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Twists on the two-affine projective line

Example

Assume the Axiom of Choice, inherited from the gluing construction. Let $k$ be a field and let $P_{k}^{1} = U_{0} \cup U_{\infty}$ be the two-affine projective line, with charts $U_{0} = Spec k [t]$ and $U_{\infty} = Spec k [u]$ , whose coordinates satisfy $u = t^{- 1}$ on the overlap $W = U_{0} \cap U_{\infty} = Spec k [t, t^{- 1}]$ , and with $O (n)$ , $n \in Z$ , the sheaves glued from the structure sheaves of the two charts by the frame relation $e_{\infty} = t^{n} e_{0}$ (Two-affine projective line and its twists).

Then:

For every $n \in Z$ the sheaf $O (n)$ is invertible, with $e_{0}$ and $e_{\infty}$ nowhere-vanishing local generators on $U_{0}$ and $U_{\infty}$ ; the relation inverts to $e_{0} = t^{- n} e_{\infty}$ , and $O (0) = O_{P_{k}^{1}}$ (Invertible sheaves).
The dual has the negated index, $O (n)^{\lor} ≅ O (- n),$ where $O (n)^{\lor} = H o m_{O_{X}} (O (n), O_{X})$ (The internal Hom sheaf of two module sheaves) and the dual frames satisfy $e_{\infty}^{\lor} = t^{- n} e_{0}^{\lor}$ on $W$ .
The tensor product adds the indices, $O (n) \otimes_{O_{X}} O (m) ≅ O (n + m),$ compatibly with the frames: $e_{0}^{(n)} \otimes e_{0}^{(m)} \mapsto e_{0}^{(n + m)}$ and $e_{\infty}^{(n)} \otimes e_{\infty}^{(m)} \mapsto e_{\infty}^{(n + m)}$ .
In particular $O (n) \otimes O (- n) ≅ O_{X}$ and $O (n)^{\lor \lor} ≅ O (n)$ , and the example contains the degenerate values $n = 0$ (trivial twist), $n = 1$ (transition $t$ ) and negative $n$ (transition $t^{n} = u^{- n}$ ).

Facts & Assumptions

Given: A field $k$ ; the two-affine projective line $P_{k}^{1} = U_{0} \cup U_{\infty}$ with $U_{0} = Spec k [t]$ , $U_{\infty} = Spec k [u]$ , $u = t^{- 1}$ on $W = U_{0} \cap U_{\infty}$ ; the sheaves $O (n)$ for $n \in Z$ with frames $e_{0}, e_{\infty}$ related by $e_{\infty} = t^{n} e_{0}$ on $W$ .

[F1]

The two-affine definition (Two-affine projective line and its twists): the two charts cover $P_{k}^{1}$ and intersect in $W = Spec k [t, t^{- 1}]$ with $t u = 1$ ; for every $n \in Z$ the sheaf $O (n)$ is glued from $O_{U_{0}}$ and $O_{U_{\infty}}$ with frames $e_{0} = 1$ on $U_{0}$ and $e_{\infty} = 1$ on $U_{\infty}$ related on $W$ by $e_{\infty} = t^{n} e_{0}$ , equivalently $e_{0} = t^{- n} e_{\infty}$ ; each $O (n)$ is free of rank one on each chart with the displayed frame, hence invertible, and $O (0) = O_{P_{k}^{1}}$ .

[F2]

Invertible means locally free of rank one (Invertible sheaves, Locally free sheaves of finite rank); the dual is $L^{\lor} = H o m_{O_{X}} (L, O_{X})$ (The internal Hom sheaf of two module sheaves), the tensor product of two invertible sheaves is invertible, and restriction to an open subscheme preserves invertibility.

[F3]

For an invertible $L$ the evaluation morphism $L^{\lor} \otimes L \to O_{X}$ is an isomorphism, and the dual is described by transition units: if $τ_{i} : L ∣_{U_{i}} \to O_{U_{i}}$ are trivialisations with $τ_{i} \circ τ_{j}^{- 1}$ equal to multiplication by the unit $u_{i j}$ , then the induced trivialisations $τ_{i}^{\lor}$ of $L^{\lor}$ have transition units $u_{i j}^{- 1}$ (Dual of a line bundle is its tensor inverse).

[F4]

A gluing datum on an open cover consists of sheaves on the members and overlap isomorphisms satisfying the cocycle condition (A gluing datum for sheaves on an open cover); the glued sheaf exists and is unique up to unique isomorphism, so two sheaves of modules on $X$ whose restrictions to the members of a common cover carry trivialisations with the same overlap identifications are canonically isomorphic (Compatible local sheaves glue uniquely up to unique isomorphism).

[F5]

Tensor product of modules and sheaves (Tensor product of sheaves of modules, Universal property of the tensor product for balanced maps into abelian groups): for a commutative ring $R$ the multiplication $R \times R \to R$ induces an isomorphism $R \otimes_{R} R \to R$ with $1 \otimes 1 \mapsto 1$ ; consequently if $F ∣_{U}$ and $G ∣_{U}$ are free of rank one with generators $e$ and $f$ , then $(F \otimes_{O_{X}} G) ∣_{U}$ is free of rank one with generator $e \otimes f$ , because restriction commutes with the tensor-product construction.

