The Stacks project

53.5 Riemann-Roch

Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. In Varieties, Section 33.44 we have defined the degree of a locally free $\mathcal{O}_ X$-module $\mathcal{E}$ of constant rank by the formula
\begin{equation} \label{curves-equation-degree} \deg (\mathcal{E}) = \chi (X, \mathcal{E}) - \text{rank}(\mathcal{E})\chi (X, \mathcal{O}_ X) \end{equation}

see Varieties, Definition 33.44.1. In the chapter on Chow Homology we defined the first Chern class of $\mathcal{E}$ as an operation on cycles (Chow Homology, Section 42.38) and we proved that
\begin{equation} \label{curves-equation-degree-c1} \deg (\mathcal{E}) = \deg (c_1(\mathcal{E}) \cap [X]_1) \end{equation}

see Chow Homology, Lemma 42.41.3. Combining ( and ( we obtain our first version of the Riemann-Roch formula
\begin{equation} \label{curves-equation-rr} \chi (X, \mathcal{E}) = \deg (c_1(\mathcal{E}) \cap [X]_1) + \text{rank}(\mathcal{E})\chi (X, \mathcal{O}_ X) \end{equation}

If $\mathcal{L}$ is an invertible $\mathcal{O}_ X$-module, then we can also consider the numerical intersection $(\mathcal{L} \cdot X)$ as defined in Varieties, Definition 33.45.3. However, this does not give anything new as
\begin{equation} \label{curves-equation-numerical-degree} (\mathcal{L} \cdot X) = \deg (\mathcal{L}) \end{equation}

by Varieties, Lemma 33.45.12. If $\mathcal{L}$ is ample, then this integer is positive and is called the degree
\begin{equation} \label{curves-equation-degree-X} \deg _\mathcal {L}(X) = (\mathcal{L} \cdot X) = \deg (\mathcal{L}) \end{equation}

of $X$ with respect to $\mathcal{L}$, see Varieties, Definition 33.45.10.

To obtain a true Riemann-Roch theorem we would like to write $\chi (X, \mathcal{O}_ X)$ as the degree of a canonical zero cycle on $X$. We refer to [F] for a fully general version of this. We will use duality to get a formula in the case where $X$ is Gorenstein; however, in some sense this is a cheat (for example because this method cannot work in higher dimension).

We first use Lemmas 53.4.1 and 53.4.2 to get a relation between the euler characteristic of $\mathcal{O}_ X$ and the euler characteristic of the dualizing complex or the dualizing module.

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. $\square$

We will use Lemma 53.5.1 to get the desired formula for $\chi (X, \mathcal{O}_ X)$ in the case that $\omega _ X$ is invertible, i.e., that $X$ is Gorenstein. The statement is that $-1/2$ of the first Chern class of $\omega _ X$ capped with the cycle $[X]_1$ associated to $X$ is a natural zero cycle on $X$ with half-integer coefficients whose degree is $\chi (X, \mathcal{O}_ X)$. The occurence of fractions in the statement of Riemann-Roch cannot be avoided.

Lemma 53.5.2 (Riemann-Roch). Let $X$ be a proper scheme over a field $k$ which is Gorenstein and equidimensional of dimension $1$. Let $\omega _ X$ be as in Lemma 53.4.1. Then

  1. $\omega _ X$ is an invertible $\mathcal{O}_ X$-module,

  2. $\deg (\omega _ X) = -2\chi (X, \mathcal{O}_ X)$,

  3. for a locally free $\mathcal{O}_ X$-module $\mathcal{E}$ of constant rank we have

    \[ \chi (X, \mathcal{E}) = \deg (\mathcal{E}) - \textstyle {\frac{1}{2}} \text{rank}(\mathcal{E}) \deg (\omega _ X) \]

    and $\dim _ k(H^ i(X, \mathcal{E})) = \dim _ k(H^{1 - i}(X, \mathcal{E}^\vee \otimes _{\mathcal{O}_ X} \omega _ X))$ for all $i \in \mathbf{Z}$.

Nonsingular (normal) curves are Gorenstein, see Duality for Schemes, Lemma 48.24.3.

Proof. Recall that Gorenstein schemes are Cohen-Macaulay (Duality for Schemes, Lemma 48.24.2) and hence $\omega _ X$ is a dualizing module on $X$, see Lemma 53.4.2. It follows more or less from the definition of the Gorenstein property that the dualizing sheaf is invertible, see Duality for Schemes, Section 48.24. By ( applied to $\omega _ X$ we have

\[ \chi (X, \omega _ X) = \deg (c_1(\omega _ X) \cap [X]_1) + \chi (X, \mathcal{O}_ X) \]

Combined with Lemma 53.5.1 this gives

\[ 2\chi (X, \mathcal{O}_ X) = - \deg (c_1(\omega _ X) \cap [X]_1) = - \deg (\omega _ X) \]

the second equality by ( Putting this back into ( for $\mathcal{E}$ gives the displayed formula of the lemma. The symmetry in dimensions is a consequence of duality for $X$, see Remark 53.4.3. $\square$

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