The Stacks project

58.32 Tricks in positive characteristic

In Piotr Achinger's paper [Achinger] it is shown that an affine scheme in positive characteristic is always a $K(\pi , 1)$. In this section we explain the more elementary parts of [Achinger]. Namely, we show that for a field $k$ of positive characteristic an affine scheme étale over $\mathbf{A}^ n_ k$ is actually finite étale over $\mathbf{A}^ n_ k$ (by a different morphism). We also show that a closed immersion of connected affine schemes in positive characteristic induces an injective map on étale fundamental groups.

Let $k$ be a field of characteristic $p > 0$. Let

\[ k[x_1, \ldots , x_ n] \longrightarrow A \]

be a surjection of finite type $k$-algebras whose source is the polynomial algebra on $x_1, \ldots , x_ n$. Denote $I \subset k[x_1, \ldots , x_ n]$ the kernel so that we have $A = k[x_1, \ldots , x_ n]/I$. We do not assume $A$ is nonzero (in other words, we allow the case where $A$ is the zero ring and $I = k[x_1, \ldots , x_ n]$). Finally, we assume given a finite étale ring map $\pi : A \to B$.

Suppose given $k, n, k[x_1, \ldots , x_ n] \to A, I, \pi : A \to B$. Let $C$ be a $k$-algebra. Consider commutative diagrams

\[ \xymatrix{ & B \\ C \ar[r] & C/\varphi (I)C \ar[u]^\tau \\ k[x_1, \ldots , x_ n] \ar[u]^\varphi \ar[r] & A \ar[u] \ar@/_3em/[uu]_\pi } \]

where $\varphi $ is an étale $k$-algebra map and $\tau $ is a surjective $k$-algebra map. Let $C, \varphi , \tau $ be given. For any $r \geq 0$ and $y_1, \ldots , y_ r \in C$ which generate $C$ as an algebra over $\mathop{\mathrm{Im}}(\varphi )$ let $s = s(r, y_1, \ldots , y_ r) \in \{ 0, \ldots , r\} $ be the maximal element such that $y_ i$ is integral over $\mathop{\mathrm{Im}}(\varphi )$ for $1 \leq i \leq s$. We define $NF(C, \varphi , \tau )$ to be the minimum value of $r - s = r - s(r, y_1, \ldots , y_ r)$ for all choices of $r$ and $y_1, \ldots , y_ r$ as above. Observe that $NF(C, \varphi , \tau )$ is $0$ if and only if $\varphi $ is finite.

Lemma 58.32.1. In the situation above, if $NF(C, \varphi , \tau ) > 0$, then there exist an étale $k$-algebra map $\varphi '$ and a surjective $k$-algebra map $\tau '$ fitting into the commutative diagram

\[ \xymatrix{ & B \\ C \ar[r] & C/\varphi '(I)C \ar[u]_{\tau '} \\ k[x_1, \ldots , x_ n] \ar[u]^{\varphi '} \ar[r] & A \ar[u] \ar@/_3em/[uu]_\pi } \]

with $NF(C, \varphi ', \tau ') < NF(C, \varphi , \tau )$.

Proof. Choose $r \geq 0$ and $y_1, \ldots , y_ r \in C$ which generate $C$ over $\mathop{\mathrm{Im}}(\varphi )$ and let $0 \leq s \leq r$ be such that $y_1, \ldots , y_ s$ are integral over $\mathop{\mathrm{Im}}(\varphi )$ such that $r - s = NF(C, \varphi , \tau ) > 0$. Since $B$ is finite over $A$, the image of $y_{s + 1}$ in $B$ satisfies a monic polynomial over $A$. Hence we can find $d \geq 1$ and $f_1, \ldots , f_ d \in k[x_1, \ldots , x_ n]$ such that

\[ z = y_{s + 1}^ d + \varphi (f_1) y_{s + 1}^{d - 1} + \ldots + \varphi (f_ d) \in J = \mathop{\mathrm{Ker}}(C \to C/\varphi (I)C \xrightarrow {\tau } B) \]

Since $\varphi : k[x_1, \ldots , x_ n] \to C$ is étale, we can find a nonzero and nonconstant polynomial $g \in k[T_1, \ldots , T_{n + 1}]$ such that

\[ g(\varphi (x_1), \ldots , \varphi (x_ n), z) = 0 \quad \text{in}\quad C \]

To see this you can use for example that $C \otimes _{\varphi , k[x_1, \ldots , x_ n]} k(x_1, \ldots , x_ n)$ is a finite product of finite separable field extensions of $k(x_1, \ldots , x_ n)$ (see Algebra, Lemmas 10.143.4) and hence $z$ satisfies a monic polynomial over $k(x_1, \ldots , x_ n)$. Clearing denominators we obtain $g$.

