The Stacks project

37.67 Affine stratifications

This material is taken from [RV]. Please read a little bit about stratifications in Topology, Section 5.28 before reading this section.

If $X$ is a scheme, then a stratification of $X$ usually means a stratification of the underlying topological space of $X$. The strata are locally closed subsets. We will view these strata as reduced locally closed subschemes of $X$ using Schemes, Remark 26.12.6.

Definition 37.67.1. Let $X$ be a scheme. An affine stratification is a locally finite stratification $X = \coprod _{i \in I} X_ i$ whose strata $X_ i$ are affine and such that the inclusion morphisms $X_ i \to X$ are affine.

The condition that a stratification $X = \coprod X_ i$ is locally finite is, in the presence of the condition that the inclusion morphisms $X_ i \to X$ are quasi-compact, equivalent to the condition that the strata are locally constructible subsets of $X$, see Properties, Lemma 28.2.7.

The condition that $X_ i \to X$ is an affine morphism is independent on the scheme structure we put on the locally closed subset $X_ i$, see Lemma 37.3.1. Moreover, if $X$ is separated (or more generally has affine diagonal) and $X = \coprod X_ i$ is a locally finite stratification with affine strata, then the morphisms $X_ i \to X$ are affine. See Morphisms, Lemma 29.11.11. This allows us to disregard the condition of affineness of the inclusion morphisms $X_ i \to X$ in most cases of interest.

We are often interested in the case where the partially ordered index set $I$ of the stratification is finite. Recall that the length of a partially ordered set $I$ is the supremum of the lengths $p$ of chains $i_0 < i_1 < \ldots < i_ p$ of elements of $I$.

Lemma 37.67.2. Let $X$ be a scheme. Let $X = \coprod _{i \in I} X_ i$ be a finite affine stratification. There exists an affine stratification with index set $\{ 0, \ldots , n\} $ where $n$ is the length of $I$.

Proof. Recall that we have a partial ordering on $I$ such that the closure of $X_ i$ is contained in $\bigcup _{j \leq i} X_ j$ for all $i \in I$. Let $I' \subset I$ be the set of maximal indices of $I$. If $i \in I'$, then $X_ i$ is open in $X$ because the union of the closures of the other strata is the complement of $X_ i$. Let $U = \bigcup _{i \in I'} X_ i$ viewed as an open subscheme of $X$ so that $U_{red} = \coprod _{i \in I'} X_ i$ as schemes. Then $U$ is an affine scheme by Schemes, Lemma 26.6.8 and Lemma 37.2.3. The morphism $U \to X$ is affine as each $X_ i \to X$, $i \in I'$ is affine by the same reasoning using Lemma 37.3.1. The complement $Z = X \setminus U$ endowed with the reduced induced scheme structure has the affine stratification $Z = \bigcup _{i \in I \setminus I'} X_ i$. Here we use that a morphism of schemes $T \to Z$ is affine if and only if the composition $T \to X$ is affine; this follows from Morphisms, Lemmas 29.11.9, 29.11.7, and 29.11.11. Observe that the partially ordered set $I \setminus I'$ has length exactly one less than the length of $I$. Hence by induction we find that $Z$ has an affine stratification $Z = Z_0 \amalg \ldots \amalg Z_{n - 1}$ with index set $\{ 1, \ldots , n\} $. Setting $Z_ n = U$ we obtain the desired stratification of $X$. $\square$

If a scheme $X$ has a finite affine stratification, then of course $X$ is quasi-compact. A bit less obvious is the fact that it forces $X$ to be quasi-separated as well.

Lemma 37.67.3. Let $X$ be a scheme. The following are equivalent

  1. $X$ has a finite affine stratification, and

  2. $X$ is quasi-compact and quasi-separated.

Proof. Let $X = \bigcup X_ i$ be a finite affine stratification. Since each $X_ i$ is affine hence quasi-compact, we conclude that $X$ is quasi-compact. Let $U, V \subset X$ be affine open. Then $U \cap X_ i$ and $V \cap X_ i$ are affine open in $X_ i$ since $X_ i \to X$ is an affine morphism. Hence $U \cap V \cap X_ i$ is an affine open of the affine scheme $X_ i$ (see Schemes, Lemma 26.21.7 for example). Therefore $U \cap V = \coprod U \cap V \cap X_ i$ is quasi-compact as a finite union of affine strata. We conclude that $X$ is quasi-separated by Schemes, Lemma 26.21.6.

