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 1st 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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