Reference: Milne’s note section 22, 23 and Litt’s note
The upshot is that Kunneth formula is a consequence of projection formula and proper base change (or smooth base change if the prime in question is invertible on the base)
The cycle class map for is just the connecting homomorphism associated to the Kummer sequence. The key input is purity, which says that for smooth (the proof hinges on the smooth pair ). For singular we can still define it thanks to the isomorphism where .
Construction of Chern class: The idea is to introduce the projectivization of a vector bundle , which allows us to reduce to the case of line bundles by Grothendieck’s axioms. For detail see the Wikipedia section. See also [here](https://en.wikipedia.org/wiki/Chern_class#In_algebraic_geometry for the algebrogeometric analogue of the cohomology ring, the Chow ring. See also Milne’s note, especially page 141 for how the cycle class map is related to Chern class and chern character (we define the Chern character with denominators because should be the correct multiplicative structure on to make into a ring homomorphism).
Digression to Grothendieck-Riemann-Roch: Essentially GRR is about the failure for the Chern character to commute with pushforward. It involves the Todd class as a correction factor, see the intuition here and also how you can discover the Todd class.
Lefschetz trace formula:
Example: where is the power- map. To see this, note that is cokernel of , so we need to understand the action of on line bundles on . But a line bundle on is given by transition functions , and essentially plug in in place of .
Reference: https://math.stanford.edu/~conrad/Weil2seminar/Notes/L20.pdf