Schreier Hypothesis Statement

53.62. The Artin-Schreier sequence

Let $p$ be a prime number. Let $S$ be a scheme in characteristic $p$. The Artin-Schreier sequence is the short exact sequence $$ 0 \longrightarrow \underline{\mathbf{Z}/p\mathbf{Z}}_S \longrightarrow \mathbf{G}_{a, S} \xrightarrow{F-1} \mathbf{G}_{a, S} \longrightarrow 0 $$ where $F - 1$ is the map $x \mapsto x^p - x$.

Lemma 53.62.1. Let $p$ be a prime. Let $S$ be a scheme of characteristic $p$.

  1. If $S$ is affine, then $H_{\acute{e}tale}^q(S, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$ for all $q \geq 2$.
  2. If $S$ is a quasi-compact and quasi-separated scheme of dimension $d$, then $H_{\acute{e}tale}^q(S, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$ for all $q \geq 2 + d$.

Proof. Recall that the étale cohomology of the structure sheaf is equal to its cohomology on the underlying topological space (Theorem 53.22.4). The first statement follows from the Artin-Schreier exact sequence and the vanishing of cohomology of the structure sheaf on an affine scheme (Cohomology of Schemes, Lemma 29.2.2). The second statement follows by the same argument from the vanishing of Cohomology, Proposition 20.23.4 and the fact that $S$ is a spectral space (Properties, Lemma 27.2.4). $\square$

Lemma 53.62.2. Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $V$ be a finite dimensional $k$-vector space. Let $F : V \to V$ be a frobenius linear map, i.e., an additive map such that $F(\lambda v) = \lambda^p F(v)$ for all $\lambda \in k$ and $v \in V$. Then $F - 1 : V \to V$ is surjective with kernel a finite dimensional $\mathbf{F}_p$-vector space of dimension $\leq \dim_k(V)$.

Proof. If $F = 0$, then the statement holds. If we have a filtration of $V$ by $F$-stable subvector spaces such that the statement holds for each graded piece, then it holds for $(V, F)$. Combining these two remarks we may assume the kernel of $F$ is zero.

Choose a basis $v_1, \ldots, v_n$ of $V$ and write $F(v_i) = \sum a_{ij} v_j$. Observe that $v = \sum \lambda_i v_i$ is in the kernel if and only if $\sum \lambda_i^p a_{ij} v_j = 0$. Since $k$ is algebraically closed this implies the matrix $(a_{ij})$ is invertible. Let $(b_{ij})$ be its inverse. Then to see that $F - 1$ is surjective we pick $w = \sum \mu_i v_i \in V$ and we try to solve $$ (F - 1)(\sum \lambda_iv_i) = \sum \lambda_i^p a_{ij} v_j - \sum \lambda_j v_j = \sum \mu_j v_j $$ This is equivalent to $$ \sum \lambda_j^p v_j - \sum b_{ij} \lambda_i v_j = \sum b_{ij} \mu_i v_j $$ in other words $$ \lambda_j^p - \sum b_{ij} \lambda_i = \sum b_{ij} \mu_i, \quad j = 1, \ldots, \dim(V). $$ The algebra $$ A = k[x_1, \ldots, x_n]/ (x_j^p - \sum b_{ij} x_i - \sum b_{ij} \mu_i) $$ is standard smooth over $k$ (Algebra, Definition 10.135.6) because the matrix $(b_{ij})$ is invertible and the partial derivatives of $x_j^p$ are zero. A basis of $A$ over $k$ is the set of monomials $x_1^{e_1} \ldots x_n^{e_n}$ with $e_i < p$, hence $\dim_k(A) = p^n$. Since $k$ is algebraically closed we see that $\mathop{\mathrm{Spec}}(A)$ has exactly $p^n$ points. It follows that $F - 1$ is surjective and every fibre has $p^n$ points, i.e., the kernel of $F - 1$ is a group with $p^n$ elements. $\square$

Lemma 53.62.3. Let $X$ be a separated scheme of finite type over a field $k$. Let $\mathcal{F}$ be a coherent sheaf of $\mathcal{O}_X$-modules. Then $\dim_k H^d(X, \mathcal{F}) < \infty$ where $d = \dim(X)$.

Proof. We will prove this by induction on $d$. The case $d = 0$ holds because in that case $X$ is the spectrum of a finite dimensional $k$-algebra $A$ (Varieties, Lemma 32.20.2) and every coherent sheaf $\mathcal{F}$ corresponds to a finite $A$-module $M = H^0(X, \mathcal{F})$ which has $\dim_k M < \infty$.

Assume $d > 0$ and the result has been shown for separated schemes of finite type of dimension $< d$. The scheme $X$ is Noetherian. Consider the property $\mathcal{P}$ of coherent sheaves on $X$ defined by the rule $$ \mathcal{P}(\mathcal{F}) \Leftrightarrow \dim_k H^d(X, \mathcal{F}) < \infty $$ We are going to use the result of Cohomology of Schemes, Lemma 29.12.4 to prove that $\mathcal{P}$ holds for every coherent sheaf on $X$.

