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two small typos
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tim-at-topos committed Sep 28, 2024
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The formula in \ref{fga1-equation-4.1} is an immediate consequence of the spectral sequence from \ref{fga1-proposition-1}, as well as \ref{fga1-proposition-5};
by the formula in \ref{fga1-equation-3.8}, we can write
##{
\begin{gathered}
\begin{aligned}
\shExt_{\OO_X}^p(\sh{F},\sh{L})
= \sh{L}\otimes(\Omega_X^n)'\otimes\shExt_{\OO_X}(\sh{F},\Omega_X^n)
\\= \sh{L}\otimes(\Omega_X^n)'\otimes\shHom_{\OO_X}(\sh{F},\omega_Y^q)
\\= \shHom_{\OO_X}(\sh{F}\otimes\sh{L}'\otimes\Omega_X^n,\omega_Y^q)
\end{gathered}
&= \sh{L}\otimes(\Omega_X^n)'\otimes\shExt_{\OO_X}(\sh{F},\Omega_X^n)
\\&= \sh{L}\otimes(\Omega_X^n)'\otimes\shHom_{\OO_X}(\sh{F},\omega_Y^q)
\\&= \shHom_{\OO_X}(\sh{F}\otimes\sh{L}'\otimes\Omega_X^n,\omega_Y^q)
\end{aligned}
}
where #{q=n-p}, whence the formula in \ref{fga1-equation-4.1bis}.
}
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\p{
The reader will find more information on the duality of coherent sheaves in \em{loc. cit.}, pp.112–115, as well as in [[GD1960](GD1960), III.2], and in [[Gro1960b]](Gro1960b).
A more systematic treatment can be found in a later chapter of [[GD1960]](GD1960) (chapter IX in the provisional plan).
A more systematic treatment can be found in a later chapter of [[GD1960]](GD1960) (Chapter IX in the provisional plan).
}
}

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