Scheme-theoretic intersection

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, the scheme-theoretic intersection of closed subschemes X, Y of a <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a> W is 
 
 
 
 X
 
 ×
 
 W
 
 
 Y
 
 
 {\displaystyle X\times _{W}Y}
 
, the <a href="/facts/Fiber_product/bC0QoX5U">fiber product</a> of the <a href="/facts/Closed_immersion/g60V5BK4">closed immersions</a> 
 
 
 
 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
 ⁡
 (
 R
 
 /
 
 I
 )
 ,
 Spec
 ⁡
 (
 R
 
 /
 
 J
 )
 
 
 {\displaystyle \operatorname {Spec} (R/I),\operatorname {Spec} (R/J)}
 
 for some ideals I, J. Thus, locally, the intersection 
 
 
 
 X
 ∩
 Y
 
 
 {\displaystyle X\cap Y}
 
 is given as

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

Here, we used 
 
 
 
 R
 
 /
 
 I
 
 ⊗
 
 R
 
 
 R
 
 /
 
 J
 ≃
 R
 
 /
 
 (
 I
 +
 J
 )
 
 
 {\displaystyle R/I\otimes _{R}R/J\simeq R/(I+J)}
 
 (for this identity, see <a href="/facts/Tensor_product_of_modules/K3oaFIS4">tensor product of modules#Examples</a>.)
Example: Let 
 
 
 
 X
 ⊂
 
 
 P
 
 
 n
 
 
 
 
 {\displaystyle X\subset \mathbb {P} ^{n}}
 
 be a <a href="/facts/Projective_variety/aB2XdB7B">projective variety</a> 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 <a href="/facts/Hypersurface/cTvtLVsV">hypersurface</a> defined by some homogeneous polynomial f in S, then

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

If f is linear (deg = 1), it is called a <a href="/facts/Hyperplane_section/o5gl0vDk">hyperplane section</a>. See also: <a href="/facts/Bertini%2527s_theorem/HE9WnFz3">Bertini's theorem</a>.
Now, a scheme-theoretic intersection may not be a correct intersection, say, from the point of view of <a href="/facts/Intersection_theory/UgckwVGF">intersection theory</a>. For example, let 
 
 
 
 W
 =
 Spec
 ⁡
 (
 k
 [
 x
 ,
 y
 ,
 z
 ,
 w
 ]
 )
 
 
 {\displaystyle W=\operatorname {Spec} (k[x,y,z,w])}
 
 be the affine 4-space and X, Y closed subschemes defined by the ideals 
 
 
 
 (
 x
 ,
 y
 )
 ∩
 (
 z
 ,
 w
 )
 
 
 {\displaystyle (x,y)\cap (z,w)}
 
 and 
 
 
 
 (
 x
 −
 z
 ,
 y
 −
 w
 )
 
 
 {\displaystyle (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 <a href="/facts/Intersection_multiplicity/ChzWmr4d">intersection multiplicity</a>, we expect X and Y intersect at the origin with multiplicity two. On the other hand, one sees the scheme-theoretic intersection 
 
 
 
 X
 ∩
 Y
 
 
 {\displaystyle 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 <a href="/facts/Intersection_number/ChzWmr4d">Serre's Tor formula</a>. Solving this disparity is one of the starting points for <a href="/facts/Derived_algebraic_geometry/FxXzQExB">derived algebraic geometry</a>, which aims to introduce the notion of derived intersection.

Scheme-theoretic intersection open-in-new

Scheme-theoretic intersection