Let $$ 0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0 $$ be a short exact sequence of coherent sheaves on $X$. Consider the long exact sequence of cohomology $$ H^d(X, \mathcal{F}_1) \to H^d(X, \mathcal{F}) \to H^d(X, \mathcal{F}_2) $$ Thus if $\mathcal{P}$ holds for $\mathcal{F}_1$ and $\mathcal{F}_2$, then it hods for $\mathcal{F}$.

Let $Z \subset X$ be an integral closed subscheme. Let $\mathcal{I}$ be a coherent sheaf of ideals on $Z$. To finish the proof have to show that $H^d(X, i_*\mathcal{I}) = H^d(Z, \mathcal{I})$ is finite dimensional. If $\dim(Z) < d$, then the result holds because the cohomology group will be zero (Cohomology, Proposition 20.21.7). In this way we reduce to the situation discussed in the following paragraph.

Assume $X$ is a variety of dimension $d$ and $\mathcal{F} = \mathcal{I}$ is a coherent ideal sheaf. In this case we have a short exact sequence $$ 0 \to \mathcal{I} \to \mathcal{O}_X \to i_*\mathcal{O}_Z \to 0 $$ where $i : Z \to X$ is the closed subscheme defined by $\mathcal{I}$. By induction hypothesis we see that $H^{d - 1}(Z, \mathcal{O}_Z) = H^{d - 1}(X, i_*\mathcal{O}_Z)$ is finite dimensional. Thus we see that it suffices to prove the result for the structure sheaf.

We can apply Chow's lemma (Cohomology of Schemes, Lemma 29.18.1) to the morphism $X \to \mathop{\mathrm{Spec}}(k)$. Thus we get a diagram $$ \xymatrix{ X \ar[rd]_g & X' \ar[d]^-{g'} \ar[l]^\pi \ar[r]_i & \mathbf{P}^n_k \ar[dl] \\ & \mathop{\mathrm{Spec}}(k) & } $$ as in the statement of Chow's lemma. Also, let $U \subset X$ be the dense open subscheme such that $\pi^{-1}(U) \to U$ is an isomorphism. We may assume $X'$ is a variety as well, see Cohomology of Schemes, Remark 29.18.2. The morphism $i' = (i, \pi) : X' \to \mathbf{P}^n_X$ is a closed immersion (loc. cit.). Hence $$ \mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^n_k}(1) \cong (i')^*\mathcal{O}_{\mathbf{P}^n_X}(1) $$ is $\pi$-relatively ample (for example by Morphisms, Lemma 28.37.7). Hence by Cohomology of Schemes, Lemma 29.16.2 there exists an $n \geq 0$ such that $R^p\pi_*\mathcal{L}^{\otimes n} = 0$ for all $p > 0$. Set $\mathcal{G} = \pi_*\mathcal{L}^{\otimes n}$. Choose any nonzero global section $s$ of $\mathcal{L}^{\otimes n}$. Since $\mathcal{G} = \pi_*\mathcal{L}^{\otimes n}$, the section $s$ corresponds to section of $\mathcal{G}$, i.e., a map $\mathcal{O}_X \to \mathcal{G}$. Since $s|_U \not = 0$ as $X'$ is a variety and $\mathcal{L}$ invertible, we see that $\mathcal{O}_X|_U \to \mathcal{G}|_U$ is nonzero. As $\mathcal{G}|_U = \mathcal{KL}^{\otimes n}|_{\pi^{-1}(U)}$ is invertible we conclude that we have a short exact sequence $$ 0 \to \mathcal{O}_X \to \mathcal{G} \to \mathcal{Q} \to 0 $$ where $\mathcal{Q}$ is coherent and supported on a proper closed subscheme of $X$. Arguing as before using our induction hypothesis, we see that it suffices to prove $\dim H^d(X, \mathcal{G}) < \infty$.

By the Leray spectral sequence (Cohomology, Lemma 20.14.6) we see that $H^d(X, \mathcal{G}) = H^d(X', \mathcal{L}^{\otimes n})$. Let $\overline{X}' \subset \mathbf{P}^n_k$ be the closure of $X'$. Then $\overline{X}'$ is a projective variety of dimension $d$ over $k$ and $X' \subset \overline{X}'$ is a dense open. The invertible sheaf $\mathcal{L}$ is the restriction of $\mathcal{O}_{\overline{X}'}(n)$ to $X$. By Cohomology, Proposition 20.23.4 the map $$ H^d(\overline{X}', \mathcal{O}_{\overline{X}'}(n)) \longrightarrow H^d(X', \mathcal{L}^{\otimes n}) $$ is surjective. Since the cohomology group on the left has finite dimension by Cohomology of Schemes, Lemma 29.14.1 the proof is complete. $\square$

Lemma 53.62.4. Let $X$ be separated of finite type over an algebraically closed field $k$ of characteristic $p > 0$. Then $H_{\acute{e}tale}^q(X, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$ for $q \geq dim(X) + 1$.

