The Stacks project

Lemma 45.11.3. Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). If there exists a smooth projective scheme $Y$ over $k$ such that $H^ i(Y)$ is nonzero for some $i < 0$, then there exists an equidimensional smooth projective scheme $X$ over $k$ such that the equivalent conditions of Lemma 45.11.2 fail for $X$.

Proof. By Lemma 45.9.9 we may assume $Y$ is irreducible and a fortiori equidimensional. If $i$ is odd, then after replacing $Y$ by $Y \times Y$ we find an example where $Y$ is equidimensional and $i = -2l$ for some $l > 0$. Set $X = Y \times (\mathbf{P}^1_ k)^ l$. Using axiom (B)(a) we obtain

\[ H^0(X) \supset H^0(Y) \oplus H^ i(Y) \otimes _ F H^2(\mathbf{P}^1_ k)^{\otimes _ F l} \]

with both summands nonzero. Thus it is clear that $H^0(X)$ cannot be isomorphic to $H^0$ of the spectrum of $\Gamma (X, \mathcal{O}_ X) = \Gamma (Y, \mathcal{O}_ Y)$ as this falls into the first summand. $\square$

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