Andre-Quillen homology of algebra retracts

الأصل · EN

Given a homomorphism of commutative noetherian rings ϕ: R → S, Daniel Quillen conjectured in 1970 that if the Andre-Quillen homology functors Dₙ(S|R,-) vanish for all n ≫ 0, then they vanish for all n ≥ 3. We prove the conjecture under the additional hypothesis that there exists a homomorphism of rings ψ: S → R such that ϕ∘ψ=ₛ. More precisely, in this case we show that ψ is complete intersection at ϕ⁻¹() for every prime ideal of S. Using these results, we describe all algebra retracts S→ R→ S for which the S-algebra Torʳ(S,S) is finitely generated.