Background: Let $X$ be a smooth complex projective algebraic variety, and let $V$ and $W$ be closed subvarieties. For simplicity, let's assume that $\dim V+\dim W=\dim X$.


Now Serre's famous Tor formula says that if $V\cap W$ has dimension zero, we have:


$$V\cdot W=\sum\_{Z\subset V\cap W}\sum\_{i=0}^\infty(-1)^i\operatorname{length}\_{\mathcal O\_{X,z}}\operatorname{Tor}\_{\mathcal O\_{X,z}}^i(\mathcal O\_{X,z}/I\_V,\mathcal O\_{X,z}/I\_W)$$


where the sum is over the irreducible components of $V\cap W$ (in this case, a finite number of points).


However (according to Wikipedia [here](http://en.wikipedia.org/wiki/Intersection_theory_%28mathematics%29)), if $Z$ is an irreducible component of $V\cap W$ with positive dimension (this is called a nonproper intersection), then the alternating sum of $\operatorname{Tor}$'s, which I'll call $\mu(Z;V,W)$, is zero. Unfortunately, this means it cannot be used in exactly the same way as before to calculate $V\cdot W$. For example, if $V=W$, then the answer would always be zero, though certainly there exist half-dimensional varieties $V$ with $V\cdot V\ne 0$ (in fact it can even be negative).


Is it possible to remedy this situation? How does one count the intersection multiplicity if the intersection is not proper?


Comment: I know how to compute the self-intersection $V\cdot V$ as the top chern class of the normal bundle evaluated on $[V]$. I'm looking here for an answer that's the most general, i.e. applies to all $V$ and $W$ of complementary dimension, regardless of the dimension of their intersection.



Is there a Serre Tor formula for nonproper intersections?
There is no formula which looks only at the generic point(s) of $V \cap W$; you need to understand the entire sheaf $\mathcal{T}or\_j^{\mathcal{O}\_X}(\mathcal{O}\_V, \mathcal{O}\_W)$. It might be worth explaining the $K$-theory perspective on this.




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Let $K\_0(X)$ be the Grothendieck group of coherent sheaves on $X$. There is a ring structure on $K\_0(X)$, where
$$[\mathcal{E}] [\mathcal{F}] = \sum (-1)^j [\mathcal{T}or\_j^{\mathcal{O}\_X}(\mathcal{E}, \mathcal{F}) ]$$
for any coherent sheaves $\mathcal{E}$ and $\mathcal{F}$. Here I am using $[\mathcal{A}]$ to mean "class of $\mathcal{A}$ in $K\_0(X)$", and I am using that $X$ is smooth to guarantee that the sum is finite.


$K\_0(X)$ has a descending filtration, $K\_0(X) \supseteq K\_0(X)\_{1} \supseteq K\_0(X)\_{2} \supseteq \cdots \supseteq (0)$ where $K\_0(X)\_i$ is spanned by classes of sheaves with support in codimension $i$. This makes $K\_0(X)$ into a filtered ring, meaning that 
$$K\_0(X)\_i K\_0(X)\_j \subseteq K\_0(X)\_{i+j} \quad (\ast)$$
Containment $(\ast)$ is NOT obvious, and we will return to this point.


Let $gr \ K\_0(X)$ be the associated graded ring $\bigoplus\_{i \geq 0} K\_0(X)\_j/K\_0(X)\_{j+1}$. Then there is a map of graded rings from $gr \ K\_0(X)$ to the Chow ring $A^{\bullet}(X)$. This map sends $[\mathcal{O}\_V]$ to $[V]$.


So, let $V$ and $W$ live in codimensions $i$ and $j$. We want to compute $[V] [W]$ in $A^{i+j}(X)$. From the above, we see that it would be enough to compute 
$$\sum (-1)^j [\mathcal{T}or\_j^{\mathcal{O}\_X}(\mathcal{O}\_V, \mathcal{O}\_W) ] \quad (\ast \ast)$$
as an element of $K\_0(X)\_{i+j}/K\_0(X)\_{i+j+1}$. 


Every summand in $(\ast \ast)$ is supported on $V \cap W$. So, if $V \cap W$ lives in codimension $i+j$, then we can just compute the image of each summand separately in the quotient $K\_0(X)\_{i+j}/K\_0(X)\_{i+j+1}$. Working this out gives Serre's formula.


Suppose now that $V \cap W$ has codimension $k$, which is less than $i+j$. Then the individual Tor terms live in $K\_0(X)\_k$ and plugging into Serre's formula gives the image of $(\ast \ast)$ in $K\_0(X)\_k/K\_0(X)\_{k+1}$. But, by containment $(\ast)$, the sum $(\ast \ast)$ actually lives farther down the filtration, in $K\_0(X)\_{i+j}$. This is why simply plugging into the formula you quote gives $0$. 




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An example might be useful. Take $X = \mathbb{P}^2$. Then $K\_0(X)$ is isomorphic as an additive group to $\mathbb{Z}^3$, and we'll take as a basis the structure sheaf of $X$, the structure sheaf of a line, and the structure sheaf of a point. The filtration is given by 
$$(\ast, \ast, \ast) \supseteq (0, \ast, \ast) \supseteq (0,0,\ast) \supseteq (0,0,0)$$


Consider intersecting a line $V$ with itself. $\mathcal{T}or\_0$ is the tensor product $\mathcal{O}\_V \otimes \mathcal{O}\_V$, whose class is $(0,1,0)$. $\mathcal{T}or\_1$, is the restriction, to $V$, of the ideal sheaf of $V$. This is $\mathcal{O}\_V(-1)$ and, as you can work out, it is $(0,1,-1)$ in the basis I chose. The other Tor terms are all zero.


So the individual Tor terms are $(0,1,0)$ and $(0,1,-1)$, which each live in $K\_0(X)\_1$ Those leading $1$ terms correspond to the lengths of the Tor modules at the generic point of $V$. In order to compute the intersection multiplicity, you have to see farther down in the filtration, to the element $(0,1,0) - (0,1,-1)$ in $K\_0(X)\_2$. Indeed, $(0,1,0) - (0,1,-1) = (0,0,1)$, showing that a line in the projective plane intersects itself in the class of a point.


