The Stacks project

57.89 Cohomological dimension

We can deduce some bounds on the cohomological dimension of schemes and on the cohomological dimension of fields using the results in Section 57.80 and one, seemingly innocuous, application of the proper base change theorem (in the proof of Proposition 57.89.6).

Definition 57.89.1. Let $X$ be a quasi-compact and quasi-separated scheme. The cohomological dimension of $X$ is the smallest element

\[ \text{cd}(X) \in \{ 0, 1, 2, \ldots \} \cup \{ \infty \} \]

such that for any abelian torsion sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $H^ i_{\acute{e}tale}(X, \mathcal{F}) = 0$ for $i > \text{cd}(X)$. If $X = \mathop{\mathrm{Spec}}(A)$ we sometimes call this the cohomological dimension of $A$.

If the scheme is in characteristic $p$, then we often can obtain sharper bounds for the vanishing of cohomology of $p$-power torsion sheaves. We will address this elsewhere (insert future reference here).

Lemma 57.89.2. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a directed limit of a system of quasi-compact and quasi-separated schemes with affine transition morphisms. Then $\text{cd}(X) \leq \max \text{cd}(X_ i)$.

Proof. Denote $f_ i : X \to X_ i$ the projections. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_{\acute{e}tale}$. Then we have $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits f_ i^{-1}f_{i, *}\mathcal{F}$ by Lemma 57.51.9. Thus $H^ q_{\acute{e}tale}(X, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ q_{\acute{e}tale}(X_ i, f_{i, *}\mathcal{F})$ by Theorem 57.51.3. The lemma follows. $\square$

Lemma 57.89.3. Let $K$ be a field. Let $X$ be a $1$-dimensional affine scheme of finite type over $K$. Then $\text{cd}(X) \leq 1 + \text{cd}(K)$.

Proof. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_{\acute{e}tale}$. Consider the Leray spectral sequence for the morphism $f : X \to \mathop{\mathrm{Spec}}(K)$. We obtain

\[ E_2^{p, q} = H^ p(\mathop{\mathrm{Spec}}(K), R^ qf_*\mathcal{F}) \]

converging to $H^{p + q}_{\acute{e}tale}(X, \mathcal{F})$. The stalk of $R^ qf_*\mathcal{F}$ at a geometric point $\mathop{\mathrm{Spec}}(\overline{K}) \to \mathop{\mathrm{Spec}}(K)$ is the cohomology of the pullback of $\mathcal{F}$ to $X_{\overline{K}}$. Hence it vanishes in degrees $\geq 2$ by Theorem 57.80.9. $\square$

Lemma 57.89.4. Let $L/K$ be a field extension. Then we have $\text{cd}(L) \leq \text{cd}(K) + \text{trdeg}_ K(L)$.

Proof. If $\text{trdeg}_ K(L) = \infty $, then this is clear. If not then we can find a sequence of extensions $L= L_ r/L_{r - 1}/ \ldots /L_1/L_0 = K$ such that $\text{trdeg}_{L_ i}(L_{i + 1}) = 1$ and $r = \text{trdeg}_ K(L)$. Hence it suffices to prove the lemma in the case that $r = 1$. In this case we can write $L = \mathop{\mathrm{colim}}\nolimits A_ i$ as a filtered colimit of its finite type $K$-subalgebras. By Lemma 57.89.2 it suffices to prove that $\text{cd}(A_ i) \leq 1 + \text{cd}(K)$. This follows from Lemma 57.89.3. $\square$

Lemma 57.89.5. Let $K$ be a field. Let $X$ be a scheme of finite type over $K$. Let $x \in X$. Set $a = \text{trdeg}_ K(\kappa (x))$ and $d = \dim _ x(X)$. Then there is a map

\[ K(t_1, \ldots , t_ a)^{sep} \longrightarrow \mathcal{O}_{X, x}^{sh} \]

such that

  1. the residue field of $\mathcal{O}_{X, x}^{sh}$ is a purely inseparable extension of $K(t_1, \ldots , t_ a)^{sep}$,

  2. $\mathcal{O}_{X, x}^{sh}$ is a filtered colimit of finite type $K(t_1, \ldots , t_ a)^{sep}$-algebras of dimension $\leq d - a$.

