(X\) is a \(\mathbb G_m\)-torsor over \(S\) with \(\pi:X\to S\) its structure map. Take the sheaf of local sections on \(S\), i.e. define \(\mathcal P(U)\) to be the set of morphisms \(s:U\to X\) such that \(\pi|_U\circ s\) is the identity on \(U\). For each \(g:U\to \mathbb G_m \) element of \(\mathbb G_m(U)\) we have an automorphism \(\varphi_g:X\to X\) given by action of \(g\), and this acts on \(\mathcal P(U)\) by composition: take a section \(s:U\to X\) and send it to \(\varphi_g\circ s\). However, \(\mathcal O_S^\times\) represents the functor of points of \(\mathbb G_m\), that is, \[\Hom_{Sch/S}(U, \mathbb G_m)\cong \mathcal O_S(U)^\times.\] Ignoring the base for a moment, a morphism \(U\to \mathbb G_m\) is the same as a morphism \(\mathcal O_{\mathbb G_m} \to \mathcal O_S|_U\). Since \(\mathbb G_m\) is affine, this in turn is the same as specifying a morphism on global sections, i.e. \(\mathbb Z[t, t^{-1}]\to \mathcal O_S(U)\). Such a morphism is determined by choosing an invertible element \(f\in \mathcal O_S(U)\) to be the image of \(t\), and thus the set of morphisms is in bijection with \(\mathcal O_S(U)^\times\).
Equivalence: Going from sheaf to scheme. If we start with an abstract sheaf \(\mathcal P\) over \(S\) which is a \(\mathcal O_S^\times\)-torsor, we can build a full \(\mathbb G_m\)-torsor \(X\) over \(S\) by gluing copies of \(\mathbb G_m\) together over the trivialization \(\{U_i\}\) of \(\mathcal P\) with transition functions given by something like the quotient \(f/g\) where \(f\) and \(g\) are the sections of \(\mathcal O_S^\times\) corresponding to the maps \(U_i \to \mathbb G_m\); it’s just descent.
Line bundles and Torsors
The basic idea here is that a torsor is a line bundle with the zero section removed.