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Rick Sanchez
Rick Sanchez
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Let $f\colon X\to Y$ be a flat morphism between two smooth projective varieties. Let $L$ be a locally free sheaf on $X$ and $\mathcal{F}$ a coherent sheaf on $Y$. How to prove $f_*(L\otimes f^*\mathcal{F})\cong f_*L\otimes\mathcal{F}$? I think it is a well-known result but I couldn't find a reference. You can assume $f$ is smooth and surjective if you want. Thank you.

Let $f\colon X\to Y$ be a flat morphism between two smooth projective varieties. Let $L$ be a locally free sheaf on $X$ and $\mathcal{F}$ a coherent sheaf on $Y$. How to prove $f_*(L\otimes f^*\mathcal{F})\cong f_*L\otimes\mathcal{F}$? I think it is a well-known result but I couldn't find a reference. Thank you.

Let $f\colon X\to Y$ be a flat morphism between two smooth projective varieties. Let $L$ be a locally free sheaf on $X$ and $\mathcal{F}$ a coherent sheaf on $Y$. How to prove $f_*(L\otimes f^*\mathcal{F})\cong f_*L\otimes\mathcal{F}$? I think it is a well-known result but I couldn't find a reference. You can assume $f$ is smooth and surjective if you want. Thank you.

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Rick Sanchez
Rick Sanchez
  • 91
  • 5

Projection formula for flat morphisms

Let $f\colon X\to Y$ be a flat morphism between two smooth projective varieties. Let $L$ be a locally free sheaf on $X$ and $\mathcal{F}$ a coherent sheaf on $Y$. How to prove $f_*(L\otimes f^*\mathcal{F})\cong f_*L\otimes\mathcal{F}$? I think it is a well-known result but I couldn't find a reference. Thank you.