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.