🚀 go-pugleaf

Let k be a field and M a graded vector space over k. Asssume that each
homogeneous part of M is finite dimensional over k. If T is an operator
on the M preserving degrees, then we can define the "trace" of T on M
as a formal power series in an indeterminate u whose coefficient of degree
n is the trace of the operator T acting on the degree n part of M.

Now let f be a homogeneous polynomial of degree q+1 in r variables x_1,...,x_r,
q>2, with coefficients in k and form the Koszul complex of f (or its dual),
i.e. the complex of differential forms  with polynomial coefficients
and where the differential is wedging with the total differential of f.

For each i, let a_i be a nonzero element of k and suppose that the substitution
T: x_i |--> a_i x_i  preserves the homogeneous polynomial f. Then T acts on
the terms of the Koszul complex and I will continue to refer to these
actions by T, as well as the action on the cohomology.

It is alleged that the "trace" of T on the cohomology is
the product over i of (u^q-a_i)/(u^{-1}-a_i)  in case f is nonsingular.

Can someone supply a proof? Presumably this is done by computing the "traces"
on the Koszul complex itself. 

I would also be intersted in knowing of places in the literature where
these kinds of computations are done (including these "traces")
so I can read about it. For example, I am under the impression that
they arise in certain approaches to index theory, but I don't know where.

Ignorantly,
Allan Adler
ara@altdorf.ai.mit.edu