The Stacks project

Proposition 59.95.6. 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 59.55.2. 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 59.95.3.

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 59.55.2. 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 59.53.1 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 $f \in \mathfrak m_ y \subset \mathcal{O}_{Y, y}$ 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 59.95.5 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 59.95.4. Thus by induction hypothesis and Lemma 59.95.2 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 59.74.5. Then the induction hypothesis kicks in to imply the desired vanishing for $\mathcal{G}_ i$1. Finally, we conclude by Theorem 59.51.3.

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 59.92.2 we see that $R^ pg_*j_*\mathcal{F} = 0$ 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 59.53.1 we have $R^2f_*\mathcal{F}_{\overline{\eta }} = H^2(\mathbf{A}^1_{\overline{\eta }}, \mathcal{F}) = 0$ (vanishing by the case of dimension $1$) where $\eta $ is the generic point of $\mathbf{A}^{d - 1}_ K$. 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 59.46.4 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 (5)

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.

Comment #8286 by Xiaolong Liu on

In "Interlude I", we should write "Choose a local equation " instead of "Choose a local equation ". (Please delete the comment above)

There are also:

  • 2 comment(s) on Section 59.95: Cohomological dimension

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F0V. Beware of the difference between the letter 'O' and the digit '0'.