The Stacks project

Proposition 58.89.6. Let $K$ be a field. Let $X$ be an affine scheme of finite type over $K$. Then we have $\text{cd}(X) \leq \dim (X) + \text{cd}(K)$.

Proof. We will prove this by induction on $\dim (X)$. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_{\acute{e}tale}$.

The case $\dim (X) = 0$. In this case the structure morphism $f : X \to \mathop{\mathrm{Spec}}(K)$ is finite. Hence we see that $R^ if_*\mathcal{F} = 0$ for $i > 0$, see Proposition 58.54.2. Thus $H^ i_{\acute{e}tale}(X, \mathcal{F}) = H^ i_{\acute{e}tale}(\mathop{\mathrm{Spec}}(K), f_*\mathcal{F})$ by the Leray spectral sequence for $f$ (Cohomology on Sites, Lemma 21.14.5) and the result is clear.

The case $\dim (X) = 1$. This is Lemma 58.89.3.

Assume $d = \dim (X) > 1$ and the proposition holds for finite type affine schemes of dimension $< d$ over fields. By Noether normalization, see for example Varieties, Lemma 33.18.2, there exists a finite morphism $f : X \to \mathbf{A}^ d_ K$. Recall that $R^ if_*\mathcal{F} = 0$ for $i > 0$ by Proposition 58.54.2. By the Leray spectral sequence for $f$ (Cohomology on Sites, Lemma 21.14.5) we conclude that it suffices to prove the result for $\pi _*\mathcal{F}$ on $\mathbf{A}^ d_ K$.

Interlude I. Let $j : X \to Y$ be an open immersion of smooth $d$-dimensional varieties over $K$ (not necessarily affine) whose complement is the support of an effective Cartier divisor $D$. The sheaves $R^ qj_*\mathcal{F}$ for $q > 0$ are supported on $D$. We claim that $(R^ qj_*\mathcal{F})_{\overline{y}} = 0$ for $a = \text{trdeg}_ K(\kappa (y)) > d - q$. Namely, by Theorem 58.52.1 we have

\[ (R^ qj_*\mathcal{F})_{\overline{y}} = H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}^{sh}) \times _ Y X, \mathcal{F}) \]

Choose a local equation $f \in \mathfrak m_ y = \mathcal{O}_{Y, y}$ for $D$. Then we have

\[ \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}^{sh}) \times _ Y X = \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}^{sh}[1/f]) \]

Using Lemma 58.89.5 we get an embedding

\[ K(t_1, \ldots , t_ a)^{sep}(x) = K(t_1, \ldots , t_ a)^{sep}[x]_{(x)}[1/x] \longrightarrow \mathcal{O}_{Y, y}^{sh}[1/f] \]

Since the transcendence degree over $K$ of the fraction field of $\mathcal{O}_{Y, y}^{sh}$ is $d$, we see that $\mathcal{O}_{Y, y}^{sh}[1/f]$ is a filtered colimit of $(d - a - 1)$-dimensional finite type algebras over the field $K(t_1, \ldots , t_ a)^{sep}(x)$ which itself has cohomological dimension $1$ by Lemma 58.89.4. Thus by induction hypothesis and Lemma 58.89.2 we obtain the desired vanishing.

Interlude II. Let $Z$ be a smooth variety over $K$ of dimension $d - 1$. Let $E_ a \subset Z$ be the set of points $z \in Z$ with $\text{trdeg}_ K(\kappa (z)) \leq a$. Observe that $E_ a$ is closed under specialization, see Varieties, Lemma 33.20.3. Suppose that $\mathcal{G}$ is a torsion abelian sheaf on $Z$ whose support is contained in $E_ a$. Then we claim that $H^ b_{\acute{e}tale}(Z, \mathcal{G}) = 0$ for $b > a + \text{cd}(K)$. Namely, we can write $\mathcal{G} = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i$ with $\mathcal{G}_ i$ a torsion abelian sheaf supported on a closed subscheme $Z_ i$ contained in $E_ a$, see Lemma 58.73.5. Then the induction hypothesis kicks in to imply the desired vanishing for $\mathcal{G}_ i$1. Finally, we conclude by Theorem 58.51.3.

