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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