Lemma 38.38.1. Let Z, X, X', E be an almost blow up square as in Example 38.37.11. Then H^ p(X', \mathcal{O}_{X'}) = 0 for p > 0 and \Gamma (X, \mathcal{O}_ X) \to \Gamma (X', \mathcal{O}_{X'}) is a surjective map of rings whose kernel is an ideal of square zero.
38.38 Absolute weak normalization and h coverings
In this section we use the criteria found in Section 38.37 to exhibit some h sheaves and we relate h sheafification of the structure sheaf to absolute weak normalization. We will need the following elementary lemma to do this.
Proof. First assume that A = \mathbf{Z}[f_1, f_2] is the polynomial ring. In this case our almost blow up square is the blowing up of X = \mathop{\mathrm{Spec}}(A) in the closed subscheme Z and in fact X' \subset \mathbf{P}^1_ X is an effective Cartier divisor cut out by the global section f_2T_0 - f_1 T_1 of \mathcal{O}_{\mathbf{P}^1_ X}(1). Thus we have a resolution
Using the description of the cohomology given in Cohomology of Schemes, Section 30.8 it follows that in this case \Gamma (X, \mathcal{O}_ X) \to \Gamma (X', \mathcal{O}_{X'}) is an isomorphism and H^1(X', \mathcal{O}_{X'}) = 0.
Next, we observe that any diagram as in Example 38.37.11 is the base change of the diagram in the previous paragraph by the ring map \mathbf{Z}[f_1, f_2] \to A. Hence by More on Morphisms, Lemmas 37.72.1, 37.72.2, and 37.72.4 we conclude that H^1(X', \mathcal{O}_{X'}) is zero in general and the surjectivity of the map H^0(X, \mathcal{O}_ X) \to H^0(X', \mathcal{O}_{X'}) in general.
Next, in the general case, let us study the kernel. If a \in A maps to zero, then looking on affine charts we see that
for some r \geq 0 and a_0, \ldots , a_ r \in A and similarly
for some s \geq 0 and b_0, \ldots , b_ s \in A. This means we have
If (a', r', a'_ i, s', b'_ j) is a second such system, then we have
as desired. \square
For an \mathbf{F}_ p-algebra A we set \mathop{\mathrm{colim}}\nolimits _ F A equal to the colimit of the system
where F : A \to A, a \mapsto a^ p is the Frobenius endomorphism.
Lemma 38.38.2. Let p be a prime number. Let S be a scheme over \mathbf{F}_ p. Let (\mathit{Sch}/S)_ h be a site as in Definition 38.34.13. There is a unique sheaf \mathcal{F} on (\mathit{Sch}/S)_ h such that
for any quasi-compact and quasi-separated object X of (\mathit{Sch}/S)_ h.
Proof. Denote \mathcal{F} the Zariski sheafification of the functor
For quasi-compact and quasi-separated schemes X we have \mathcal{F}(X) = \mathop{\mathrm{colim}}\nolimits _ F \Gamma (X, \mathcal{O}_ X). by Sheaves, Lemma 6.29.1 and the fact that \mathcal{O} is a sheaf for the Zariski topology. Thus it suffices to show that \mathcal{F} is a h sheaf. To prove this we check conditions (1), (2), (3), and (4) of Lemma 38.37.12. Condition (1) holds because we performed an (almost unnecessary) Zariski sheafification. Condition (2) holds because \mathcal{O} is an fppf sheaf (Descent, Lemma 35.8.1) and if A is the equalizer of two maps B \to C of \mathbf{F}_ p-algebras, then \mathop{\mathrm{colim}}\nolimits _ F A is the equalizer of the two maps \mathop{\mathrm{colim}}\nolimits _ F B \to \mathop{\mathrm{colim}}\nolimits _ F C.
We check condition (3). Let A, f, J be as in Example 38.37.10. We have to show that
This reduces to the following algebra question: suppose a', a'' \in A are such that F^ n(a' - a'') \in fA + J. Find a \in A and m \geq 0 such that a - F^ m(a') \in J and a - F^ m(a'') \in fA and show that the pair (a, m) is uniquely determined up to a replacement of the form (a, m) \mapsto (F(a), m + 1). To do this just write F^ n(a' - a'') = f h + g with h \in A and g \in J and set a = F^ n(a') - g = F^ n(a'') + fh and set m = n. To see uniqueness, suppose (a_1, m_1) is a second solution. By a replacement of the form given above we may assume m = m_1. Then we see that a - a_1 \in J and a - a_1 \in fA. Since J is annihilated by a power of f we see that a - a_1 is a nilpotent element. Hence F^ k(a - a_1) is zero for some large k. Thus after doing more replacements we get a = a_1.
