Lemma 43.14.3. Let $X$ be a nonsingular variety. Let $V, W \subset X$ be closed subvarieties which intersect properly. Let $Z$ be an irreducible component of $V \cap W$ and assume that the multiplicity (in the sense of Section 43.4) of $Z$ in the closed subscheme $V \cap W$ is $1$. Then $e(X, V \cdot W, Z) = 1$ and $V$ and $W$ are smooth in a general point of $Z$.

Proof. Let $(A, \mathfrak m, \kappa ) = (\mathcal{O}_{X, \xi }, \mathfrak m_\xi , \kappa (\xi ))$ where $\xi \in Z$ is the generic point. Then $\dim (A) = \dim (X) - \dim (Z)$, see Varieties, Lemma 33.20.3. Let $I, J \subset A$ cut out the trace of $V$ and $W$ in $\mathop{\mathrm{Spec}}(A)$. Set $\overline{I} = I + \mathfrak m^2/\mathfrak m^2$. Then $\dim _\kappa \overline{I} \leq \dim (X) - \dim (V)$ with equality if and only if $A/I$ is regular (this follows from the lemma cited above and the definition of regular rings, see Algebra, Definition 10.60.10 and the discussion preceding it). Similarly for $\overline{J}$. If the multiplicity is $1$, then $\text{length}_ A(A/I + J) = 1$, hence $I + J = \mathfrak m$, hence $\overline{I} + \overline{J} = \mathfrak m/\mathfrak m^2$. Then we get equality everywhere (because the intersection is proper). Hence we find $f_1, \ldots , f_ a \in I$ and $g_1, \ldots g_ b \in J$ such that $\overline{f}_1, \ldots , \overline{g}_ b$ is a basis for $\mathfrak m/\mathfrak m^2$. Then $f_1, \ldots , g_ b$ is a regular system of parameters and a regular sequence (Algebra, Lemma 10.106.3). The same lemma shows $A/(f_1, \ldots , f_ a)$ is a regular local ring of dimension $\dim (X) - \dim (V)$, hence $A/(f_1, \ldots , f_ a) \to A/I$ is an isomorphism (if the kernel is nonzero, then the dimension of $A/I$ is strictly less, see Algebra, Lemmas 10.106.2 and 10.60.13). We conclude $I = (f_1, \ldots , f_ a)$ and $J = (g_1, \ldots , g_ b)$ by symmetry. Thus the Koszul complex $K_\bullet (A, f_1, \ldots , f_ a)$ on $f_1, \ldots , f_ a$ is a resolution of $A/I$, see More on Algebra, Lemma 15.30.2. Hence

\begin{align*} \text{Tor}_ p^ A(A/I, A/J) & = H_ p(K_\bullet (A, f_1, \ldots , f_ a) \otimes _ A A/J) \\ & = H_ p(K_\bullet (A/J, f_1 \bmod J, \ldots , f_ a \bmod J)) \end{align*}

Since we've seen above that $f_1 \bmod J, \ldots , f_ a \bmod J$ is a regular system of parameters in the regular local ring $A/J$ we conclude that there is only one cohomology group, namely $H_0 = A/(I + J) = \kappa$. This finishes the proof. $\square$

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