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AndreQuillen Homology Of Algebra Retracts
, 2001
"... this paper. Let K k be a surjective morphism of DG algebras, where k is a eld concentrated in degree 0, and let k ,! l be a eld extension. If K ,! KhX i k is a factorization as in 1.7, then the unique DGalgebra structure on lhX i = KhX K l is admissible. Thus, Tor de nes a functor from th ..."
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Cited by 5 (3 self)
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this paper. Let K k be a surjective morphism of DG algebras, where k is a eld concentrated in degree 0, and let k ,! l be a eld extension. If K ,! KhX i k is a factorization as in 1.7, then the unique DGalgebra structure on lhX i = KhX K l is admissible. Thus, Tor de nes a functor from the category of diagrams K k ,! l, with the obvious morphisms, to the category ofalgebras and their morphisms. 2. Factorizations of local homomorphisms Let ' : (R; m; k) ! (S; n; l) be a homomorphism of local rings, which is local in the sense that '(m) n. A regular factorization of ' is a commutative diagram # # # # # # # # ;; # # # # # # # # of local homomorphisms such that the Rmodule R is at, the ring R is regular, and the map R ! S is surjective. Regular factorizations are often easily found, for instance when ' is essentially of nite type (in particular, surjective), or when ' is the canonical embedding of R in its completion with respect to the maximal ideal. In this paper they are mostly used through the following construction of Avramov, Foxby, and B. Herzog [10]
Gaps in Hochschild cohomology imply smoothness for commutative algebra
 Math. Res. Letters
"... Abstract. The paper concerns Hochschild cohomology of a commutative algebra S, which is essentially of finite type over a commutative noetherian ring K and projective as a Kmodule. For a finite Smodule M it is proved that vanishing of HH n (S K; M) in sufficiently long intervals imply the smoothn ..."
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Abstract. The paper concerns Hochschild cohomology of a commutative algebra S, which is essentially of finite type over a commutative noetherian ring K and projective as a Kmodule. For a finite Smodule M it is proved that vanishing of HH n (S K; M) in sufficiently long intervals imply the smoothness of Sq over K for all prime ideals q in the support of M. In particular, S is smooth if HH n (S K; S) = 0 for (dim S + 2) consecutive n ≥ 0.
Nilpotency in the homotopy of simplicial commutative algebras
 Department of Mathematics, Purdue University, West Lafayette
, 2001
"... Abstract. In this paper, we continue a study of simplicial commutative algebras with finite AndréQuillen homology, that was begun in [19]. Here we restrict our focus to simplicial algebras having characteristic 2. Our aim is to find a generalization of the main theorem in [19]. In particular, we re ..."
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Abstract. In this paper, we continue a study of simplicial commutative algebras with finite AndréQuillen homology, that was begun in [19]. Here we restrict our focus to simplicial algebras having characteristic 2. Our aim is to find a generalization of the main theorem in [19]. In particular, we 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 AndréQuillen homology, this nilpotency condition provides a way to bound the height at which the homology vanishes. As a consequence, we establish a special case of an open conjecture of Quillen.
ON SIMPLICIAL COMMUTATIVE ALGEBRAS WITH FINITE ANDRÉQUILLEN HOMOLOGY
, 2008
"... Abstract. In [22, 4] a conjecture was posed to the effect that if R → A is a homomorphism ..."
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Abstract. In [22, 4] a conjecture was posed to the effect that if R → A is a homomorphism
HOCHSCHILD (CO)HOMOLOGY IN COMMUTATIVE ALGEBRA. A SURVEY
"... Originally introduced for associative algebras over commutative rings, Hochschild (co)homology theory first played a big role in commutative algebra by the famous theorem obtained by Hochschild, Kostant and Rosenberg. For some time, the interest in this theory was not so big, but was reinitialized a ..."
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Originally introduced for associative algebras over commutative rings, Hochschild (co)homology theory first played a big role in commutative algebra by the famous theorem obtained by Hochschild, Kostant and Rosenberg. For some time, the interest in this theory was not so big, but was reinitialized around 1990, due