Assume $X$ is quasi-compact and quasi-separated. We may use the induction principle of Cohomology of Schemes, Lemma 30.4.1 to prove the assertion that $X$ has a finite affine stratification. If $X$ is empty, then it has an empty affine stratification. If $X$ is nonempty affine then it has an affine stratification with one stratum. Next, asssume $X = U \cup V$ where $U$ is quasi-compact open, $V$ is affine open, and we have a finite affine stratifications $U = \bigcup _{i \in I} U_ i$ and $U \cap V = \coprod _{j \in J} W_ j$. Denote $Z = X \setminus V$ and $Z' = X \setminus U$. Note that $Z$ is closed in $U$ and $Z'$ is closed in $V$. Observe that $U_ i \cap Z$ and $U_ i \cap W_ j = U_ i \times _ U W_ j$ are affine schemes affine over $U$. (Hints: use that $U_ i \times _ U W_ j \to W_ j$ is affine as a base change of $U_ i \to U$, hence $U_ i \cap W_ j$ is affine, hence $U_ i \cap W_ j \to U_ i$ is affine, hence $U_ i \cap W_ j \to U$ is affine.) It follows that

\[ U = \coprod \nolimits _{i \in I} (U_ i \cap Z) \amalg \coprod \nolimits _{(i, j) \in I \times J} (U_ i \cap W_ j) \]

is a finite affine stratification with partial ordering on $I \amalg I \times J$ given by $i' \leq (i, j) \Leftrightarrow i' \leq i$ and $(i', j') \leq (i, j) \Leftrightarrow i' \leq i$ and $j' \leq j$. Observe that $(U_ i \cap Z) \times _ X V = \emptyset $ and $(U_ i \cap W_ j) \times _ X V = U_ i \cap W_ j$ are affine. Hence the morphisms $U_ i \cap Z \to X$ and $U_ i \cap W_ j \to X$ are affine because we can check affineness of a morphism locally on the target (Morphisms, Lemma 29.11.3) and we have affineness over both $U$ and $V$. To finish the proof we take the stratification above and we add one additional stratum, namely $Z'$, whose index we add as a minimal element to the partially ordered set. $\square$

Definition 37.67.4. Let $X$ be a nonempty quasi-compact and quasi-separated scheme. The affine stratification number is the smallest integer $n \geq 0$ such that the following equivalent conditions are satisfied

  1. there exists a finite affine stratification $X = \coprod _{i \in I} X_ i$ where $I$ has length $n$,

  2. there exists an affine stratification $X = X_0 \amalg X_1 \amalg \ldots \amalg X_ n$ with index set $\{ 0, \ldots , n\} $.

The equivalence of the conditions holds by Lemma 37.67.2. The existence of a finite affine stratification is proven in Lemma 37.67.3.

Lemma 37.67.5. Let $X$ be a separated scheme which has an open covering by $n + 1$ affines. Then the affine stratification number of $X$ is at most $n$.

Proof. Say $X = U_0 \cup \ldots \cup U_ n$ is an affine open covering. Set

\[ X_ i = (U_ i \cup \ldots \cup U_ n) \setminus (U_{i + 1} \cup \ldots \cup U_ n) \]

Then $X_ i$ is affine as a closed subscheme of $U_ i$. The morphism $X_ i \to X$ is affine by Morphisms, Lemma 29.11.11. Finally, we have $\overline{X_ i} \subset X_ i \cup X_{i - 1} \cup \ldots X_0$. $\square$

Lemma 37.67.6. Let $X$ be a Noetherian scheme of dimension $\infty > d \geq 0$. Then the affine stratification number of $X$ is at most $d$.

