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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
FREE RESOLUTIONS OVER COMMUTATIVE KOSZUL ALGEBRAS
, 904
"... Abstract. For R = Q/J with Q a commutative graded algebra over a field and J = 0, we relate the slopes of the minimal resolutions of R over Q and of k = R/R+ over R. When Q and R are Koszul and J1 = 0 we prove Tor Q i (R, k)j = 0 for j> 2i ≥ 0, and also for j = 2i when i> dim Q − dim R and pd Q R i ..."
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Abstract. For R = Q/J with Q a commutative graded algebra over a field and J = 0, we relate the slopes of the minimal resolutions of R over Q and of k = R/R+ over R. When Q and R are Koszul and J1 = 0 we prove Tor Q i (R, k)j = 0 for j> 2i ≥ 0, and also for j = 2i when i> dim Q − dim R and pd Q R is finite. Let K be a field and Q a commutative Ngraded Kalgebra with Q0 = K. Each graded Qmodule M with Mj = 0 for j ≪ 0 has a unique up to isomorphism minimal graded free resolution, F M. The module F M i has a basis element in degree j if and only if Tor Q i (k, M)j = 0 holds, where k = Q/Q + for Q + = ⊕ j�1 Qj. Important structural information on F M is encoded in the sequence of numbers t Q i (M) = sup{j ∈ Z  TorQi (k, M)j = 0}. It is distilled through the notion of CastelnuovoMumford regularity, defined by regQ M = sup{t i�0 Q i (M) − i}. One has regQ k ≥ 0, and equality means that Q is Koszul; see, for instance, [17]. When the Kalgebra Q is finitely generated, every finitely genetrated graded Qmodule M has finite regularity if and only if Q is a polynomial ring over some Koszul algebra, see [6]; by contrast, the slope of M over Q, defined to be the real number slope Q M = sup i�1 t Q