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