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\centerline{\Kopfont Schemes}

\smallskip
\centerline{Exercises 6}

\bigskip\noindent
{\bf Schemes -- Examples again}

\bigskip\noindent
Feel free to quote results from Hartshorne or EGA.

\bigskip\item{\bf 1.} {\v C}ech cohomology. Here $k$ is a field.
\itemitem{\bf (a)} Let $X$ be a scheme with an open covering
${\cal U} : X = U_1 \cup U_2$, with $U_1 = \Spec k[x]$, $U_2= \Spec k[y]$
with $U_1 \cap U_2 = \Spec k[z,1/z]$ and with open immersions
$U_1 \cap U_2 \to U_1$ resp.\ $U_1 \cap U_2 \to U_2$ determined
by $x \mapsto z$ resp.\ $y \mapsto z$ (and I really mean this).
(We've seen in the lectures that such an $X$ exists; it is the affine
line zith zero doubled.) Compute ${\mathaccent 20 H}^1({\cal U}, {\cal O})$;
eg.\ give a basis for it as a $k$-vectorspace.
\itemitem{\bf (b)} For each element in
${\mathaccent 20 H}^1({\cal U}, {\cal O})$
construct an exact sequence of sheaves of ${\cal O}_X$-modules
$$
0 \to {\cal O}_X \to E \to {\cal O}_X \to 0
$$ 
such that the boundary $\delta(1) \in {\mathaccent 20 H}^1({\cal U}, {\cal O})$
equals the given element. (Part of the problem is to make sense of this. 
It is also OK to show abstractly such a thing has to exist.)

\medskip\noindent
{\bf Definition of delta.} Suppose that 
$$
0 \to {\cal F}_1 \to {\cal F}_2 \to {\cal F}_3 \to 0
$$
is a short exact sequence of abelian sheaves on any topological space $X$.
The boundary map
$\delta : H^0(X, {\cal F}_3) \to {\mathaccent 20 H}^1(X, {\cal F}_1)$
is defined as follows. Take an element $\tau \in H^0(X, {\cal F}_3)$.
Choose an open covering ${\cal U} : X = \bigcup_{i\in I} U_i$ such
that for each $i$ there exists a section $\tilde \tau_i \in {\cal F}_2$
lifting the restriction of $\tau$ to $U_i$. Then consider the assignment
$$
(i_0, i_1) \longmapsto
\tilde \tau_{i_0}|_{U_{i_0i_1}} - \tilde \tau_{i_1}|_{U_{i_0i_1}}.
$$
This is clearly a 1-coboundary in the {\v C}ech complex
${\mathaccent 20 C}^\ast({\cal U}, {\cal F}_2)$. But we observe that
(thinking of ${\cal F}_1$ as a subsheaf of ${\cal F}_2$) the RHS
always is a section of ${\cal F}_1$ over $U_{i_0i_1}$. Hence we
see that the assignment defines a 1-cochain in the complex
${\mathaccent 20 C}^\ast({\cal U}, {\cal F}_2)$. The cohomology class of
this 1-cochain is by definition $\delta(\tau)$.

\medskip\item{\bf 2.} Algebra. (Silly and should be easy.)
\itemitem{\bf (a)} Give an example of a ring $A$ and a nonsplit
short exact sequence of $A$-modules
$$
0 \to M_1 \to M_2 \to M_3 \to 0.
$$
\itemitem{\bf (b)} Give an example of a nonsplit sequence of $A$-modules
as above and a faithfully flat $A \to B$ such that 
$$
0 \to M_1\otimes_AB \to M_2\otimes_AB \to M_3\otimes_AB \to 0.
$$
is split as a sequence of $B$-modules.

\medskip\item{\bf 3.} Maps of $\Proj$. Let $R$ and $S$ be graded rings. So
$R = \oplus_{d \geq 0} R_d$ and $R_a \cdot R_b \subset R_{a+b}$.
Suppose we have a ring map
$$
\varphi : R \to S
$$
such that there exists an integer $e \geq 1$ such that
$\varphi( R_d ) \subset S_{de}$.
\itemitem{\bf (a)} For which elements $\gp \in \Proj(S)$ is
there a well-defined corresponding point in $\Proj(R)$? In other words,
find a suitable open $U \subset \Proj(S)$ such that $\varphi$ defines
a continuous map $\Proj(\varphi) : U \to \Proj(R)$.
\itemitem{\bf (b)} Give an example where $U \not = \Proj(S)$.
\itemitem{\bf (c)} Give an example where $U = \Proj(S)$.
\itemitem{\bf (d)} (Do not write this down.) Convince yourself that 
the continuous map $U \to \Proj(R)$ comes canonically with
a map on sheaves so that $\Proj(\varphi)$ is a morphism of schemes:
$$
\Proj(S) \supset U \longrightarrow \Proj(R)
$$

\medskip\noindent{\bf Notation.} Let $R$ be a graded ring as above and
let $n \geq 0$ be an integer. Let $X = \Proj(R)$. Then there is a unique
quasi-coherent ${\cal O}_X$-module ${\cal O}_X(n)$ on $X$ such that
for every homogeneous element $f \in R$ of positive degree we have
${\cal O}_X |_{D_{+}(f)}$ is the quasi-coherent sheaf associated to the
$R_{(f)} = (R_f)_0$-module $(R_f)_n$ ($=$elements homogenous of degree
$n$ in $R_f = R[1/f]$). See Hartshorne, page 116+. Note that there are
natural maps
$$
{\cal O}_X(n_1) \otimes_{{\cal O}_X} {\cal O}_X(n_2) \longrightarrow
{\cal O}_X(n_1+n_2)
$$

\medskip\item{\bf 4.} Pathologies in $\Proj$. 
Give examples of $R$ as above such that
\itemitem{\bf (a)} ${\cal O}_X(1)$ is not an invertible ${\cal O}_X$-module.
\itemitem{\bf (b)} ${\cal O}_X(1)$ is invertible, but the
natural map ${\cal O}_X(1) \otimes_{{\cal O}_X} {\cal O}_X(1) \to
{\cal O}_X(2)$ is NOT an isomorphism.

\bye
