The Stacks project

37.71 Weightings and affine stratification numbers

In this section we give a bound for the affine stratification number of a scheme which has a certain kind of cover by an affine scheme.

Lemma 37.71.1. Let $f : X \to Y$ be a morphism of affine schemes which is quasi-finite and of finite presentation. Let $w : X \to \mathbf{Z}_{> 0}$ be a postive weighting of $f$. Let $d < \infty $ be the maximum value of $\int _ f w$. The open

\[ Y_ d = \{ y \in Y \mid (\textstyle {\int }_ f w)(y) = d \} \]

of $Y$ is affine.

Proof. Observe that $\int _ f w$ attains its maximum by Lemma 37.70.5. The set $Y_ d$ is open by Lemma 37.70.4. Thus the statement of the lemma makes sense.

Reduction to the Noetherian case; please skip this paragraph. Recall that a finite type morphism is quasi-finite if and only if it has relative dimension $0$, see Morphisms, Lemma 29.29.5. By Lemma 37.31.9 applied with $d = 0$ we can find a quasi-finite morphism $f_0 : X_0 \to Y_0$ of affine Noetherian schemes and a morphism $Y \to Y_0$ such that $f$ is the base change of $f_0$. Then we can write $Y = \mathop{\mathrm{lim}}\nolimits Y_ i$ as a directed limit of affine schemes of finite type over $Y_0$, see Algebra, Lemma 10.127.2. By Lemma 37.70.9 we can find an $i$ such that our weighting $w$ descends to a weighting $w_ i$ of the base change $f_ i : X_ i \to Y_ i$ of $f_0$. Now if the lemma holds for $f_ i, w_ i$, then it implies the lemma for $f$ as formation of $\int _ f w$ commutes with base change, see Lemma 37.69.1.

Assume $X$ and $Y$ Noetherian. Let $X' \to Y'$ be the base change of $f$ by a morphism $g : Y' \to Y$. The formation of $\int _ f w$ and hence the open $Y_ d$ commute with base change. If $g$ is finite and surjective, then $Y'_ d \to Y_ d$ is finite and surjective. In this case proving that $Y_ d$ is affine is equivalent to showing that $Y'_ d$ is affine, see Cohomology of Schemes, Lemma 30.13.3.

We may choose an immersion $X \to T$ with $T$ finite over $Y$, see Lemma 37.39.3. We are going to apply Morphisms, Lemma 29.48.6 to the finite morphism $T \to Y$. This lemma tells us that there is a finite surjective morphism $Y' \to Y$ such that $Y' \times _ Y T$ is a closed subscheme of a scheme $T'$ finite over $Y'$ which has a special form. By the discussion in the first paragraph, we may replace $Y$ by $Y'$, $T$ by $T'$, and $X$ by $Y' \times _ Y X$. Thus we may assume there is an immersion $X \to T$ (not necessarily open or closed) and closed subschemes $T_ i \subset T$, $i = 1, \ldots , n$ where

  1. $T \to Y$ is finite (and locally free),

  2. $T_ i \to Y$ is an isomorphism, and

  3. $T = \bigcup _{i = 1, \ldots , n} T_ i$ set theoretically.

Let $Y' = \coprod Y_ k$ be the disjoint union of the irreducible components of $Y$ (viewed as integral closed subschemes of $Y$). Then we may base change once more by $Y' \to Y$; here we are using that $Y$ is Noetherian. Thus we may in addition assume $Y$ is integral and Noetherian.

We also may and do assume that $T_ i \not= T_ j$ if $i \not= j$ by removing repeats. Since $Y$ and hence all $T_ i$ are integral, this means that if $T_ i$ and $T_ j$ intersect, then they intersect in a closed subset which maps to a proper closed subset of $Y$.

