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 $n$.


Comments (0)


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 0F2R. Beware of the difference between the letter 'O' and the digit '0'.