Proof. We may assume $X$ is affine. By Noether normalization, after possibly shrinking $X$ again, we can choose a finite morphism $\pi : X \to \mathbf{A}^ d_ K$, see Algebra, Lemma 10.114.5. Since $\kappa (x)$ is a finite extension of the residue field of $\pi (x)$, this residue field has transcendence degree $a$ over $K$ as well. Thus we can find a finite morphism $\pi ' : \mathbf{A}^ d_ K \to \mathbf{A}^ d_ K$ such that $\pi '(\pi (x))$ corresponds to the generic point of the linear subspace $\mathbf{A}^ a_ K \subset \mathbf{A}^ d_ K$ given by setting the last $d - a$ coordinates equal to zero. Hence the composition

\[ X \xrightarrow {\pi ' \circ \pi } \mathbf{A}^ d_ K \xrightarrow {p} \mathbf{A}^ a_ K \]

of $\pi ' \circ \pi $ and the projection $p$ onto the first $a$ coordinates maps $x$ to the generic point $\eta \in \mathbf{A}^ a_ K$. The induced map

\[ K(t_1, \ldots , t_ a)^{sep} = \mathcal{O}_{\mathbf{A}^ a_ k, \eta }^{sh} \longrightarrow \mathcal{O}_{X, x}^{sh} \]

on étale local rings satisfies (1) since it is clear that the residue field of $\mathcal{O}_{X, x}^{sh}$ is an algebraic extension of the separably closed field $K(t_1, \ldots , t_ a)^{sep}$. On the other hand, if $X = \mathop{\mathrm{Spec}}(B)$, then $\mathcal{O}_{X, x}^{sh} = \mathop{\mathrm{colim}}\nolimits B_ j$ is a filtered colimit of étale $B$-algebras $B_ j$. Observe that $B_ j$ is quasi-finite over $K[t_1, \ldots , t_ d]$ as $B$ is finite over $K[t_1, \ldots , t_ d]$. We may similarly write $K(t_1, \ldots , t_ a)^{sep} = \mathop{\mathrm{colim}}\nolimits A_ i$ as a filtered colimit of étale $K[t_1, \ldots , t_ a]$-algebras. For every $i$ we can find an $j$ such that $A_ i \to K(t_1, \ldots , t_ a)^{sep} \to \mathcal{O}_{X, x}^{sh}$ factors through a map $\psi _{i, j} : A_ i \to B_ j$. Then $B_ j$ is quasi-finite over $A_ i[t_{a + 1}, \ldots , t_ d]$. Hence

\[ B_{i, j} = B_ j \otimes _{\psi _{i, j}, A_ i} K(t_1, \ldots , t_ a)^{sep} \]

has dimension $\leq d - a$ as it is quasi-finite over $K(t_1, \ldots , t_ a)^{sep}[t_{a + 1}, \ldots , t_ d]$. The proof of (2) is now finished as $\mathcal{O}_{X, x}^{sh}$ is a filtered colimit1 of the algebras $B_{i, j}$. Some details omitted. $\square$

Proposition 57.89.6. 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 57.54.2. 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 57.89.3.

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 57.54.2. 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 57.52.1 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 57.89.5 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 57.89.4. Thus by induction hypothesis and Lemma 57.89.2 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 57.73.5. Then the induction hypothesis kicks in to imply the desired vanishing for $\mathcal{G}_ i$2. Finally, we conclude by Theorem 57.51.3.

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 57.88.2 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 57.52.1 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. $\square$

Lemma 57.89.7. Let $K$ be a field. Let $X$ be an affine scheme of finite type over $K$. Let $E_ a \subset X$ be the set of points $x \in X$ with $\text{trdeg}_ K(\kappa (x)) \leq a$. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_{\acute{e}tale}$ whose support is contained in $E_ a$. Then $H^ b_{\acute{e}tale}(X, \mathcal{F}) = 0$ for $b > a + \text{cd}(K)$.

Proof. We can write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ with $\mathcal{F}_ i$ a torsion abelian sheaf supported on a closed subscheme $Z_ i$ contained in $E_ a$, see Lemma 57.73.5. Then Proposition 57.89.6 gives the desired vanishing for $\mathcal{F}_ i$. Details omitted; hints: first use Proposition 57.46.4 to write $\mathcal{F}_ i$ as the pushforward of a sheaf on $Z_ i$, use the vanishing for this sheaf on $Z_ i$, and use the Leray spectral sequence for $Z_ i \to Z$ to get the vanishing for $\mathcal{F}_ i$. Finally, we conclude by Theorem 57.51.3. $\square$

Lemma 57.89.8. Let $f : X \to Y$ be an affine morphism of schemes of finite type over a field $K$. Let $E_ a(X)$ be the set of points $x \in X$ with $\text{trdeg}_ K(\kappa (x)) \leq a$. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_{\acute{e}tale}$ whose support is contained in $E_ a$. Then $R^ qf_*\mathcal{F}$ has support contained in $E_{a - q}(Y)$.

