André–Quillen homology of algebra retracts

Avramov, Luchézar L.; Iyengar, Srikanth B.

doi:10.1016/s0012-9593(03)00014-4

Cited by 11 publications

(5 citation statements)

References 22 publications

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“…Finally, Avramov [4] gave a complete solution to (2) while simultaneously establishing general properties for locally complete intersection homomorphisms. Following that work, Avramov and S. Iyengar [9] proved part (1) for R → A an algebra retract. The general case of (1) is still open.…”

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confidence: 82%

On the rigidity of the cotangent complex at the prime 2

Turner¹

2009

Journal of Pure and Applied Algebra

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Communicated by E.M. Friedlander MSC: Primary: 13D03 secondary: 13D07 13H10 18G30 55U35 a b s t r a c t In [D. Quillen, On the (co)homology of commutative rings, Proc. Symp. Pure Math. 17 (1970) 65-87; L. Avramov, Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology, Annals of Math. 2 (150) (1999) 455-487] a conjecture was posed to the effect that if R → A is a homomorphism of Noetherian commutative rings then the flat dimension, as defined in the derived category of Amodules, of the associated cotangent complex L A/R satisfies: fd AThe aim of this paper is to initiate an approach for solving this conjecture when R has characteristic 2 using simplicial algebra techniques. To that end, we obtain two results. First, we prove that the conjecture can be reframed in terms of certain nilpotence properties for the divided square γ 2 and the André operation ϑ as it acts on Tor R (A, ), any residue field of A. Second, we prove the conjecture is valid in two cases: when fd R A < ∞ and when R is a Cohen-Macaulay ring.

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confidence: 82%

On the rigidity of the cotangent complex at the prime 2

Turner¹

2009

Journal of Pure and Applied Algebra

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“…An affirmative answer to (iv) was conjectured by Quillen [34, 5.6] when β is of finite type and in [8, p. 459] in general. That conjecture was proved in [8, 1.3] in case C q has finite flat dimension over B, and in [16,Theorem 1] in case β admits a right inverse ring homomorphism.…”

Section: Proofmentioning

confidence: 96%

Quasi-Complete Intersection Homomorphisms

Avramov

Bonacho

Şega

2013

Pure and Applied Mathematics Quarterly

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“…In fact, the hypothesis on the characteristic is not necessary since by [12, Theorem I], for a noetherian supplemented -algebra , implies for all (a direct proof of this fact was communicated to me by Rodicio: we can assume that the rings are local and complete; take a regular factorization ; we have and so by Lemma 19; using this same Lemma, we have for all ; in the commutative diagram with exact row…”

Section: Complete Intersectionmentioning

confidence: 99%

A Descent Theorem for Formal Smoothness

Majadas

2016

Nagoya Math. J.

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Abstract. Let u be a local homomorphism of noetherian local rings forming part of a commutative square vf = gu. We give some conditions on the square which imply that u is formally smooth. This theorem encapsulates a variety of (apparently unrelated) results in commutative algebra greatly improving some of them: Greco's theorem on descent of quasi-excellence property by finite surjective morphisms, Kunz's characterization of regular local rings in positive characteristic by means of the Frobenius homomorphism (and in fact the relative version obtained by André and Radu), etc. In the second part of the paper, we study a similar question for the complete intersection property instead of formal smoothness, giving also some applications.In this paper we obtain a descent result for formal smoothness (and regularity). This single result (Theorem 1) has as particular cases a variety of (a priori unconnected) results in commutative algebra. Some of these particular cases greatly improve known important results (the descent of quasi-excellence by finite surjective morphisms [20] (Corollary 25). Since the main result is rather technical, we confine ourselves in this introduction to discuss these applications in more detail.A. Extension of Greco's theorem. If f : R → S is a finite local homomorphism of noetherian local rings such that Spec(S) → Spec(R) is surjective, Greco [20] showed that if S is quasi-excellent then so is R (in particular, quasi-excellence descends along finite flat local homomorphisms). He also obtained an example [20, Proposition 4.1] which shows that this theorem cannot be extended to the case where f is of finite type instead of finite, even when R and S are local domains of dimension 1 (the problem is concentrated in the local condition, i.e., the geometric regularity of the formal fibers, since the J-2 property always descends and ascends by surjective morphisms of finite type [20, Corollary 2.4]). So the problem for finite type homomorphisms was left aside and he proceeded instead to consider proper surjective morphisms of schemes instead, a problem which was finally solved in [14], [24].We reconsider here the problem for homomorphisms of finite type (but with an adequate formulation of the hypothesis on the surjectivity between spectra), and we will prove that it holds. Note first that ifR andŜ are the completions of R and S at their maximal ideals, the surjectivity of Spec(S) → Spec(R) is equivalent to

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André–Quillen homology of algebra retracts

Cited by 11 publications

References 22 publications

On the rigidity of the cotangent complex at the prime 2

On the rigidity of the cotangent complex at the prime 2

Quasi-Complete Intersection Homomorphisms

A Descent Theorem for Formal Smoothness

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