Proof. Let $d = \dim(X)$. By the vanishing established in Lemma 53.62.1 it suffices to show that $H_{\acute{e}tale}^{d + 1}(X, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$. By Lemma 53.62.3 we see that $H^d(X, \mathcal{O}_X)$ is a finite dimensional $k$-vector space. Hence the long exact cohomology sequence associated to the Artin-Schreier sequence ends with $$ H^d(X, \mathcal{O}_X) \xrightarrow{F - 1} H^d(X, \mathcal{O}_X) \to H^{d + 1}_{\acute{e}tale}(X, \mathbf{Z}/p\mathbf{Z}) \to 0 $$ By Lemma 53.62.2 the map $F - 1$ in this sequence is surjective. This proves the lemma. $\square$

Lemma 53.62.5. Let $X$ be a proper scheme over an algebraically closed field $k$ of characteristic $p > 0$. Then

  1. $H_{\acute{e}tale}^q(X, \underline{\mathbf{Z}/p\mathbf{Z}})$ is a finite $\mathbf{Z}/p\mathbf{Z}$-module for all $q$, and
  2. $H^q_{\acute{e}tale}(X, \underline{\mathbf{Z}/p\mathbf{Z}}) \to H^q_{\acute{e}tale}(X_{k'}, \underline{\mathbf{Z}/p\mathbf{Z}}))$ is an isomorphism if $k \subset k'$ is an extension of algebraically closed fields.

Proof. By Cohomology of Schemes, Lemma 29.19.2) and the comparison of cohomology of Theorem 53.22.4 the cohomology groups $H^q_{\acute{e}tale}(X, \mathbf{G}_a) = H^q(X, \mathcal{O}_X)$ are finite dimensional $k$-vector spaces. Hence by Lemma 53.62.2 the long exact cohomology sequence associated to the Artin-Schreier sequence, splits into short exact sequences $$ 0 \to H_{\acute{e}tale}^q(X, \underline{\mathbf{Z}/p\mathbf{Z}}) \to H^q(X, \mathcal{O}_X) \xrightarrow{F - 1} H^q(X, \mathcal{O}_X) \to 0 $$ and moreover the $\mathbf{F}_p$-dimension of the cohomology groups $H_{\acute{e}tale}^q(X, \underline{\mathbf{Z}/p\mathbf{Z}})$ is equal to the $k$-dimension of the vector space $H^q(X, \mathcal{O}_X)$. This proves the first statement. The second statement follows as $H^q(X, \mathcal{O}_X) \otimes_k k' \to H^q(X_{k'}, \mathcal{O}_{X_{k'}})$ is an isomorphism by flat base change (Cohomology of Schemes, Lemma 29.5.2). $\square$

Step 5: Hypothesis Statement

Hypothesis Statement

(will be worked on in class prior to due date)

Your hypothesis statement will be turned in during science class, reviewed by the teacher and returned. Below is a short explanation of a hypothesis statement and some examples of hypothesis statements.

Hypothesis statement--a prediction that can be tested or an educated guess.

In a hypothesis statement, students make a prediction about what they think will happen or is happening in their experiment. They try to answer their question or problem.

EXAMPLES:

Question: Why do leaves change colors in the fall?

Hypothesis: I think that leaves change colors in the fall because they are not being exposed to as much sunlight.

Hypothesis: Bacterial growth may be affected by temperature.

Hypothesis: Chocolate may cause pimples

 

All of these are examples of hypotheses because they use the tentative word "may." However, their form in not particularly useful. Using the word does not suggest how you would go about proving it. If these statements had not been written carefully, they may not have been a hypotheses at all.

A better way to write a hypotheses is to use a formalized hypotheses

Example: If skin cancer is related to ultraviolet light, then people with a high exposure to uv light will have a higher frequency of skin cancer.

Example: If leaf color change is related to temperature, then exposing plants to low temperatures will result in changes in leaf color.

Example: If the rate of photosynthesis is related to wave lengths of light, then exposing a plant to different colors of light will produce different amounts of oxygen.

Example: If the volume of a gas is related to temperature, then increasing the temperature will increase the volume.

These examples contain the words, if and then. Formalized hypotheses contain two variables. One is "independent" and the other is "dependent." The independent variable is the one you, the scientist control and the dependent variable is the one that you observe and/or measure the results.

The ultimate value of a formalized hypotheses is it forces us to think about what results we should look for in an experiment.

Example: If the diffusion rate (dependent variable) through a membrane is related to molecular size (independent variable), then the smaller the molecule the faster it will pass through the membrane.

 

 

 

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