The author generalizes a theorem of Fujiwara that gives a Lefschetz-Verdier trace formula for the fixed point set for the action of a correspondence on the compact support cohomology of an \(\ell\)-adic sheaf \(\mathcal{F}\) on a separated scheme of finite type over a finite field, \(\mathbb{F}_q\). When \(X\) is proper, this is the well-known formula that can be found in SGA V, but when \(X\) is not proper, the situation is more complicated. Deligne conjectured that things are much better if we twist the correspondence by a sufficiently high power of Frobenius. The desired formula was then announced in the case of constant coefficients by E. Shpiz [PhD thesis, Harvard University (1990)] and in general by R. Pink [Ann. Math. (2) 135, No. 3, 483–525 (1992; Zbl 0769.14007)], both assuming resolution of singularities. Using rigid geometry, K. Fujiwara [Invent. Math. 127, No. 3, 489–533 (1997; Zbl 0920.14005)] was able to prove this formula without using resolution.
In the paper under review, the author proves a more general result using algebro-geometric techniques. His general strategy is similar to that of Pink and Fujiwara, but there are two crucial differences. First, the author gives what he feels is a simpler definition of contracting correspondence, and this gives a sharper bound on the power of Frobenius required to prove the formula. Second, he works locally, proving the vanishing of trace maps, from which local terms are obtained by integration.