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In algebraic geometry, the scheme-theoretic intersection of closed subschemes X, Y of a scheme W is X × W Y {\displaystyle X\times _{W}Y} , the fiber product of the closed immersions X ↪ W , Y ↪ W {\displaystyle X\hookrightarrow W,Y\hookrightarrow W}. It is denoted by X ∩ Y {\displaystyle X\cap Y}.
Locally, W is given as Spec R {\displaystyle \operatorname {Spec} R} for some ring R and X, Y as Spec , Spec {\displaystyle \operatorname {Spec} ,\operatorname {Spec} } for some ideals I, J. Thus, locally, the intersection X ∩ Y {\displaystyle X\cap Y} is given as
Here, we used R / I ⊗ R R / J ≃ R / {\displaystyle R/I\otimes _{R}R/J\simeq R/}
Example: Let X ⊂ P n {\displaystyle 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 } ⊂ P n {\displaystyle H=\{f=0\}\subset \mathbb {P} ^{n}} is a hypersurface defined by some homogeneous polynomial f in S, then