I was reading about cotangent complexes in derived algebraic geometry. I saw the following two surprising conjectures by Quillen (from "Cohomology of commutative rings").
My question is: are these conjectures proved? Is anything known using derived or spectral algebraic geometry?
Conjecture 5.7 was proven by Avramov. See:
Avramov, Luchezar L., Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology, Ann. Math. (2) 150, No. 2, 455-487 (1999). ZBL0968.13007.
This is also available on at arXiv:9909192. As far as I am aware, various special cases of Conjecture 5.6 are know (for example, some results in Avramov's paper), but the general conjecture remains open.
