Lemma 110.47.1. There exists an affine scheme X = \mathop{\mathrm{Spec}}(A) and an injective A-module J such that \widetilde{J} is not a flasque sheaf on X. Even the restriction \Gamma (X, \widetilde{J}) \to \Gamma (U, \widetilde{J}) with U a standard open need not be surjective.
110.47 Non flasque quasi-coherent sheaf associated to injective module
For more examples of this type see [Exposé II, Appendix I, SGA6] where Illusie explains some examples due to Verdier.
Consider the affine scheme X = \mathop{\mathrm{Spec}}(A) where
A = k[x, y, z_1, z_2, \ldots ]/(x^ nz_ n)
is the ring from Properties, Example 28.25.2. Set I = (x) \subset A. Consider the quasi-compact open U = D(x) of X. We have seen in loc. cit. that there is a section s \in \mathcal{O}_ X(U) which does not come from an A-module map I^ n \to A for any n \geq 0.
Let \alpha : A \to J be the embedding of A into an injective A-module. Let Q = J/\alpha (A) and denote \beta : J \to Q the quotient map. We claim that the map
\Gamma (X, \widetilde{J}) \longrightarrow \Gamma (U, \widetilde{J})
is not surjective. Namely, we claim that \alpha (s) is not in the image. To see this, we argue by contradiction. So assume that x \in J is an element which restricts to \alpha (s) over U. Then \beta (x) \in Q is an element which restricts to 0 over U. Hence we know that I^ n\beta (x) = 0 for some n, see Properties, Lemma 28.25.1. This implies that we get a morphism \varphi : I^ n \to A, h \mapsto \alpha ^{-1}(hx). It is easy to see that this morphism \varphi gives rise to the section s via the map of Properties, Lemma 28.25.1 which is a contradiction.
Proof. See above. \square
In fact, we can use a similar construction to get an example of an injective module whose associated quasi-coherent sheaf has nonzero cohomology over a quasi-compact open. Namely, we start with the ring
A = k[x, y, w_1, u_1, w_2, u_2, \ldots ]/(x^ nw_ n, y^ nu_ n, u_ n^2, w_ n^2)
where k is a field. Choose an injective map A \to I where I is an injective A-module. We claim that the element 1/xy in A_{xy} \subset I_{xy} is not in the image of I_ x \oplus I_ y \to I_{xy}. Arguing by contradiction, suppose that
\frac{1}{xy} = \frac{i}{x^ n} + \frac{j}{y^ n}
for some n \geq 1 and i, j \in I. Clearing denominators we obtain
(xy)^{n + m - 1} = x^ my^{n + m}i + x^{n + m}y^ mj
for some m \geq 0. Multiplying with u_{n + m}w_{n + m} we see that u_{n + m}w_{n + m}(xy)^{n + m - 1} = 0 in A which is the desired contradiction. Let U = D(x) \cup D(y) \subset X = \mathop{\mathrm{Spec}}(A). For any A-module M we have an exact sequence
0 \to H^0(U, \widetilde{M}) \to M_ x \oplus M_ y \to M_{xy} \to H^1(U, \widetilde{M}) \to 0
by Mayer-Vietoris. We conclude that H^1(U, \widetilde{I}) is nonzero.
Lemma 110.47.2. There exists an affine scheme X = \mathop{\mathrm{Spec}}(A) whose underlying topological space is Noetherian and an injective A-module I such that \widetilde{I} has nonvanishing H^1 on some quasi-compact open U of X.
Proof. See above. Note that \mathop{\mathrm{Spec}}(A) = \mathop{\mathrm{Spec}}(k[x, y]) as topological spaces. \square
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