4 papers this by time, one by Bauer and three by Jim Turner. Mark Hovey New papers appearing on hopf between 02/11/02 and 03/05/02 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BauerK/bauer1 Title: Higher Hochschild homology and its decompositions Author: Kristine Bauer Department of Mathematics Johns Hopkins University E-mail: kbbauer@math.jhu.edu Let k be a field of characteristic 0, A a k-algebra and M an A-module. In this paper we seek to provide a decomposition of a generalization of Hochschild homology. The construction is as follows: Let F_A be the functor from the category of finite pointed sets to k-vector spaces which takes [n]={0,1,...,n} to the tensor product of M with the n-fold tensor product of A with itself. Now consider the homology of the chain complex associated to F_A(S^1\wedge Y) where S^1\wedge Y is a simplicial finite pointed set. The special case where the realization of Y is an (n-1)-dimensional sphere is the n-th order higher Hochschild homology. To obtain the decomposition, we show that F_A(S^1\wedge Y) is a Hopf algebra under maps whose existence is suggested by the pinch and fold maps on the circle. We are then able to apply the methods which Loday and Gerstenhaber and Schack used to obtain a decomposition of Hochschild homology, which is the case F_A(S^1). Finally, we show that this decomposition recovers the decomposition of higher Hochschild homology recently obtained by Pirashvili. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Turner/nilpotence Title: Nilpotency in the homotopy of simplicial commutative algebras Authors: James M Turner Address: Calvin College E-mail: jturner@calvin.edu ArXiv id. no.: math.AT/0201064 MSC-class: 13D03, 13D05, 13H10, 18G30, 55S99 Abstract: In this paper, we study simplicial commutative algebras with finite Andr\'e-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 Andr\'e-Quillen 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. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Turner/Noetherian Title: On simplicial commutative algebras with Noetherian homotopy Authors: James M Turner Address: Calvin College E-mail: jturner@calvin.edu ArXiv id. no.: math.AT/0201063 MSC-class: 13D03, 13D05, 18G30, 55S45, 55U99 Abstract: In this paper, a strategy is developed for 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 general form of a conjecture posed by Quillen. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Turner/vanishing (This is the final version of a paper that has been annnounced before, the last time in 1998) Title: Simplicial commutative algebras with vanishing Andre-Quillen homology Author: James M Turner Address: Calvin College E-mail: jturner@calvin.edu MSC-class: 13D03, 13D40, 18G30, 18G55 Journal-ref: Inventiones mathematicae 142 (3) (2000), pp. 547-558 Abstract: In this paper, we study the Andr\'e-Quillen homology of simplicial commutative $\ell$-algebras, $\ell$ a field, having certain vanishing properties. When $\ell$ has non-zero characteristic, we obtain an algebraic version of a theorem of J.-P. Serre and Y. Umeda that characterizes such simplicial algebras having bounded homotopy groups. We further discuss how this theorem fails in the rational case and, as an application, indicate how the algebraic Serre theorem can be used to resolve a conjecture of D. Quillen for algebras of finite type over Noetherian rings, which have non-zero characteristic.