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\centerline{\Kopfont Schemes}

\bigskip\noindent
{\bf Definitions and results.} K\"ahler differentials.
\item{\bf (a)} Let $R \to A$ be a ring map. The module of K\"ahler
differentials of $A$ over $R$ is 
$$
\Omega^1_{A/R} = \bigoplus\nolimits_{a\in A} A \cdot {\rm d}a \Big/
\big\langle {\rm d}(a_1a_2)-a_1{\rm d}a_2-a_2{\rm d}a_1, {\rm d}r\big\rangle.
$$
The canonical universal $R$-derivation ${\rm d} : A \to \Omega^1_{A/R}$ 
maps $a\mapsto {\rm d}a$.
\item{\bf (b)} Consider the short exact sequence
$$
0 \to I \to A\otimes_R A \to A \to 0
$$
which defines the ideal $I$. There is a canonical derivation
${\rm d} : A \to I/I^2$ which maps $a$ to the class of
$a\otimes 1 - 1 \otimes a$. This is another presentation of
the module of derivations of $A$ over $R$, in other words
$$
(I/I^2, {\rm d}) \cong (\Omega^1_{A/R}, {\rm d}).
$$
\item{\bf (c)} For multiplicative subsets $S_R \subset R$ and
$S_A \subset A$ such that $S_R$ maps into $S_A$ we have
$$
\Omega^1_{S_A^{-1}A / S_R^{-1}R} =
S_A^{-1}\Omega^1_{A/R}.
$$
\item{\bf (d)} If $A$ is a finitely presented $R$-algebra then
$\Omega^1_{A/R}$ is a finitely presented $A$-module. Hence in
this case the {\it fitting} ideals of $\Omega^1_{A/R}$ are defined.
(See exercise set 6 of last semester.)
\item{\bf (e)} Let $f : X \to S$ be a morphism of schemes. There is
a quasi-coherent sheaf of ${\cal O}_X$-modules $\Omega^1_{X/S}$
and a ${\cal O}_S$-linear derivation 
$$
{\rm d} : {\cal O}_X \longrightarrow \Omega^1_{X/S}
$$
such that for any affine opens $\Spec A \subset X$, $\Spec R \subset S$
with $f(\Spec A) \subset \Spec R$ we have
$$
\Gamma(\Spec A, \Omega^1_{X/S}) = \Omega^1_{A/R}
$$
compatibly with ${\rm d}$.

\medskip\item{\bf 1.} Let $k[\epsilon]$ be the ring of dual numbers
over the field $k$, i.e., $\epsilon^2=0$.
\itemitem{\bf (a)} Consider the ring map
$$
R = k[\epsilon] \to A = k[x,\epsilon]/(\epsilon x)
$$
Show that the fitting ideals of $\Omega^1_{A/R}$ are (starting with the
zeroth fitting ideal)
$$
(\epsilon), A, A,\ldots
$$
\itemitem{\bf (b)} Consider the map $R=k[t] \to 
A=k[x,y,t]/(x(y-t)(y-1),x(x-t))$. Show that the fitting ideals of
of $\Omega^1_{A/R}$ in $A$ are (assume characteristic $k$ is zero
for simplicity)
$$
x(2x-t)(2y-t-1)A,\ (x,y,t)\cap (x,y-1,t),\ A,\ A,\ldots
$$
So the $0$-the fitting ideal is cut out by a single element of $A$,
the $1$st fitting ideal defines two closed points of $\Spec A$, and
the others are all trivial.
\itemitem{\bf (c)} Consider the map $R=k[t] \to A=k[x,y,t]/(xy-t^n)$.
Compute the fitting ideals of $\Omega^1_{A/R}$.

\medskip\noindent
{\bf Remark.} The $k$th fitting ideal of $\Omega^1_{X/S}$ is commonly used
to define the singular scheme of the morphism $X \to S$ when $X$ has relative
dimension $k$ over $S$. But as part (a) shows, you have to be careful doing
this when your family does not have ``constant'' fibre dimension, e.g., when 
it is not flat. As part (b) shows, flatness doesn't garantee it works either
(and yes this is a flat family). In ``good cases'' -- such as in (c) -- for
families of curves you expect the $0$-th fitting ideal to be zero and
the $1$st fitting ideal to define (scheme-theoretically) the singular locus.

\medskip\item{\bf 2.} Suppose that $R$ is a ring and 
$$
A = k[x_1,\ldots,x_n]/(f_1,\ldots,f_n).
$$
Note that we are assuming that $A$ is presented by the same
number of equations as variables. Thus the matrix of partial
derivatives
$$
( \partial f_i / \partial x_j )
$$
is $n\times n$, i.e., a square matrix. Assume that
its determinant is invertible as an element in $A$. Note that
this is exactly the condition that says that $\Omega^1_{A/R} = (0)$
in this case of $n$-generators and $n$ relations.
Let $\pi : B' \to B$ be a surjection of $R$-algebras 
whose kernel $J$ has square zero (as an ideal in $B'$).
Let $\varphi : A \to B$ be a homomorphism of $R$-algebras. 
Show there exists a unique homomorphism of $R$-algebras
$\varphi' : A \to B'$ such that $\varphi = \pi \circ \varphi'$.

\medskip\item{\bf 3.} Find a generalization
of the result of the previous exercise to the case where $A=R[x,y]/(f)$.

\bye