Proof. By induction on $d$. If $d = 0$, then $X$ is affine, see Properties, Lemma 28.10.5. Assume $d > 0$. Let $\eta _1, \ldots , \eta _ n$ be the generic points of the irreducible components of $X$ (Properties, Lemma 28.5.7). We can cover $X$ by affine opens containing $\eta _1, \ldots , \eta _ n$, see Properties, Lemma 28.29.4. Since $X$ is quasi-compact we can find a finite affine open covering $X = \bigcup _{j = 1, \ldots , m} U_ j$ with $\eta _1, \ldots , \eta _ n \in U_ j$ for all $j = 1, \ldots , m$. Choose an affine open $U \subset U_1 \cap \ldots \cap U_ m$ containing $\eta _1, \ldots , \eta _ n$ (possible by the lemma already quoted). Then the morphism $U \to X$ is affine because $U \to U_ j$ is affine for all $j$, see Morphisms, Lemma 29.11.3. Let $Z = X \setminus U$. By construction $\dim (Z) < \dim (X)$. By induction hypothesis we can find an affine stratification $Z = \bigcup _{i \in \{ 0, \ldots , n\} } Z_ i$ of $Z$ with $n \leq \dim (Z)$. Setting $U = X_{n + 1}$ and $X_ i = Z_ i$ for $i \leq n$ we conclude. $\square$

Proposition 37.67.7. Let $X$ be a nonempty quasi-compact and quasi-separated scheme with affine stratification number $n$. Then $H^ p(X, \mathcal{F}) = 0$, $p > n$ for every quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$.

Proof. We will prove this by induction on the affine stratification number $n$. If $n = 0$, then $X$ is affine and the result is Cohomology of Schemes, Lemma 30.2.2. Assume $n > 0$. By Definition 37.67.4 there is an affine scheme $U$ and an affine open immersion $j : U \to X$ such that the complement $Z$ has affine stratification number $n - 1$. As $U$ and $j$ are affine we have $H^ p(X, j_*(\mathcal{F}|_ U)) = 0$ for $p > 0$, see Cohomology of Schemes, Lemmas 30.2.4 and 30.2.3. Denote $\mathcal{K}$ and $\mathcal{Q}$ the kernel and cokernel of the map $\mathcal{F} \to j_*(\mathcal{F}|_ U)$. Thus we obtain an exact sequence

\[ 0 \to \mathcal{K} \to \mathcal{F} \to j_*(\mathcal{F}|_ U) \to \mathcal{Q} \to 0 \]

of quasi-coherent $\mathcal{O}_ X$-modules (see Schemes, Section 26.24). A standard argument, breaking our exact sequence into short exact sequences and using the long exact cohomology sequence, shows it suffices to prove $H^ p(X, \mathcal{K}) = 0$ and $H^ p(X, \mathcal{Q}) = 0$ for $p \geq n$. Since $\mathcal{F} \to j_*(\mathcal{F}|_ U)$ restricts to an isomorphism over $U$, we see that $\mathcal{K}$ and $\mathcal{Q}$ are supported on $Z$. By Properties, Lemma 28.22.3 we can write these modules as the filtered colimits of their finite type quasi-coherent submodules. Using the fact that cohomology of sheaves on $X$ commutes with filtered colimits, see Cohomology, Lemma 20.19.1, we conclude it suffices to show that if $\mathcal{G}$ is a finite type quasi-coherent module whose support is contained in $Z$, then $H^ p(X, \mathcal{G}) = 0$ for $p \geq n$. Let $Z' \subset X$ be the scheme theoretic support of $\mathcal{G} \oplus \mathcal{O}_ Z$; we may and do think of $\mathcal{G}$ as a quasi-coherent module on $Z'$, see Morphisms, Section 29.5. Then $Z'$ and $Z$ have the same underlying topological space and hence the same affine stratification number, namely $n - 1$. Hence $H^ p(X, \mathcal{G}) = H^ p(Z', \mathcal{G})$ (equality by Cohomology of Schemes, Lemma 30.2.4) vanishes for $p \geq n$ by induction hypothesis. $\square$

Example 37.67.8. Let $k$ be a field and let $X = \mathbf{P}^ n_ k$ be $n$-dimensional projective space over $k$. Lemma 37.67.5 applies to this by Constructions, Lemma 27.13.3. Hence the affine stratification number of $\mathbf{P}^ n_ k$ is at most $n$. On the other hand, we have nonzero cohomology in degree $n$ for some quasi-coherent modules on $\mathbf{P}^ n_ k$, see Cohomology of Schemes, Lemma 30.8.1. Using Proposition 37.67.7 we conclude that the affine stratification number of $\mathbf{P}^ n_ k$ is equal to $n$.

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