The existence of $g$ and Algebra, Lemma 10.115.2 produce integers $e_1, e_2, \ldots , e_ n \geq 1$ such that $z$ is integral over the subring $C'$ of $C$ generated by $t_1 = \varphi (x_1) + z^{pe_1}, \ldots , t_ n = \varphi (x_ n) + z^{pe_ n}$. Of course, the elements $\varphi (x_1), \ldots , \varphi (x_ n)$ are also integral over $C'$ as are the elements $y_1, \ldots , y_ s$. Finally, by our choice of $z$ the element $y_{s + 1}$ is integral over $C'$ too.

Consider the ring map

\[ \varphi ' : k[x_1, \ldots , x_ n] \longrightarrow C, \quad x_ i \longmapsto t_ i \]

with image $C'$. Since $\text{d}(\varphi (x_ i)) = \text{d}(t_ i) = \text{d}(\varphi '(x_ i))$ in $\Omega _{C/k}$ (and this is where we use the characteristic of $k$ is $p > 0$) we conclude that $\varphi '$ is étale because $\varphi $ is étale, see Algebra, Lemma 10.151.9. Observe that $\varphi '(x_ i) - \varphi (x_ i) = t_ i - \varphi (x_ i) = z^{pe_ i}$ is in the kernel $J$ of the map $C \to C/\varphi (I)C \to B$ by our choice of $z$ as an element of $J$. Hence for $f \in I$ the element

\[ \varphi '(f) = f(t_1, \ldots , t_ n) = f(\varphi (x_1) + z^{pe_1}, \ldots , \varphi (x_ n) + z^{pe_ n}) = \varphi (f) + \text{element of }(z) \]

is in $J$ as well. In other words, $\varphi '(I)C \subset J$ and we obtain a surjection

\[ \tau ' : C/\varphi '(I)C \longrightarrow C/J \cong B \]

of algebras étale over $A$. Finally, the algebra $C$ is generated by the elements $\varphi (x_1), \ldots , \varphi (x_ n), y_1, \ldots , y_ r$ over $C' = \mathop{\mathrm{Im}}(\varphi ')$ with $\varphi (x_1), \ldots , \varphi (x_ n), y_1, \ldots , y_{s + 1}$ integral over $C' = \mathop{\mathrm{Im}}(\varphi ')$. Hence $NF(C, \varphi ', \tau ') < r - s = NF(C, \varphi , \tau )$. This finishes the proof. $\square$

Lemma 58.32.2. Let $k$ be a field of characteristic $p > 0$. Let $X \to \mathbf{A}^ n_ k$ be an étale morphism with $X$ affine. Then there exists a finite étale morphism $X \to \mathbf{A}^ n_ k$.

Proof. Write $X = \mathop{\mathrm{Spec}}(C)$. Set $A = 0$ and denote $I = k[x_1, \ldots , x_ n]$. By assumption there exists some étale $k$-algebra map $\varphi : k[x_1, \ldots , x_ n] \to C$. Denote $\tau : C/\varphi (I)C \to 0$ the unique surjection. We may choose $\varphi $ and $\tau $ such that $N(C, \varphi , \tau )$ is minimal. By Lemma 58.32.1 we get $N(C, \varphi , \tau ) = 0$. Hence $\varphi $ is finite étale. $\square$

Lemma 58.32.3. Let $k$ be a field of characteristic $p > 0$. Let $Z \subset \mathbf{A}^ n_ k$ be a closed subscheme. Let $Y \to Z$ be finite étale. There exists a finite étale morphism $f : U \to \mathbf{A}^ n_ k$ such that there is an open and closed immersion $Y \to f^{-1}(Z)$ over $Z$.

