Example 43.14.4. In this example we show that it is necessary to use the higher tors in the formula for the intersection multiplicities above. Let X be a nonsingular variety of dimension 4. Let p \in X be a closed point. Let V, W \subset X be closed subvarieties in X. Assume that there is an isomorphism
such that the ideal of V is (xz, xw, yz, yw) and the ideal of W is (x - z, y - w). Then a computation shows that
On the other hand, the multiplicity e(X, V \cdot W, p) = 2 as can be seen from the fact that formal locally V is the union of two smooth planes x = y = 0 and z = w = 0 at p, each of which has intersection multiplicity 1 with the plane x - z = y - w = 0 (Lemma 43.14.3). To make an actual example, take a general morphism f : \mathbf{P}^2 \to \mathbf{P}^4 given by 5 homogeneous polynomials of degree > 1. The image V \subset \mathbf{P}^4 = X will have singularities of the type described above, because there will be p_1, p_2 \in \mathbf{P}^2 with f(p_1) = f(p_2). To find W take a general plane passing through such a point.
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