Differentials of a polynomial quotient and the Jacobian cokernel

Define ${\mathrm{\partial }}_i\text{\hspace{0.17em}⁣}:P\to P$ on the monomial basis of [F3] by ${\mathrm{\partial }}_i(x^a):=a_i{x}^{a-e_i}$ when $a_i\ge 1$, and ${\mathrm{\partial }}_i(x^a):=0$ when $a_i=0$, extended $A$-linearly. Since exponents add under multiplication and $\mathbb{N}$-multiplication distributes, ${\mathrm{\partial }}_i(fg)=f{\mathrm{\partial }}_i(g)+g{\mathrm{\partial }}_i(f)$ for monomials and hence, by $A$-bilinearity of multiplication, for all $f,g\in P$; also ${\mathrm{\partial }}_i(A)=0$. So each ${\mathrm{\partial }}_i$ is an $A$-derivation of $P$. By [F1] there are $P$-linear ${\alpha }_i\text{\hspace{0.17em}⁣}:{\mathrm{\Omega }}_{P\mathrm{/}A}\to P$ with ${\alpha }_i(\mathrm{d}{x}_j)={\mathrm{\partial }}_i(x_j)={\delta }_{ij}$. The $P$-linear map $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:P^n\to {\mathrm{\Omega }}_{P\mathrm{/}A}$, $\mathrm{\Phi }(e_i)=\mathrm{d}{x}_i$, is surjective: by additivity and the Leibniz rule $\mathrm{d}(x^a)={\sum }_i{a}_i{x}^{a-e_i}\mathrm{d}{x}_i$, and an arbitrary element of $P$ is a finite $A$-linear combination of monomials, so every $\mathrm{d}f$ lies in ${\sum }_{i}P\mathrm{d}{x}_i$, and these elements generate ${\mathrm{\Omega }}_{P\mathrm{/}A}$ by construction. The $P$-linear endomorphism $\mathrm{\Phi }\circ ({\alpha }_1,\dots ,{\alpha }_n)$ of ${\mathrm{\Omega }}_{P\mathrm{/}A}$ fixes each generator $\mathrm{d}{x}_j$, hence is the identity; therefore $\mathrm{\Phi }$ is injective as well, and $\mathrm{d}{x}_1,\dots ,\mathrm{d}{x}_n$ are a basis.

Let $Q:=\mathrm{coker}(B{\otimes }_{P}I\to B{\otimes }_P{\mathrm{\Omega }}_{P\mathrm{/}A})$ where the map sends $1\otimes f$ to $1\otimes \mathrm{d}f$. This is well defined: the map $I\to B{\otimes }_P{\mathrm{\Omega }}_{P\mathrm{/}A}$, $f\mapsto 1\otimes \mathrm{d}f$, is $P$-linear, and the class of $1\otimes f$ depends only on $f$, so extension of scalars gives the displayed $B$-linear map. There is a $B$-linear surjection $B{\otimes }_P{\mathrm{\Omega }}_{P\mathrm{/}A}\to {\mathrm{\Omega }}_{B\mathrm{/}A}$ with $1\otimes \mathrm{d}f\mapsto \mathrm{d}(f+I)$, induced by the $A$-derivation $P\to {\mathrm{\Omega }}_{B\mathrm{/}A}$, $f\mapsto \mathrm{d}(f+I)$; it kills the image of $B{\otimes }_{P}I$ because $\mathrm{d}f$ maps to the class of $\mathrm{d}(f+I)$ with $f\in I$. Hence it factors through a surjection $Q\to {\mathrm{\Omega }}_{B\mathrm{/}A}$. Conversely define $D\text{\hspace{0.17em}⁣}:B\to Q$ by $D(g+I):=$ the class of $1\otimes \mathrm{d}g$. This is well defined because $g\in I$ makes $1\otimes \mathrm{d}g$ lie in the image defining $Q$; it is additive, $A$-constant, and satisfies Leibniz because $\mathrm{d}$ does and the $B$-module structure on $B{\otimes }_P{\mathrm{\Omega }}_{P\mathrm{/}A}$ is that of $B$. By [F1] applied to the $A$-algebra $B$ and the $B$-module $Q$, $D$ induces a $B$-linear map ${\mathrm{\Omega }}_{B\mathrm{/}A}\to Q$, and the two displayed maps are inverse on the generating classes of $1\otimes \mathrm{d}{x}_i$ and of $\mathrm{d}{x}_i$. Therefore ${\mathrm{\Omega }}_{B\mathrm{/}A}\cong Q$, which is exactness of the sequence of (2); right exactness of $B{\otimes }_P-$ is the statement of [F2] applied to $I\to P\to B\to 0$, and it is what makes $B{\otimes }_P{\mathrm{\Omega }}_{P\mathrm{/}A}$ the receptacle of this cokernel presentation.

Suppose $I=(f_1,\dots ,f_c)$. Since $I={\sum }_{j}P{f}_j$, every class in $I\mathrm{/}{I}^2$ is a $B$-linear combination of the classes of $f_1,\dots ,f_c$: from $f={\sum }_j{g}_j{f}_j$ and $I^2\ni$ (terms with two factors of $I$) one gets $f\equiv {\sum }_j(g_j+I)f_j$ modulo $I^2$. Hence the image of the first map of (2) is generated by the classes of $1\otimes \mathrm{d}{f}_j$, and $\mathrm{d}{f}_j={\sum }_i{\mathrm{\partial }}_i{f}_j\mathrm{d}{x}_i$ by step 1.1 and the Leibniz rule. Under the basis identification of step 1.1, $1\otimes \mathrm{d}{f}_j$ corresponds to the $j$-th column $({\mathrm{\partial }}_i{f}_j{)}_i$ of the Jacobian matrix, so the cokernel of $B^c\to B^n$, $e_j\mapsto ({\mathrm{\partial }}_i{f}_j{)}_i$, is exactly $Q\cong {\mathrm{\Omega }}_{B\mathrm{/}A}$ of step 2.1.

The first map of (2) is not injective in general: take $A=\mathbb{Z}$, $P=\mathbb{Z}[x]$ and $I=(2)$. Then $I\mathrm{/}{I}^2=2\mathbb{Z}[x]\mathrm{/}4\mathbb{Z}[x]\cong {\mathbb{F}}_2[x]\mathrm{\ne }0$, while $\mathrm{d}(2)=0$ in ${\mathrm{\Omega }}_{P\mathrm{/}A}$ because $2=\phi (2)$ for the structure map $\phi \text{\hspace{0.17em}⁣}:\mathbb{Z}\to \mathbb{Z}[x]$, so the first map has nonzero kernel. This proves the final clause and, with steps 2.1 and 3.1, the whole statement.