[F6]

The Axiom of Choice as used by the gluing construction of [F1] and the existence theorem of [F4] (The Axiom of Choice).

Proof technique: direct; read off the transition units of frames, multiply them for the tensor product and invert them for the dual, then apply the uniqueness of glued sheaves.

Proof

1.1F1F2

The transition units: on the overlap $W = Spec k [t, t^{- 1}]$ the element $t$ is a unit with inverse $u$ , so for every $n \in Z$ the relation $e_{\infty} = t^{n} e_{0}$ of [F1] is a relation between generators of free rank-one modules and inverts to $e_{0} = t^{- n} e_{\infty}$ ; the two charts cover $P_{k}^{1}$ , so each $O (n)$ is locally free of rank one with the displayed frames, hence invertible, and $O (0) = O_{P_{k}^{1}}$ .

2.1F1F4F5step 1.1

The tensor product adds the indices: fix $n, m \in Z$ and write $e_{0}^{(n)}$ and $e_{\infty}^{(n)}$ for the frames of $O (n)$ , and similarly for $m$ ; by [F5] the tensor product $O (n) \otimes O (m)$ is free of rank one on $U_{0}$ with generator $g_{0} = e_{0}^{(n)} \otimes e_{0}^{(m)}$ , and free of rank one on $U_{\infty}$ with generator $g_{\infty} = e_{\infty}^{(n)} \otimes e_{\infty}^{(m)}$ . On $W$ the frame relations give $g_{\infty} = (t^{n} e_{0}^{(n)}) \otimes (t^{m} e_{0}^{(m)}) = t^{n + m} g_{0}$ , and inverting gives $g_{0} = t^{- (n + m)} g_{\infty}$ , so the trivialisations of $O (n) \otimes O (m)$ on the two charts have exactly the overlap identification $e_{\infty}^{(n + m)} = t^{n + m} e_{0}^{(n + m)}$ that defines $O (n + m)$ in [F1]; by the uniqueness clause of [F4] there is a canonical isomorphism $O (n) \otimes O (m) ≅ O (n + m)$ carrying $g_{0}$ to $e_{0}^{(n + m)}$ and $g_{\infty}$ to $e_{\infty}^{(n + m)}$ .

2.2F1F2F3F4step 1.1

The dual negates the index: since $O (n)$ is invertible with the trivialisations determined by the frames $e_{0}$ and $e_{\infty}$ , its dual $O (n)^{\lor}$ is invertible and free of rank one on each chart with the dual frames $e_{0}^{\lor}$ and $e_{\infty}^{\lor}$ characterised by $e_{0}^{\lor} (e_{0}) = 1$ and $e_{\infty}^{\lor} (e_{\infty}) = 1$ [F2, F3]. On $W$ one has $e_{\infty} = t^{n} e_{0}$ , hence $(t^{- n} e_{0}^{\lor}) (e_{\infty}) = t^{- n} e_{0}^{\lor} (t^{n} e_{0}) = t^{- n} t^{n} = 1$ , and since the dual is free of rank one there the section with this property is unique, so $e_{\infty}^{\lor} = t^{- n} e_{0}^{\lor}$ ; equivalently $e_{0}^{\lor} = t^{n} e_{\infty}^{\lor}$ , which is the overlap identification defining $O (- n)$ in [F1], so the uniqueness clause of [F4] gives a canonical isomorphism $O (n)^{\lor} ≅ O (- n)$ carrying $e_{0}^{\lor}$ to the frame of $O (- n)$ on $U_{0}$ and $e_{\infty}^{\lor}$ to its frame on $U_{\infty}$ .

3.1F1F2F3F6step 1.1step 2.1step 2.2∎

The degenerate values and choice accounting: taking $n = 0$ in step 2.1 gives $O (0) \otimes O (m) ≅ O (m)$ , and $O (0) = O_{P_{k}^{1}}$ by step 1.1, so $O (0)$ is the tensor unit; taking $m = - n$ and using step 2.2 gives $O (n) \otimes O (- n) ≅ O (n) \otimes O (n)^{\lor} ≅ O (0) = O_{X}$ , in agreement with the evaluation isomorphism $O (n)^{\lor} \otimes O (n) \to O_{X}$ of [F3], whose transition unit is $t^{n} t^{- n} = 1$ ; applying step 2.2 twice gives $O (n)^{\lor \lor} ≅ O (- n)^{\lor} ≅ O (n)$ ; for $n = 1$ the transition is $t$ and for negative $n$ it is $t^{n} = u^{- n}$ with $u = t^{- 1}$ , both units on $W$ . All frames, transition units and isomorphisms used above come from the fixed two-chart cover and the canonical dual and tensor structures, so no selection is made and the only use of the Axiom of Choice is the inherited one recorded in [F6].

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Sources

The Stacks Project, Schemes, §§26.5, 26.7, 26.24 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, 29 August 2022, Chapters 6, 14, 17 (standard reference, not scraped)
Gao and Zhang, Lectures on Algebraic Geometry, projective-line gluing (standard reference, not scraped)