The Stacks project

Proof. By composition we mean the map

of Cohomology, Lemma 20.42.5. This induces multiplication maps

Please compare with More on Algebra, Equation (15.63.0.1). The statement of the lemma means that the induced map

factors through the wedge product and then induces an isomorphism. To see this is true we may work locally on $Y$. Hence we may assume that we have global sections $f_1, \ldots , f_ r$ of $\mathcal{O}_ Y$ which generate $\mathcal{I}$ and which form a Koszul regular sequence. Denote

the sheaf of strictly commutative differential graded $\mathcal{O}_ Y$-algebras which is a (divided power) polynomial algebra on $\xi _1, \ldots , \xi _ r$ in degree $-1$ over $\mathcal{O}_ Y$ with differential $\text{d}$ given by the rule $\text{d}\xi _ i = f_ i$. Let us denote $\mathcal{A}^\bullet$ the underlying complex of $\mathcal{O}_ Y$-modules which is the Koszul complex mentioned above. Thus the canonical map $\mathcal{A}^\bullet \to \mathcal{O}_ X$ is a quasi-isomorphism. We obtain quasi-isomorphisms

by Cohomology, Lemma 20.46.9. The differentials of the latter complex are zero, and hence

For $j \in \{ 1, \ldots , r\}$ let $\delta _ j : \mathcal{A} \to \mathcal{A}$ be the derivation of degree $1$ with $\delta _ j(\xi _ i) = \delta _{ij}$ (Kronecker delta). A computation shows that $\delta _ j \circ \text{d} = - \text{d} \circ \delta _ j$ which shows that we get a morphism of complexes

Whence $\delta _ j$ defines a section of the corresponding $\mathop{\mathcal{E}\! \mathit{xt}}\nolimits$-sheaf. Another computation shows that $\delta _1, \ldots , \delta _ r$ map to a basis for $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(\mathcal{A}^{-1}, \mathcal{O}_ X)$ over $\mathcal{O}_ X$. Since it is clear that $\delta _ j \circ \delta _ j = 0$ and $\delta _ j \circ \delta _{j'} = - \delta _{j'} \circ \delta _ j$ as endomorphisms of $\mathcal{A}$ and hence in the $\mathop{\mathcal{E}\! \mathit{xt}}\nolimits$-sheaves we obtain the statement that our map above factors through the exterior power. To see we get the desired isomorphism the reader checks that the elements

for $j_1 < \ldots < j_ i$ map to a basis of the sheaf $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(\mathcal{A}^{-i}, \mathcal{O}_ X)$ over $\mathcal{O}_ X$. $\square$