Characterizing Simplicial Commutative Algebras with Vanishing André-Quillen Homology

André-Quillen homology via functor homology

We obtain André-Quillen homology for commutative algebras using relative homological algebra in the category of functors on finite pointed sets

André-Quillen同调与完全相交平坦维数的上升

André-Quillen homology and ascent of complete intersection flat dimensions

本文利用André-Quillen同调，证明了多种完全相交平坦维数沿具有完全相交闭纤维的本性有限型平坦局部同态的上升定理，并将前人关于满射局部同态维数的结果推广至本性有限型情形。

On Simplicial Commutative Rings with Vanishing André-Quillen Homology

We propose a generalization of a conjecture of D. Quillen, on the vanishing of André-Quillen homology, to simplicial commutative rings. This conjecture characterizes a notion of local complete intersection, extended to the simplicial setting, under a suitable hypothesis on the local characteristic. Further, under the condition of finite-type homology, we then prove the conjecture in the case of a simplicial commutative algebra augmented over a field of non-zero characteristic. As a consequence, we obtain a proof of Quillen's conjecture for a Noetherian commutative algebra - again augmented ove…

通过André-Quillen同调得到的交换环同态的幽灵引理

A Ghost Lemma for Commutative Ring Homomorphisms via André-Quillen Homology

本文利用André-Quillen同调将导出范畴中的幽灵映射理论推广到交换环，证明了环的幽灵引理，并由此得到Kunz定理的特征无关推广及完全交环的类似结果。

Andre-Quillen cohomology of monoid algebras

We compute the Andre-Quillen cohomology of an affine toric variety. The best results are obtained either in the general case for the first three cohomology groups, or in the case of isolated singularities for all cohomology groups, respectively.

Nilpotency in the homotopy of simplicial commutative algebras

In this paper, we study simplicial commutative algebras with finite André-Quillen homology. Here we restrict our focus to simplicial algebras having characteristic 2. Our aim is to find a generalization of results established by the author. Our goal is to replace the finiteness condition on homotopy with a weaker condition expressed in terms of nilpotency for the action of the homotopy operations. Coupled with the finiteness assumption on André-Quillen homology, this nilpotency condition provides a way to bound the height at which the homology vanishes. As a consequence, we establish a special…

Andre-Quillen homology of algebra retracts

Given a homomorphism of commutative noetherian rings $ϕ: R \to S$, Daniel Quillen conjectured in 1970 that if the Andre-Quillen homology functors $D_n(S|R,-)$ vanish for all $n \gg 0$, then they vanish for all $n \ge 3$. We prove the conjecture under the additional hypothesis that there exists a homomorphism of rings $ψ: S \to R$ such that $ϕ\circψ=\id_S$. More precisely, in this case we show that $ψ$ is complete intersection at $ϕ^{-1}(\fn)$ for every prime ideal $\fn$ of $S$. Using these results, we describe all algebra retracts $S\to R\to S$ for which the $S$-algebra $Tor^R(S,S)$ is finitel…

On simplicial commutative algebras with Noetherian homotopy

In this paper, a strategy is developed studying a simplicial commutative algebra A whose zeroth homotopy group is a Noetherian ring B and whose higher homotopy groups are finite over B. The strategy replaces A with a connected simplicial supplemented k(q)-algebra, for each prime ideal q in B, which preserves much of the Andre-Quillen homology of A. The methods for this construction involves a mixture of methods of homotopy theory (e.g. Postnikov towers) with methods of commutative algebras (e.g. completions, Cohen factorizations). We finish by indicating how these methods resolve a more genera…

交换幺半群的 André-Quillen 同调

The André-Quillen cohomology of commutative monoids

本文指出交换幺半群的 Beck 模块等价于相关分次交换环上的模，证明其 Quillen 同调是 André-Quillen 同调的特例，并发展分次形式化理论验证了 Grillet 部分上链复形与 Barr 修正的 Harrison 上链复形的联系。

重访 Quillen 基本谱序列

Quillen's Fundamental Spectral Sequences Revisited

本文针对 Quillen 基本谱序列证明缺失或不完整的问题，系统梳理相关微妙结构，给出可读、严谨且详尽的完整证明。

Rigidity properties of the cotangent complex

This work concerns maps $φ\colon R\to S$ of commutative noetherian rings, locally of finite flat dimension. It is proved that the André-Quillen homology functors are rigid, namely, if $\mathrm{D}_n(S/R;-)=0$ for some $n\ge 2$, then $\mathrm{D}_n(S/R;-)=0$ for all $n\ge 2$ and $φ$ is locally complete intersection. This extends Avramov's theorem that draws the same conclusion assuming $\mathrm{D}_n(S/R;-)$ vanishes for all $n\gg 0$, confirming a conjecture of Quillen. The rigidity of André-Quillen functors is deduced from a more general result about the higher cotangent modules which answers a q…

Rees代数与扩张Rees代数的André-Quillen同调

André-Quillen homology of Rees algebras and extended Rees algebras

本文研究Rees代数与扩张Rees代数的André-Quillen同调，证明了Proj R(I)为完全交的等价条件，给出扩张Rees代数的对应结论，并计算了Cohen-Macaulay整环情形下H₁(J)的秩与自由轨迹。

On the cohomology of highly connected covers of finite complexes

Relying on the computation of the Andre-Quillen homology groups for unstable Hopf algebras, we prove that the mod p cohomology of the n-connected cover of a finite H-space is always finitely generated as algebra over the Steenrod algebra.

Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology

Classical definitions of locally complete intersection (l.c.i.) homomorphisms of commutative rings are limited to maps that are essentially of finite type, or flat. The concept introduced in this paper is meaningful for homomorphisms phi : R \longrightarrow S of commutative noetherian rings. It is defined in terms of the structure of phi in a formal neighborhood of each point of Spec S. We characterize the l.c.i. property by different conditions on the vanishing of the André-Quillen homology of the R-algebra S. One of these descriptions establishes a very general form of a conjecture of Quille…