111.50 Differentials
Definitions and results. Kähler differentials.
Let $R \to A$ be a ring map. The module of Kähler differentials of $A$ over $R$ is denoted $\Omega _{A/R}$. It is generated by the elements $\text{d}a$, $a \in A$ subject to the relations:
\[ \text{d}(a_1 + a_2) = \text{d}a_1 + \text{d}a_2,\quad \text{d}(a_1a_2) = a_1\text{d}a_2 + a_2\text{d}a_1,\quad \text{d}r = 0 \]
The canonical universal $R$-derivation $\text{d} : A \to \Omega _{A/R}$ maps $a\mapsto \text{d}a$.
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 $\text{d} : A \to I/I^2$ 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
\[ (I/I^2, \text{d}) \cong (\Omega _{A/R}, \text{d}). \]
For multiplicative subsets $S_ R \subset R$ and $S_ A \subset A$ such that $S_ R$ maps into $S_ A$ we have
\[ \Omega _{S_ A^{-1}A / S_ R^{-1}R} = S_ A^{-1}\Omega _{A/R}. \]
If $A$ is a finitely presented $R$-algebra then $\Omega _{A/R}$ is a finitely presented $A$-module. Hence in this case the fitting ideals of $\Omega _{A/R}$ are defined.
Let $f : X \to S$ be a morphism of schemes. There is a quasi-coherent sheaf of ${\mathcal O}_ X$-modules $\Omega _{X/S}$ and a ${\mathcal O}_ S$-linear derivation
\[ \text{d} : {\mathcal O}_ X \longrightarrow \Omega _{X/S} \]
such that for any affine opens $\mathop{\mathrm{Spec}}(A) = U \subset X$, $\mathop{\mathrm{Spec}}(R) = V \subset S$ with $f(U) \subset V$ we have
\[ \Gamma (\mathop{\mathrm{Spec}}(A), \Omega _{X/S}) = \Omega _{A/R} \]
compatibly with $\text{d}$.
Exercise 111.50.1. Let $k[\epsilon ]$ be the ring of dual numbers over the field $k$, i.e., $\epsilon ^2 = 0$.
Consider the ring map
\[ R = k[\epsilon ] \to A = k[x, \epsilon ]/(\epsilon x) \]
Show that the Fitting ideals of $\Omega _{A/R}$ are (starting with the zeroth Fitting ideal)
\[ (\epsilon ), A, A, \ldots \]
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 $\Omega _{A/R}$ 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 $\mathop{\mathrm{Spec}}(A)$, and the others are all trivial.
Consider the map $R = k[t] \to A = k[x, y, t]/(xy-t^ n)$. Compute the Fitting ideals of $\Omega _{A/R}$.
Exercise 111.50.3. Suppose that $R$ is a ring and
\[ A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ n). \]
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 $\Omega _{A/R} = (0)$ 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 $\varphi = \pi \circ \varphi '$.
Exercise 111.50.4. Find a generalization of the result of Exercise 111.50.3 to the case where $A = R[x, y]/(f)$.
Exercise 111.50.5. Let $k$ be a field, let $f_1, \ldots , f_ c \in k[x_1, \ldots , x_ n]$, and let $A = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$. Assume that $f_ j(0, \ldots , 0) = 0$. This means that $\mathfrak m = (x_1, \ldots , x_ n)A$ is a maximal ideal. Prove that the local ring $A_\mathfrak m$ is regular if the rank of the matrix
\[ (\partial f_ j/ \partial x_ i)|_{(x_1, \ldots , x_ n) = (0, \ldots , 0)} \]
is $c$. What is the dimension of $A_\mathfrak m$ in this case? Show that the converse is false by giving an example where $A_\mathfrak m$ is regular but the rank is less than $c$; what is the dimension of $A_\mathfrak m$ in your example?
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