Consider the commutative diagram

\[ \xymatrix{ \mathbf{A}^ d_ K \ar[rd]_ f \ar[rr]_-j & & \mathbf{P}^1_ K \times _ K \mathbf{A}^{d - 1}_ K \ar[ld]^ g \\ & \mathbf{A}^{d - 1}_ K } \]

Observe that $j$ is an open immersion of smooth $d$-dimensional varieties whose complement is an effective Cartier divisor $D$. Thus we may use the results obtained in interlude I. We are going to study the relative Leray spectral sequence

\[ E_2^{p, q} = R^ pg_*R^ qj_*\mathcal{F} \Rightarrow R^{p + q}f_*\mathcal{F} \]

Since $R^ qj_*\mathcal{F}$ for $q > 0$ is supported on $D$ and since $g|_ D : D \to \mathbf{A}^{d - 1}_ K$ is an isomorphism, we find $R^ pg_*R^ qj_*\mathcal{F} = 0$ for $p > 0$ and $q > 0$. Moreover, we have $R^ qj_*\mathcal{F} = 0$ for $q > d$. On the other hand, $g$ is a proper morphism of relative dimension $1$. Hence by Lemma 58.88.2 we see that $R^ pj_*\mathcal{F} = 0$ for $p > 2$. Thus the $E_2$-page of the spectral sequence looks like this

\[ \begin{matrix} g_*R^ dj_*\mathcal{F} & 0 & 0 \\ \ldots & \ldots & \ldots \\ g_*R^2j_*\mathcal{F} & 0 & 0 \\ g_*R^1j_*\mathcal{F} & 0 & 0 \\ g_*j_*\mathcal{F} & R^1g_*j_*\mathcal{F} & R^2g_*j_*\mathcal{F} \end{matrix} \]

We conclude that $R^ qf_*\mathcal{F} = g_*R^ qj_*\mathcal{F}$ for $q > 2$. By interlude I we see that the support of $R^ qf_*\mathcal{F}$ for $q > 2$ is contained in the set of points of $\mathbf{A}^{d - 1}_ K$ whose residue field has transcendence degree $\leq d - q$. By interlude II

\[ H^ p(\mathbf{A}^{d - 1}_ K, R^ qf_*\mathcal{F}) = 0 \text{ for }p > d - q + \text{cd}(K)\text{ and }q > 2 \]

On the other hand, by Theorem 58.52.1 we have $R^2f_*\mathcal{F}_{\overline{\eta }} = H^2(\mathbf{A}^1_{\overline{\eta }}, \mathcal{F}) = 0$ (vanishing by the case of dimension $1$). Hence by interlude II again we see

\[ H^ p(\mathbf{A}^{d - 1}_ K, R^2f_*\mathcal{F}) = 0 \text{ for }p > d - 2 + \text{cd}(K) \]

Finally, we have

\[ H^ p(\mathbf{A}^{d - 1}_ K, R^ qf_*\mathcal{F}) = 0 \text{ for }p > d - 1 + \text{cd}(K)\text{ and }q = 0, 1 \]

by induction hypothesis. Combining everything we just said with the Leray spectral sequence $H^ p(\mathbf{A}^{d - 1}_ K, R^ qf_*\mathcal{F}) \Rightarrow H^{p + q}(\mathbf{A}^ d_ K, \mathcal{F})$ we conclude. $\square$

[1] Here we first use Proposition 58.46.4 to write $\mathcal{G}_ i$ as the pushforward of a sheaf on $Z_ i$, the induction hypothesis gives the vanishing for this sheaf on $Z_ i$, and the Leray spectral sequence for $Z_ i \to Z$ gives the vanishing for $\mathcal{G}_ i$.

Comments (2)

Comment #4985 by Noah Olander on

There's a typo right before you write down the E2 page of the spectral sequence: You should write for instead of .

Comment #4995 by Noah Olander on

I think when you write "On the other hand, by Theorem 03Q9 we have , you mean is the generic point of but you don't say what is.


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