What is "a" homology "theory"?
It’s impossible to get far in reading 20th-century mathematics without encountering the word
cohomology. Cohomology & schemes are the subject of Hartshorne’s classic, where you can find out (Appendix C) that the Weil conjectures were resolved by defining a thing called l-adic cohomology.
Schemes are like varieties = cycles. And cohomology is a way (ok, apparently various