Behrend's trace formula


In algebraic geometry, Behrend's trace formula is a generalization of the Grothendieck–Lefschetz trace formula to a smooth algebraic stack over a finite field, conjectured in 1993 and proven in 2003 by Kai Behrend. Unlike the classical one, the formula counts points in the "stacky way"; it takes into account the presence of nontrivial automorphisms.
The desire for the formula comes from the fact that it applies to the moduli stack of principal bundles on a curve over a finite field See the moduli stack of principal bundles and references therein for the precise formulation in this case.
Deligne found an example that shows the formula may be interpreted as a sort of the Selberg trace formula.
A proof of the formula in the context of the six operations formalism developed by Laszlo and Olsson is given by Shenghao Sun.

Formulation

By definition, if C is a category in which each object has finitely many automorphisms, the number of points in is denoted by
with the sum running over representatives p of all isomorphism classes in C. The formula states: for a smooth algebraic stack X of finite type over a finite field and the "arithmetic" Frobenius, i.e., the inverse of the usual geometric Frobenius in Grothendieck's formula,
Here, it is crucial that the cohomology of a stack is with respect to the smooth topology.
When X is a variety, the smooth cohomology is the same as etale one and, via the Poincaré duality, this is equivalent to Grothendieck's trace formula.

Simple example

Consider, the classifying stack of the multiplicative group scheme. By definition, is the category of principal -bundles over, which has only one isomorphism class. Its group of automorphisms is, which means that the number of -isomorphisms is.
On the other hand, we may compute the l-adic cohomology of directly. We remark that in the topological setting, we have , whose rational cohomology ring is a polynomial ring in one generator, but we shall not use this directly. If we wish to stay in the world of algebraic geometry, we may instead "approximate" by projective spaces of larger and larger dimension. Thus we consider the map induced by the -bundle corresponding to This map induces an isomorphism in cohomology in degrees up to 2N. Thus the even Betti numbers of are 1, and the l-adic Galois representation on the th cohomology group is the nth power of the cyclotomic character. The second part is a consequence of the fact that the cohomology of is generated by algebraic cycle classes. This shows that
Note that
Multiplying by, one obtains the predicted equality.