We check condition (4). Let X, X', Z, E be as in Example 38.37.11. By Lemma 38.38.1 we see that
is bijective. Since E = \mathbf{P}^1_ Z in this case we also see that \mathcal{F}(Z) \to \mathcal{F}(E) is bijective. Thus the conclusion holds in this case as well. \square
Let p be a prime number. For an \mathbf{F}_ p-algebra A we set \mathop{\mathrm{lim}}\nolimits _ F A equal to the limit of the inverse system
where F : A \to A, a \mapsto a^ p is the Frobenius endomorphism.
Lemma 38.38.3. Let p be a prime number. Let S be a scheme over \mathbf{F}_ p. Let (\mathit{Sch}/S)_ h be a site as in Definition 38.34.13. The rule
defines a sheaf on (\mathit{Sch}/S)_ h.
Proof. To prove \mathcal{F} is a sheaf, let's check conditions (1), (2), (3), and (4) of Lemma 38.37.12. Condition (1) holds because limits of sheaves are sheaves and \mathcal{O} is a Zariski sheaf. Condition (2) holds because \mathcal{O} is an fppf sheaf (Descent, Lemma 35.8.1) and if A is the equalizer of two maps B \to C of \mathbf{F}_ p-algebras, then \mathop{\mathrm{lim}}\nolimits _ F A is the equalizer of the two maps \mathop{\mathrm{lim}}\nolimits _ F B \to \mathop{\mathrm{lim}}\nolimits _ F C.
We check condition (3). Let A, f, J be as in Example 38.37.10. We have to show that
is bijective. Since J is annihilated by a power of f we see that \mathfrak a = fA \cap J is a nilpotent ideal, i.e., there exists an n such that \mathfrak a^ n = 0. It is straightforward to verify that in this case \mathop{\mathrm{lim}}\nolimits _ F A \to \mathop{\mathrm{lim}}\nolimits _ F A/\mathfrak a is bijective.
We check condition (4). Let X, X', Z, E be as in Example 38.37.11. By Lemma 38.38.1 and the same argument as above we see that
is bijective. Since E = \mathbf{P}^1_ Z in this case we also see that \mathcal{F}(Z) \to \mathcal{F}(E) is bijective. Thus the conclusion holds in this case as well. \square
In the following lemma we use the absolute weak normalization X^{awn} of a scheme X, see Morphisms, Section 29.47.
Lemma 38.38.4. Let (\mathit{Sch}/S)_{ph} be a site as in Topologies, Definition 34.8.11. The rule
is a sheaf on (\mathit{Sch}/S)_{ph}.
Proof. To prove \mathcal{F} is a sheaf, let's check conditions (1) and (2) of Topologies, Lemma 34.8.15. Condition (1) holds because formation of X^{awn} commutes with open coverings, see Morphisms, Lemma 29.47.7 and its proof.
Let \pi : Y \to X be a surjective proper morphism. We have to show that the equalizer of the two maps
is equal to \Gamma (X^{awn}, \mathcal{O}_{X^{awn}}). Let f be an element of this equalizer. Then we consider the morphism
Since Y^{awn} \to X is universally closed, the scheme theoretic image Z of f is a closed subscheme of \mathbf{A}^1_ X proper over X and f : Y^{awn} \to Z is surjective. See Morphisms, Lemma 29.41.10. Thus Z \to X is finite (Morphisms, Lemma 29.44.11) and surjective.
Let k be a field and let z_1, z_2 : \mathop{\mathrm{Spec}}(k) \to Z be two morphisms equalized by Z \to X. We claim that z_1 = z_2. It suffices to show the images \lambda _ i = z_ i^*f \in k agree (as the structure sheaf of Z is generated by f over the structure sheaf of X). To see this we choose a field extension K/k and morphisms y_1, y_2 : \mathop{\mathrm{Spec}}(K) \to Y^{awn} such that z_ i \circ (\mathop{\mathrm{Spec}}(K) \to \mathop{\mathrm{Spec}}(k)) = f \circ y_ i. This is possible by the surjectivity of the map Y^{awn} \to Z. Choose an algebraically closed extension \Omega /k of very large cardinality. For any k-algebra maps \sigma _ i : K \to \Omega we obtain
Since the canonical morphism (Y \times _ X Y)^{awn} \to Y^{awn} \times _ X Y^{awn} is a universal homeomorphism and since \Omega is algebraically closed, we can lift the composition above uniquely to a morphism \mathop{\mathrm{Spec}}(\Omega ) \to (Y \times _ X Y)^{awn}. Since f is in the equalizer above, this proves that \sigma _1(\lambda _1) = \sigma _2(\lambda _2). An easy lemma about field extensions shows that this implies \lambda _1 = \lambda _2; details omitted.
