From 6ff63525db6913844b203f196b3b83f81ea561ff Mon Sep 17 00:00:00 2001 From: Tim Hosgood Date: Sat, 28 Sep 2024 16:22:31 +0100 Subject: [PATCH] two small typos --- trees/fga1/fga1-4.tree | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/trees/fga1/fga1-4.tree b/trees/fga1/fga1-4.tree index e286cad..f5e7302 100644 --- a/trees/fga1/fga1-4.tree +++ b/trees/fga1/fga1-4.tree @@ -53,12 +53,12 @@ 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}. } @@ -319,6 +319,6 @@ \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). } }