The Stacks project

Lemma 53.5.1. Let $X$ be a proper scheme of dimension $\leq 1$ over a field $k$. With $\omega _ X^\bullet $ and $\omega _ X$ as in Lemma 53.4.1 we have

\[ \chi (X, \mathcal{O}_ X) = \chi (X, \omega _ X^\bullet ) \]

If $X$ is Cohen-Macaulay and equidimensional of dimension $1$, then

\[ \chi (X, \mathcal{O}_ X) = - \chi (X, \omega _ X) \]

Proof. We define the right hand side of the first formula as follows:

\[ \chi (X, \omega _ X^\bullet ) = \sum \nolimits _{i \in \mathbf{Z}} (-1)^ i\dim _ k H^ i(X, \omega _ X^\bullet ) \]

This is well defined because $\omega _ X^\bullet $ is in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$, but also because

\[ H^ i(X, \omega _ X^\bullet ) = \mathop{\mathrm{Ext}}\nolimits ^ i(\mathcal{O}_ X, \omega _ X^\bullet ) = H^{-i}(X, \mathcal{O}_ X) \]

which is always finite dimensional and nonzero only if $i = 0, -1$. This of course also proves the first formula. The second is a consequence of the first because $\omega _ X^\bullet = \omega _ X[1]$ in the CM case, see Lemma 53.4.2. $\square$

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View Lemma 53.5.1 as pdf