Scheme-theoretic intersection

In algebraic geometry, the scheme-theoretic intersection of closed subschemes X, Y of a scheme W is $X\times _{W}Y$ , the fiber product of the closed immersions $X\hookrightarrow W,Y\hookrightarrow W$ . It is denoted by $X\cap Y$ .

Locally, W is given as $\operatorname {Spec}R$ for some ring R and X, Y as $\operatorname {Spec} (R/I),\operatorname {Spec} (R/J)$ for some ideals I, J. Thus, locally, the intersection $X\cap Y$ is given as

\operatorname {Spec} (R/(I+J)).

Here, we used $R/I\otimes _{R}R/J\simeq R/(I+J)$ (for this identity, see tensor product of modules#Examples.)

Example: Let $X\subset \mathbb {P} ^{n}$ be a projective variety with the homogeneous coordinate ring S/I, where S is a polynomial ring. If $H=\{f=0\}\subset \mathbb {P} ^{n}$ is a hypersurface defined by some homogeneous polynomial f in S, then

X\cap H=\operatorname {Proj} (S/(I,f)).

If f is linear (deg = 1), it is called a hyperplane section. See also: Bertini's theorem.

Now, a scheme-theoretic intersection may not be a correct intersection, say, from the point of view of intersection theory. For example,^[1] let $W=\operatorname {Spec} (k[x,y,z,w])$ = the affine 4-space and X, Y closed subschemes defined by the ideals $(x,y)\cap (z,w)$ and $(x-z,y-w)$ . Since X is the union of two planes, each intersecting with Y at the origin with multiplicity one, by the linearity of intersection multiplicity, we expect X and Y intersect at the origin with multiplicity two. On the other hand, one sees the scheme-thereoric intersection $X\cap Y$ consists of the origin with multiplicity three. That is, a scheme-theoretic multiplicity of an intersection may differ from an intersection-theoretic multiplicity, the latter given by Serre's Tor formula.

References

↑ Hartshorne, Appendix A: Example 1.1.1.

Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

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