# stacks/stacks-project

Fix mistake in dualizing

 @@ -4415,10 +4415,17 @@ \section{Upper shriek algebraically} be the $R$-linear map which picks off the coefficient of $x^i$ with respect to the given basis. Then $\delta_0, \ldots, \delta_{d - 1}$ is a basis for $\Hom_R(B, R)$. Finally, $x^i \delta_{d - 1} = \delta_{d - 1 - i}$ for $i \leq d - 1$. Hence $\Hom_R(B, R)$ is a principal $B$-module, and by looking at ranks we conclude that it is a free $B$-module of rank $1$ (with basis element $\delta_{d - 1}$). Finally, for $0 \leq i \leq d - 1$ a computation shows that $$x^i \delta_{d - 1} = \delta_{d - 1 - i} + b_1 \delta_{d - i} + \ldots + b_i \delta_{d - 1}$$ for some $c_1, \ldots, c_d \in R$\footnote{If $f = x^d + a_1 x^{d - 1} + \ldots + a_d$, then $c_1 = -a_1$, $c_2 = a_1^2 - a_2$, $c_3 = -a_1^3 + 2a_1a_2 -a_3$, etc.}. Hence $\Hom_R(B, R)$ is a principal $B$-module with generator $\delta_{d - 1}$. By looking at ranks we conclude that it is a rank $1$ free $B$-module. \end{proof} \begin{lemma}