Editing Intersection number (section)

== Serre's Tor formula ==
Let ''V'' and ''W'' be two subvarieties of a [[nonsingular variety|nonsingular]] [[projective variety]] ''X'' such that dim(''V'') + dim(''W'') = dim(''X''). Then we expect the intersection ''V'' ∩ ''W'' to be a finite set of points. If we try to count them, two kinds of problems may arise. First, even if the expected dimension of ''V'' ∩ ''W'' is zero, the actual intersection may be of a large dimension: for example the self-intersection number of a [[projective line]] in a [[projective plane]]. The second potential problem is that even if the intersection is zero-dimensional, it may be non-transverse, for example, if ''V'' is a plane curve and ''W'' is one of its [[tangent line|tangent lines]].

The first problem requires the machinery of [[intersection theory]], discussed above in detail, which replaces ''V'' and ''W'' by more convenient subvarieties using the [[moving lemma]]. On the other hand, the second problem can be solved directly, without moving ''V'' or ''W''. In 1965 [[Jean-Pierre Serre]] described how to find the multiplicity of each intersection point by methods of [[commutative algebra]] and [[homological algebra]].<ref>{{cite book| first = Jean-Pierre | last = Serre | author-link = Jean-Pierre Serre| title=Algèbre locale, multiplicités | series= Lecture Notes in Mathematics | volume = 11 | publisher = Springer-Verlag | year = 1965 | pages = x+160 }}</ref> This connection between a geometric notion of intersection and a homological notion of a [[Tor functor|derived tensor product]] has been influential and led in particular to several [[homological conjectures in commutative algebra]].

'''Serre's Tor formula''' states: let ''X'' be a [[regular local ring|regular]] variety, ''V'' and ''W'' two subvarieties of complementary dimension such that ''V'' ∩ ''W'' is zero-dimensional. For any point ''x'' ∈ ''V'' ∩ ''W'', let ''A'' be the [[local ring]] <math>\mathcal{O}_{X, x}</math> of ''x''. The [[structure sheaf|structure sheaves]] of ''V'' and ''W'' at ''x'' correspond to ideals ''I'', ''J'' ⊆ ''A''. Then the multiplicity of ''V'' ∩ ''W'' at the point ''x'' is
:<math>e(X; V, W; x) = \sum_{i=0}^{\infty} (-1)^i \mathrm{length}_A(\operatorname{Tor}_i^A(A/I, A/J))</math>
where length is the [[length of a module]] over a local ring, and Tor is the [[Tor functor]]. When ''V'' and ''W'' can be moved into a transverse position, this homological formula produces the expected answer. So, for instance, if ''V'' and ''W'' meet transversely at ''x'', the multiplicity is 1. If ''V'' is a tangent line at a point ''x'' to a [[parabola]] ''W'' in a plane at a point ''x'', then the multiplicity at ''x'' is 2.

If both ''V'' and ''W'' are locally cut out by [[regular sequence]]s, for example if they are [[nonsingular variety|nonsingular]], then in the formula above all higher Tor's vanish, hence the multiplicity is positive. The positivity in the arbitrary case is one of [[Serre's multiplicity conjectures]].