Interlude I: Serre’s formula (1957)
Consider the intersection of the axis in 4-dimensions \(R=\mathbb {C}[x,y,z,w]\), with the diagonal
Geometrically, we find an intersection of multiplicity 2, as seen through the following picture:
![\includegraphics[scale=0.2]{Images/all8circles/all8circles-0.png}](images/diagonal.gif)
Any small perturbation of the diagonal leads to two intersection points.
Now, the problem is that the algebraic computation, leads to a different number. Indeed, by computing the tensor product we obtain
which as a \(\mathbb {C}\)-vector space, has dimension 3
The reason for the different results lies in redundancies, in the following sense: take the following function \(f\)
and observe that it vanishes for two reasons, because it simultaneously belongs to the two ideals (of the diagonal and the axis)
but the two reasons are different, in the sense that \(f\) does not live in the product ideal
In fact, if we compute the quotient we find an isomorphism
showing us that \(f\) is in fact very special: it is the only function that vanishes non-trivially for two different reasons. Now, when we computed the tensor product
algebraically solving the intersection, we discarded such ambiguous outcomes. In order to correct the counting we need to account for such phenomena. More precisely:
More generally, the tensor product
and the quotient
form the first two layers of an hierarchy of redundancies captured by the homology groups of a chain complex
computing the derived tensor product. More precisely, its homology groups encode
etc.
Finally, Serre’s formula for the multiplicity of intersections is given by adding the dimensions of homologies of even degree and subtracting dimensions of odd degrees.