## Locally Complete Intersection Homomorphisms And A Conjecture Of Quillen On The Vanishing Of Cotangent Homology (1999)

Venue: | Ann. of Math |

Citations: | 16 - 6 self |

### BibTeX

@ARTICLE{Avramov99locallycomplete,

author = {Luchezar L. Avramov},

title = {Locally Complete Intersection Homomorphisms And A Conjecture Of Quillen On The Vanishing Of Cotangent Homology},

journal = {Ann. of Math},

year = {1999},

volume = {2},

pages = {15--0}

}

### OpenURL

### Abstract

. Classical definitions of 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 ' : R \Gamma! S of commutative noetherian rings. It is defined in terms of the structure of ' 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 Andr'eQuillen homology of the R-algebra S. One of these descriptions establishes a very general form of a conjecture of Quillen that was open even for homomorphisms of finite type: If S has a finite resolution by flat R--modules and the cotangent complex L(S jR) is quasi-isomorphic to a bounded complex of flat S--modules, then ' is l.c.i. The proof uses a mixture of methods from commutative algebra, differential graded homological algebra, and homotopy theory. The l.c.i. property is shown to be stable under a variety of operations, including composition, decomp...

### Citations

30 | Structure of local homomorphisms
- Avramov, Foxby, et al.
- 1994
(Show Context)
Citation Context ...r each prime ideal q ` S. For homomorphisms of noetherian rings we introduce an l.c.i. notion locally, by using a relative version of Cohen structure theory developed jointly with Foxby and B. Herzog =-=[14]-=-. We show that these maps coincide with the weakly c.i. homomorphisms, then characterize them by the vanishing of D 2 (S j R; \Gamma), and by various other vanishing conditions. In particular, we prov... |

30 |
Through the looking glass: A dictionary between rational homotopy theory and local algebra
- Avramov, Halperin
- 1986
(Show Context)
Citation Context ...malized chain complexes. We produce a substitute for simplicial modules over a simplicial ring. On the way, we introduce some notation for DG and simplicial algebra. (2.1) DG algebra (cf. [34], [27], =-=[15]-=-). DG objects have differentials of degree \Gamma1, denoted ubiquitously @. Morphisms of DG objects are chain maps of degree 0 that preserve the appropriate structure; quasi-isomorphisms are morphisms... |

25 |
Méthode simpliciale en algèbre homologique et algèbre commutative. Lecture
- André
(Show Context)
Citation Context ...ty is well established: a homomorphism ' : R \Gamma! S is regular if it is flat and has geometrically regular fibers. Foundational work of Grothendieck [26], Lichtenbaum and Schlessinger [32], Andr'e =-=[1]-=-, [3], and Quillen [37], characterized regularity by the vanishing for ns1 of the functors D n (S j R; \Gamma) of Andr'eQuillen (or: cotangent) homology. In contrast, no general notion of l.c.i. homom... |

23 |
Locally Gorenstein homomorphisms
- Avramov, Foxby
- 1992
(Show Context)
Citation Context ...Gamma) for any single ns1 implies that ' is l.c.i. An interplay of structural and homological arguments allows for a comprehensive study of l.c.i. maps, in the framework of a joint program with Foxby =-=[11]-=-, [12], [13] to classify ring homomorphisms according to their local structure. We establish the stability of the new class under composition, decomposition, flat base change, localization, completion... |

12 |
Local algebra and rational homotopy, Homotopie algébrique et algèbre locale
- Avramov
- 1984
(Show Context)
Citation Context ... with the projection U \Gamma! U=(u 1 ; : : : ; u e ) = `hXi has the desired properties.sFinally, we establish the exponential growth of deviations: this is a `looking glass' version, in the sense of =-=[8]-=- and [15], of a result of F'elix, Halperin, and Thomas [23] on the ranks of homotopy groups of finite CW complexes. The strategy of their proof goes through, but its engine---the `mapping theorem' of ... |

8 | Foxby, Cohen-Macaulay properties of ring homomorphisms
- Avramov, H-B
- 1998
(Show Context)
Citation Context ...ny single ns1 implies that ' is l.c.i. An interplay of structural and homological arguments allows for a comprehensive study of l.c.i. maps, in the framework of a joint program with Foxby [11], [12], =-=[13]-=- to classify ring homomorphisms according to their local structure. We establish the stability of the new class under composition, decomposition, flat base change, localization, completion, and clarif... |

6 |
Infinite free resolutions
- Avramov
- 1998
(Show Context)
Citation Context ...perin's result [30] on the rigidity of deviations of local rings. We extend his argument to the relative case by using Cohen factorizations, and develop shortcuts based on the study of derivations in =-=[10]-=-. First, we record how conditions on the flat dimension of ' pass through factorizations. (3.5) Remark. If R \Gamma! R 0 \Gamma! b S is a Cohen factorization of ` ' then fd R S = fd R b Sspd R 0 b Ssf... |

