ag.algebraic geometry – Subrings of Chow rings

Let $$X$$ be a smooth projective variety over $$mathbf{F}_p$$, call $$overline{X}$$ the base change to $$overline{mathbf{F}}_p$$, and denote by $$F$$ the base change to $$overline{X}$$ of the absolute Frobenius of $$X$$ over $$mathbf{F}_p$$.

Call $$A$$ the Chow ring of cycles up to homological equivalence (defined using, say, $$ell$$-adic cohomology, i.e. the image of the cycle map $$Z^*(overline{X}times overline{X})otimes_{mathbf{Z}}mathbf{Q}to H^{2*}(overline{X}times overline{X},mathbf{Q}_{ell})(*)$$).

Call $$R$$ the $$mathbf{Q}$$-subalgebra of $$A$$ generated by the classes of the graphs of the endomorphisms $${F^n, nge 1}$$ of $$overline{X}$$.

Is anything at all known about $$R$$?

Examples of questions I’d be interested in are

• is $$R$$ known to not be $$A$$?
• is $$R$$ a domain, or even a field?
• is $$R$$ normal?
• does $$R$$ contain the Lefschetz class?
• is $$R$$ expected to contain the inverse of the Lefschetz class, perhaps assuming some standard conjectures.