For context, I am rather new to the whole business of abstract Weil cohomology theories and motives in general, so if I am not making sense somewhere, do let me know!
- In many of the literature that I am consulting while trying to learn about Weil cohomology theories and motives, it is often said that these cohomology theories must, in particular, satisfy the Weak and Hard Leftschetz Conditions. However, The Stacks Project apparently does not impose this axiom upon Weil cohomology theories (see their Tag 0FHA). Am I misinterpreting The Stacks Project, or can the Lefschetz Conditions be deduced from the other Weil cohomology axioms ?
- The other thing that I have come across is that the theory of crystalline cohomology is a Weil cohomology theory, but for some reason I can not find any source which confirms that the Lefschetz Conditions are satisfied here. Can anyone point me to such a source, or alternatively, does crystalline cohomology even satisfy the Lefschetz Conditions at all ? (I should note that I'm not yet too familiar with crystalline cohomology.)