We conclude that Z \to X is universally injective, i.e., Z \to X is injective on points and induces purely inseparated residue field extensions (Morphisms, Lemma 29.10.2). All in all we conclude that Z \to X is a universal homeomorphism, see Morphisms, Lemma 29.45.5.
Let g : X^{awn} \to Z be the map obtained from the universal property of X^{awn}. Then Y^{awn} \to X^{awn} \to Z and f : Y^{awn} \to Z are two morphisms over X. By the universal property of Y^{awn} \to Y the two corresponding morphisms Y^{awn} \to Y \times _ X Z over Y have to be equal. This implies that g \circ \pi ^{wan} = f as morphisms into \mathbf{A}^1_ X and we conclude that g \in \Gamma (X^{awn}, \mathcal{O}_{X^{awn}}) is the element we were looking for. \square
Lemma 38.38.5. Let S be a scheme. Choose a site (\mathit{Sch}/S)_ h as in Definition 38.34.13. The rule
is the sheafification of the “structure sheaf” \mathcal{O} on (\mathit{Sch}/S)_ h. Similarly for the ph topology.
Proof. In Lemma 38.38.4 we have seen that the rule \mathcal{F} of the lemma defines a sheaf in the ph topology and hence a fortiori a sheaf for the h topology. Clearly, there is a canonical map of presheaves of rings \mathcal{O} \to \mathcal{F}. To finish the proof, it suffices to show
if f \in \mathcal{O}(X) maps to zero in \mathcal{F}(X), then there is a h covering \{ X_ i \to X\} such that f|_{X_ i} = 0, and
given f \in \mathcal{F}(X) there is a h covering \{ X_ i \to X\} such that f|_{X_ i} is the image of f_ i \in \mathcal{O}(X_ i).
Let f be as in (1). Then f|_{X^{awn}} = 0. This means that f is locally nilpotent. Thus if X' \subset X is the closed subscheme cut out by f, then X' \to X is a surjective closed immersion of finite presentation. Hence \{ X' \to X\} is the desired h covering. Let f be as in (2). After replacing X by the members of an affine open covering we may assume X = \mathop{\mathrm{Spec}}(A) is affine. Then f \in A^{awn}, see Morphisms, Lemma 29.47.6. By Morphisms, Lemma 29.46.11 we can find a ring map A \to B of finite presentation such that \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A) is a universal homeomorphism and such that f is the image of an element b \in B under the canonical map B \to A^{awn}. Then \{ \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)\} is an h covering and we conclude. The statement about the ph topology follows in the same manner (or it can be deduced from the statement for the h topology). \square
Let p be a prime number. An \mathbf{F}_ p-algebra A is called perfect if the map F : A \to A, x \mapsto x^ p is an automorphism of A.
Lemma 38.38.6. Let p be a prime number. An \mathbf{F}_ p-algebra A is absolutely weakly normal if and only if it is perfect.
Proof. It is immediate from condition (2)(b) in Morphisms, Definition 29.47.1 that if A is absolutely weakly normal, then it is perfect.
Assume A is perfect. Suppose x, y \in A with x^3 = y^2. If p > 3 then we can write p = 2n + 3m for some n, m > 0. Choose a, b \in A with a^ p = x and b^ p = y. Setting c = a^ n b^ m we have
and hence c^2 = x. Similarly c^3 = y. If p = 2, then write x = a^2 to get a^6 = y^2 which implies a^3 = y. If p = 3, then write y = a^3 to get x^3 = a^6 which implies x = a^2.
Suppose x, y \in A with \ell ^\ell x = y^\ell for some prime number \ell . If \ell \not= p, then a = y/\ell satisfies a^\ell = x and \ell a = y. If \ell = p, then y = 0 and x = a^ p for some a. \square
Lemma 38.38.7. Let p be a prime number.
If A is an \mathbf{F}_ p-algebra, then \mathop{\mathrm{colim}}\nolimits _ F A = A^{awn}.
If S is a scheme over \mathbf{F}_ p, then the h sheafification of \mathcal{O} sends a quasi-compact and quasi-separated X to \mathop{\mathrm{colim}}\nolimits _ F \Gamma (X, \mathcal{O}_ X).
Proof. Proof of (1). Observe that A \to \mathop{\mathrm{colim}}\nolimits _ F A induces a universal homeomorphism on spectra by Algebra, Lemma 10.46.7. Thus it suffices to show that B = \mathop{\mathrm{colim}}\nolimits _ F A is absolutely weakly normal, see Morphisms, Lemma 29.47.6. Note that the ring map F : B \to B is an automorphism, in other words, B is a perfect ring. Hence Lemma 38.38.6 applies.
Proof of (2). This follows from (1) and Lemmas 38.38.2 and 38.38.5 by looking affine locally. \square
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