Lemma 45.10.4 (0FHS)—The Stacks project

Lemma 45.10.4. Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let $X$ be a nonempty smooth projective scheme over $k$ which is equidimensional of dimension $d$. We have

\[ \sum \nolimits _ i (-1)^ i\dim _ F H^ i(X) = \deg (\Delta \cdot \Delta ) = \deg (c_ d(\mathcal{T}_{X/k})) \]

Proof. Equality on the right. We have $[\Delta ] \cdot [\Delta ] = \Delta _*(\Delta ^![\Delta ])$ (Chow Homology, Lemma 42.62.6). Since $\Delta _*$ preserves degrees of $0$-cycles it suffices to compute the degree of $\Delta ^![\Delta ]$. The class $\Delta ^![\Delta ]$ is given by capping $[\Delta ]$ with the top Chern class of the normal sheaf of $\Delta \subset X \times X$ (Chow Homology, Lemma 42.54.5). Since the conormal sheaf of $\Delta $ is $\Omega _{X/k}$ (Morphisms, Lemma 29.32.7) we see that the normal sheaf is equal to the tangent sheaf $\mathcal{T}_{X/k} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\Omega _{X/k}, \mathcal{O}_ X)$ as desired.

Equality on the left. By Lemma 45.10.3 we have

\begin{align*} \deg ([\Delta ] \cdot [\Delta ]) & = \int _{X \times X} \gamma ([\Delta ]) \cup \gamma ([\Delta ]) \\ & = \int _{X \times X} \Delta _*1 \cup \gamma ([\Delta ]) \\ & = \int _{X \times X} \Delta _*(\Delta ^*\gamma ([\Delta ])) \\ & = \int _ X \Delta ^*\gamma ([\Delta ]) \end{align*}

We have used Lemmas 45.9.6 and 45.9.1. Write $\gamma ([\Delta ]) = \sum e_{i, j} \otimes e'_{2d - i , j}$ as in Lemma 45.9.7. Recalling that $\Delta ^*$ is given by cup product (Remark 45.9.3) we obtain

\[ \int _ X \sum \nolimits _{i, j} e_{i, j} \cup e'_{2d - i, j} = \sum \nolimits _{i, j} \int _ X e_{i, j} \cup e'_{2d - i, j} = \sum \nolimits _{i, j} (-1)^ i = \sum (-1)^ i\beta _ i \]

as desired. $\square$

The Stacks project

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