Lemma 15.9.12. Let $A$ be a ring. Let $0 \to K \to A^{\oplus m} \to M \to 0$ be a sequence of $A$-modules. Consider the $A$-algebra $C = \text{Sym}^*_ A(M)$ with its presentation $\alpha : A[y_1, \ldots , y_ m] \to C$ coming from the surjection $A^{\oplus m} \to M$. Then

\[ \mathop{N\! L}\nolimits (\alpha ) = (K \otimes _ A C \to \bigoplus \nolimits _{j = 1, \ldots , m} C \text{d}y_ j) \]

(see Algebra, Section 10.134) in particular $\Omega _{C/A} = M \otimes _ A C$.

**Proof.**
Let $J = \mathop{\mathrm{Ker}}(\alpha )$. The lemma asserts that $J/J^2 \cong K \otimes _ A C$. Note that $\alpha $ is a homomorphism of graded algebras. We will prove that in degree $d$ we have $(J/J^2)_ d = K \otimes _ A C_{d - 1}$. Note that

\[ J_ d = \mathop{\mathrm{Ker}}(\text{Sym}^ d_ A(A^{\oplus m}) \to \text{Sym}^ d_ A(M)) = \mathop{\mathrm{Im}}(K \otimes _ A \text{Sym}^{d - 1}_ A(A^{\oplus m}) \to \text{Sym}^ d_ A(A^{\oplus m})), \]

see Algebra, Lemma 10.13.2. It follows that $(J^2)_ d = \sum _{a + b = d} J_ a \cdot J_ b$ is the image of

\[ K \otimes _ A K \otimes _ A \text{Sym}^{d - 2}_ A(A^{\otimes m}) \to \text{Sym}^ d_ A(A^{\oplus m}). \]

The cokernel of the map $K \otimes _ A \text{Sym}^{d - 2}_ A(A^{\otimes m}) \to \text{Sym}^{d - 1}_ A(A^{\oplus m})$ is $\text{Sym}^{d - 1}_ A(M)$ by the lemma referenced above. Hence it is clear that $(J/J^2)_ d = J_ d/(J^2)_ d$ is equal to

\begin{align*} \mathop{\mathrm{Coker}}( K \otimes _ A K \otimes _ A \text{Sym}^{d - 2}_ A(A^{\otimes m}) \to K \otimes _ A \text{Sym}^{d - 1}_ A(A^{\otimes m})) & = K \otimes _ A \text{Sym}^{d - 1}_ A(M) \\ & = K \otimes _ A C_{d -1} \end{align*}

as desired.
$\square$

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