5 |
Flat morphisms of complete intersections, Dokl
- Avramov
- 1975
(Show Context)
Citation Context ...eld statements on the transfer of the l.c.i. property of rings along '; in the flat case, by (5.2) they show that S is l.c.i. if and only if R and the non-trivial fibers of ' are l.c.i., as proved in =-=[6]-=-. (5.9) Ascent. If R and ' are l.c.i., then so is S.s(5.10) Descent. If S is l.c.i. and ' is locally of finite flat dimension, then R is l.c.i. on the image of a ' and ' is l.c.i.sThe functorial prope... |

4 |
Homological dimension in noetherian rings
- Auslander, Buchsbaum
- 1958
(Show Context)
Citation Context ...t dimension if fd R S q is finite for all q 2 Spec S. This condition is clearly implied by the finiteness of fd R S, and is equivalent to it in many cases, e.g. when R has finite Krull dimension, cf. =-=[5]-=-. For maps essentially of finite type the `only if' part of the next theorem is well known. Quillen [37; (5.7)] conjectured that the converse holds as well. This was proved by Avramov and Halperin [16... |

4 |
Local rings of finite simplicial dimension
- Avramov
(Show Context)
Citation Context ... [37; (7.3)]. When char ` = p ? 0 the edge homomorphism fi n is bijective for 2sns2p by the argument of Andr'e [4].sNext we prove a local version of Quillen's conjecture by replaying an argument from =-=[9]-=-. (4.4) Theorem. Let ' : R \Gamma! (S; n; `) be a local homomorphism, such that fd R S ! 1. If D n (S j R; `) = 0 for n AE 0 then ' is c.i. at n. Proof. By Lemma (1.7) and Remark (3.5), we may assume ... |

2 |
Descente des déviations par homomorphismes locaux et génération des ideaux de dimension projective finie, C.R.Acad.Sci.Paris295
- Avramov
- 1982
(Show Context)
Citation Context ... give D n (S j R; k(q)) = 0 for ns3 and each q 2 Spec S, and we conclude that the homology functors vanish by [3; (S.29)].s22 L. L. AVRAMOV We finish this section with an application of the result of =-=[7]-=-, on maps of vector spaces of indecomposables in Tor's, to the proof of a vanishing theorem for connecting homomorphisms in some Jacobi-Zariski exact sequences. For surjective ring homomorphisms and i... |

1 |
L'alg`ebre de Lie d'un anneau local, Symp
- Andr'e
- 1970
(Show Context)
Citation Context ...ocal ring. (1) For an ideal a ( R 0 the following are equivalent: (i) a is generated by a regular sequence; (ii) D 2 (R 0 =a j R 0 ; `) = 0; (iii) D n (R 0 =a j R 0 ; `) = 0 for ns2, cf. [3; (6.25)]. =-=(2)-=- The ring R 0 is regular if and only if D 2 (` j R 0 ; `) = 0, cf. [3; (6.26)]. (3) If k \Gamma! ` is a field extension, then fd ` L(` jk)s1, cf. [3; (7.4)]. (1.7) Lemma. If ' : (R; m; k) \Gamma! (S; ... |

1 |
Homologie des Alg`ebres
- Andr'e
(Show Context)
Citation Context ... well established: a homomorphism ' : R \Gamma! S is regular if it is flat and has geometrically regular fibers. Foundational work of Grothendieck [26], Lichtenbaum and Schlessinger [32], Andr'e [1], =-=[3]-=-, and Quillen [37], characterized regularity by the vanishing for ns1 of the functors D n (S j R; \Gamma) of Andr'eQuillen (or: cotangent) homology. In contrast, no general notion of l.c.i. homomorphi... |

1 |
La (2p + 1)`eme d'eviation d'un anneau local
- Andr'e
- 1977
(Show Context)
Citation Context .... When S 0 and k 0 are fields of characteristic 0 Quillen [37; (7.3)] proves that fi n is an isomorphism for 1sn ! 1, cf. also [3; (19.21)]. When S 0 and k 0 are fields of characteristic p ? 0 Andr'e =-=[4]-=- proves that fi n is an isomorphism for 1sns2p, but not necessarily for n = 2p + 1. The spectral sequence in the next theorem takes an input analogous to that of Quillen's sequence described above, an... |

1 |
Grothendieck's localization problem, in: Commutative Algebra
- Avramov, Foxby
- 1992
(Show Context)
Citation Context ... for any single ns1 implies that ' is l.c.i. An interplay of structural and homological arguments allows for a comprehensive study of l.c.i. maps, in the framework of a joint program with Foxby [11], =-=[12]-=-, [13] to classify ring homomorphisms according to their local structure. We establish the stability of the new class under composition, decomposition, flat base change, localization, completion, and ... |