Scheme-theoretic intersection

rdf:langString Scheme-theoretic intersection

rdf:langString In algebraic geometry, the scheme-theoretic intersection of closed subschemes X, Y of a scheme W is , the fiber product of the closed immersions . It is denoted by . Locally, W is given as for some ring R and X, Y as for some ideals I, J. Thus, locally, the intersection is given as Here, we used (for this identity, see tensor product of modules#Examples.) Example: Let be a projective variety with the homogeneous coordinate ring S/I, where S is a polynomial ring. If is a hypersurface defined by some homogeneous polynomial f in S, then 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, let = the affine 4-space and X, Y closed subschemes defined by the ideals and . 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-theoretic intersection 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. Solving this disparity is one of the starting points for derived algebraic geometry, which aims to introduce the notion of .