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tim-at-topos committed Sep 12, 2024
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To define the #{\varphi_x}, we consider the complex #{K_x} whose underlying #{A}-modules are the #{\wedge A^p}, and whose differential is the interior product #{i_x} by #{x}, considered as a linear form on #{A^p} with components #{x_1,\ldots,x_p}.
The differential is of degree #{-1}, the degrees of the complex are positive, and the cohomology of this complex in dimension #{0} is #{A/(x_1A+\ldots+x_pA)}.
Since the #{x_i} are in #{\mathfrak{J}}, we obtain an augmentation #{K_{x,0}\to A/\mathfrak{J}}.
Thus #{K_x} is a _free_ augmented complex, with augmentation module #{A/\mathfrak{J}}.
Thus #{K_x} is a \em{free} augmented complex, with augmentation module #{A/\mathfrak{J}}.
We thus obtain known homomorphisms
##{
\begin{aligned}
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