Twisting sheaves and a skyscraper sequence on the projective line

(A skyscraper sequence)

Let \(X = \PP^1_k = \Proj k[x_0, x_1]\) and \(p = [1:0]\), the zero of \(x_1\). Multiplication by \(x_1\) gives \(\OO(-1) \to \OO\), with image the ideal sheaf \(\mathcal{I}_p\) of the reduced point \(p\).

1. On \(D_+(x_1)\), \(x_1\) is a unit, so the map is an isomorphism and both \(\mathcal{I}_p\) and the quotient are trivial away from \(p\).

2. On \(D_+(x_0) = \Spec k[t]\) with \(t = x_1/x_0\), the map is \(k[t] \xrightarrow{t} k[t]\), which is injective with cokernel \(k[t]/(t) = k\) supported at \(t = 0\), that is, at \(p\).

3. So \(0 \to \OO(-1) \xrightarrow{x_1} \OO \to k_p \to 0\) is exact on stalks, hence exact, where \(k_p\) is the skyscraper sheaf at \(p\) with value \(k\).

(Tensor products and duals of twists)

On each chart \(D_+(x_i)\) the sheaf \(\OO(d)\) is free with generator \(x_i^d\), and on overlaps the generators differ by the transition functions \((x_i/x_j)^d\). Multiplication \(\OO(d_1) \otimes \OO(d_2) \to \OO(d_1 + d_2)\), \(s \otimes s' \mapsto s s'\), sends the local generator \(x_i^{d_1} \otimes x_i^{d_2}\) to the local generator \(x_i^{d_1 + d_2}\), so it is an isomorphism. Taking \(d_2 = -d_1\) gives \(\OO(d) \otimes \OO(-d) \cong \OO\), and a perfect pairing of invertible sheaves identifies \(\OO(-d) \cong \sheafhom(\OO(d), \OO) = \OO(d)^\vee\).