Proof. Let us turn the problem into algebra. Write $\mathbf{A}^ n_ k = \mathop{\mathrm{Spec}}(k[x_1, \ldots , x_ n])$. Then $Z = \mathop{\mathrm{Spec}}(A)$ where $A = k[x_1, \ldots , x_ n]/I$ for some ideal $I \subset k[x_1, \ldots , x_ n]$. Write $Y = \mathop{\mathrm{Spec}}(B)$ so that $Y \to Z$ corresponds to the finite étale $k$-algebra map $A \to B$.

By Algebra, Lemma 10.143.10 there exists an étale ring map

\[ \varphi : k[x_1, \ldots , x_ n] \to C \]

and a surjective $A$-algebra map $\tau : C/\varphi (I)C \to B$. (We can even choose $C, \varphi , \tau $ such that $\tau $ is an isomorphism, but we won't use this). We may choose $\varphi $ and $\tau $ such that $N(C, \varphi , \tau )$ is minimal. By Lemma 58.32.1 we get $N(C, \varphi , \tau ) = 0$. Hence $\varphi $ is finite étale.

Let $f : U = \mathop{\mathrm{Spec}}(C) \to \mathbf{A}^ n_ k$ be the finite étale morphism corresponding to $\varphi $. The morphism $Y \to f^{-1}(Z) = \mathop{\mathrm{Spec}}(C/\varphi (I)C)$ induced by $\tau $ is a closed immersion as $\tau $ is surjective and open as it is an étale morphism by Morphisms, Lemma 29.36.18. This finishes the proof. $\square$

Here is the main result.

Proposition 58.32.4. Let $p$ be a prime number. Let $i : Z \to X$ be a closed immersion of connected affine schemes over $\mathbf{F}_ p$. For any geometric point $\overline{z}$ of $Z$ the map

\[ \pi _1(Z, \overline{z}) \to \pi _1(X, \overline{z}) \]

is injective.

Proof. Let $Y \to Z$ be a finite étale morphism. It suffices to construct a finite étale morphism $f : U \to X$ such that $Y$ is isomorphic to an open and closed subscheme of $f^{-1}(Z)$, see Lemma 58.4.4. Write $Y = \mathop{\mathrm{Spec}}(A)$ and $X = \mathop{\mathrm{Spec}}(R)$ so the closed immersion $Y \to X$ is given by a surjection $R \to A$. We may write $A = \mathop{\mathrm{colim}}\nolimits A_ i$ as the filtered colimit of its $\mathbf{F}_ p$-subalgebras of finite type. By Lemma 58.14.1 we can find an $i$ and a finite étale morphism $Y_ i \to Z_ i = \mathop{\mathrm{Spec}}(A_ i)$ such that $Y = Z \times _{Z_ i} Y_ i$.

Choose a surjection $\mathbf{F}_ p[x_1, \ldots , x_ n] \to A_ i$. This determines a closed immersion

\[ Z_ i = \mathop{\mathrm{Spec}}(A_ i) \longrightarrow X_ i = \mathbf{A}^ n_{\mathbf{F}_ p} = \mathop{\mathrm{Spec}}(\mathbf{F}_ p[x_1, \ldots , x_ n]) \]

By the universal property of polynomial algebras and since $R \to A$ is surjective, we can find a commutative diagram

\[ \xymatrix{ \mathbf{F}_ p[x_1, \ldots , x_ n] \ar[r] \ar[d] & A_ i \ar[d] \\ R \ar[r] & A } \]

of $\mathbf{F}_ p$-algebras. Thus we have a commutative diagram

\[ \xymatrix{ Y_ i \ar[r] & Z_ i \ar[r] & X_ i \\ Y \ar[u] \ar[r] & Z \ar[u] \ar[r] & X \ar[u] } \]

whose right square is cartesian. Clearly, if we can find $f_ i : U_ i \to X_ i$ finite étale such that $Y_ i$ is isomorphic to an open and closed subscheme of $f_ i^{-1}(Z_ i)$, then the base change $f : U \to X$ of $f_ i$ by $X \to X_ i$ is a solution to our problem. Thus we conclude by applying Lemma 58.32.3 to $Y_ i \to Z_ i \to X_ i = \mathbf{A}^ n_{\mathbf{F}_ p}$. $\square$

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