Observe that $V_ i = X \cap T_ i$ is a locally closed subset which is in addition a closed subscheme of $X$ hence affine. Let $\eta \in Y$ and $\eta _ i \in T_ i$ be the generic points. If $\eta \not\in Y_ d$, then $Y_ d = \emptyset $ and we're done. Assume $\eta \in Y_ d$. Denote $I \in \{ 1, \ldots , n\} $ the subset of indices $i$ such that $\eta _ i \in V_ i$. For $i \in I$ the locally closed subset $V_ i \subset T_ i$ contains the generic point of the irreducible space $T_ i$ and hence is open. On the other hand, since $f$ is open (Lemma 37.69.6), for any $x \in X$ we can find an $i \in I$ and a specialization $\eta _ i \leadsto x$. It follows that $x \in T_ i$ and hence $x \in V_ i$. In other words, we see that $X = \bigcup _{i \in I} V_ i$ set theoretically. We claim that $Y_ d = \bigcap _{i \in I} \mathop{\mathrm{Im}}(V_ i \to Y)$; this will finish the proof as the intersection of affine opens $\mathop{\mathrm{Im}}(V_ i \to Y)$ of $Y$ is affine.

For $y \in Y$ let $f^{-1}(\{ y\} ) = \{ x_1, \ldots , x_ r\} $ in $X$. For each $i \in I$ there is at most one $j(i) \in \{ 1, \ldots , x_ r\} $ such that $\eta _ i \leadsto x_{j(i)}$. In fact, $j(i)$ exists and is equal to $j$ if and only if $x_ j \in V_ i$. If $i \in I$ is such that $j = j(i)$ exists, then $V_ i \to Y$ is an isomorphism in a neighbourhood of $x_ j \mapsto y$. Hence $\bigcup _{i \in I,\ j(i) = j} V_ i \to Y$ is finite after replacing source and target by neighbourhoods of $x_ j \mapsto y$. Thus the definition of a weighting tells us that $w(x_ j) = \sum _{i \in I,\ j(i) = j} w(\eta _ i)$. Thus we see that

\[ (\textstyle {\int }_ f w)(\eta ) = \sum \nolimits _{i \in I} w(\eta _ i) \geq \sum \nolimits _{j(i)\text{ exists}} w(\eta _ i) = \sum \nolimits _ j w(x_ j) = (\textstyle {\int }_ f w)(y) \]

Thus equality holds if and only if $y$ is contained in $\bigcap _{i \in I} \mathop{\mathrm{Im}}(V_ i \to Y)$ which is what we wanted to show. $\square$

Proposition 37.71.2. Let $f : X \to Y$ be a surjective quasi-finite morphism of schemes. Let $w : X \to \mathbf{Z}_{> 0}$ be a positive weighting of $f$. Assume $X$ affine and $Y$ separated1. Then the affine stratification number of $Y$ is at most the number of distinct values of $\int _ f w$.

Proof. Note that since $Y$ is separated, the morphism $X \to Y$ is affine (Morphisms, Lemma 29.11.11). The function $\int _ f w$ attains its maximum $d$ by Lemma 37.70.5. We will use induction on $d$. Consider the open subscheme $Y_ d = \{ y \in Y \mid (\int _ f w)(y) = d\} $ of $Y$ and recall that $f^{-1}(Y_ d) \to Y_ d$ is finite, see Lemma 37.70.6. By Lemma 37.71.1 for every affine open $W \subset Y$ we have that $Y_ d \cap W$ is affine (this uses that $W \times _ Y X$ is affine, being affine over $X$). Hence $Y_ d \to Y$ is an affine morphism of schemes. We conclude that $f^{-1}(Y_ d) = Y_ d \times _ Y X$ is an affine scheme being affine over $X$. Then $f^{-1}(Y_ d) \to Y_ d$ is surjective and hence $Y_ d$ is affine by Limits, Lemma 32.11.1. Set $X' = X \setminus f^{-1}(Y_ d)$ and $Y' = Y \setminus Y_ d$ viewed as closed subschemes of $X$ and $Y$. Since $X'$ is closed in $X$ it is affine. Since $Y'$ is closed in $Y$ it is separated. The morphism $f' : X' \to Y'$ is surjective and $w$ induces a weighting $w'$ of $f'$, see Lemma 37.69.3. By induction $Y'$ has an affine stratification of length $\leq $ the number of distinct values of $\int _{f'} w'$ and the proof is complete. $\square$

[1] It suffices if the diagonal of $Y$ is affine.

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 0F3Q. Beware of the difference between the letter 'O' and the digit '0'.