Proof. The question is local on $Y$ hence we can assume $Y$ is affine. Then $X$ is affine too and we can choose a diagram

\[ \xymatrix{ X \ar[d]_ f \ar[r]_ i & \mathbf{A}^{n + m}_ K \ar[d]^{\text{pr}} \\ Y \ar[r]^ j & \mathbf{A}^ n_ K } \]

where the horizontal arrows are closed immersions and the vertical arrow on the right is the projection (details omitted). Then $j_*R^ qf_*\mathcal{F} = R^ q\text{pr}_*i_*\mathcal{F}$ by the vanishing of the higher direct images of $i$ and $j$, see Proposition 57.54.2. Moreover, the description of the stalks of $j_*$ in the proposition shows that it suffices to prove the vanishing for $j_*R^ qf_*\mathcal{F}$. Thus we may assume $f$ is the projection morphism $\text{pr} : \mathbf{A}^{n + m}_ K \to \mathbf{A}^ n_ K$ and an abelian torsion sheaf $\mathcal{F}$ on $\mathbf{A}^{n + m}_ K$ satisfying the assumption in the statement of the lemma.

Let $y$ be a point in $\mathbf{A}^ n_ K$. By Theorem 57.52.1 we have

\[ (R^ q\text{pr}_*\mathcal{F})_{\overline{y}} = H^ q(\mathbf{A}^{n + m}_ K \times _{A^ n_ K} \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}^{sh}), \mathcal{F}) = H^ q(\mathbf{A}^ m_{\mathcal{O}_{Y, y}^{sh}}, \mathcal{F}) \]

Say $b = \text{trdeg}_ K(\kappa (y))$. From Lemma 57.89.5 we get an embedding

\[ L = K(t_1, \ldots , t_ b)^{sep} \longrightarrow \mathcal{O}_{Y, y}^{sh} \]

Write $\mathcal{O}_{Y, y}^{sh} = \mathop{\mathrm{colim}}\nolimits B_ i$ as the filtered colimit of finite type $L$-subalgebras $B_ i \subset \mathcal{O}_{Y, y}^{sh}$ containing the ring $K[T_1, \ldots , T_ n]$ of regular functions on $\mathbf{A}^ n_ K$. Then we get

\[ \mathbf{A}^ m_{\mathcal{O}_{Y, y}^{sh}} = \mathop{\mathrm{lim}}\nolimits \mathbf{A}^ m_{B_ i} \]

If $z \in \mathbf{A}^ m_{B_ i}$ is a point in the support of $\mathcal{F}$, then the image $x$ of $z$ in $\mathbf{A}^{m + n}_ K$ satisfies $\text{trdeg}_ K(\kappa (x)) \leq a$ by our assumption on $\mathcal{F}$ in the lemma. Since $\mathcal{O}_{Y, y}^{sh}$ is a filtered colimit of étale algebras over $K[T_1, \ldots , T_ n]$ and since $B_ i \subset \mathcal{O}_{Y, y}^{sh}$ we see that $\kappa (z)/\kappa (x)$ is algebraic (some details omitted). Then $\text{trdeg}_ K(\kappa (z)) \leq a$ and hence $\text{trdeg}_ L(\kappa (z)) \leq a - b$. By Lemma 57.89.7 we see that

\[ H^ q(\mathbf{A}^ m_{B_ i}, \mathcal{F}) = 0\text{ for }q > a - b \]

Thus by Theorem 57.51.3 we get $(Rf_*\mathcal{F})_{\overline{y}} = 0$ for $q > a - b$ as desired. $\square$

[1] Let $R$ be a ring. Let $A = \mathop{\mathrm{colim}}\nolimits _{i \in I} A_ i$ be a filtered colimit of finitely presented $R$-algebras. Let $B = \mathop{\mathrm{colim}}\nolimits _{j \in J} B_ j$ be a filtered colimit of $R$-algebras. Let $A \to B$ be an $R$-algebra map. Assume that for all $i \in I$ there is a $j \in J$ and an $R$-algebra map $\psi _{i, j} : A_ i \to B_ j$. Say $(i', j', \psi _{i', j'}) \geq (i, j, \psi _{i, j})$ if $i' \geq i$, $j' \geq j$, and $\psi _{i, j}$ and $\psi _{i', j'}$ are compatible. Then the collection of triples forms a directed set and $B = \mathop{\mathrm{colim}}\nolimits B_ j \otimes _{\psi _{i, j} A_ i} A$.
[2] Here we first use Proposition 57.46.4 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 $\mathcal{G}_ i$.

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