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 33.43 we have defined the degree of a locally free $\mathcal{O}_ X$-module $\mathcal{E}$ of constant rank by the formula

53.5.0.1
\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 33.43.1. In the chapter on Chow Homology we defined the first Chern class of $\mathcal{E}$ as an operation on cycles (Chow Homology, Section 42.37) and we proved that

53.5.0.2
\begin{equation} \label{curves-equation-degree-c1} \deg (\mathcal{E}) = \deg (c_1(\mathcal{E}) \cap [X]_1) \end{equation}

see Chow Homology, Lemma 42.40.3. Combining (53.5.0.1) and (53.5.0.2) we obtain our first version of the Riemann-Roch formula

53.5.0.3
\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 33.44.3. However, this does not give anything new as

53.5.0.4
\begin{equation} \label{curves-equation-numerical-degree} (\mathcal{L} \cdot X) = \deg (\mathcal{L}) \end{equation}

by Varieties, Lemma 33.44.12. If $\mathcal{L}$ is ample, then this integer is positive and is called the degree

53.5.0.5
\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 33.44.10.

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 (53.5.0.3) 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 (53.5.0.2). Putting this back into (53.5.0.3) 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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