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Main Teaching Research Computing |
Abstracts of John Palmieri's papersFor a finite module M over the Steenrod algebra A, we prove the existence of a non-nilpotent element in ExtA(M,M) ``parallel to the vanishing line.'' We use this result to give a proof of Margolis' construction to kill Pst-homology groups, at all primes. We prove a conjecture of Adams: the mod 2 Steenrod algebra A satisfies the ``detection'' property; i.e., every non-nilpotent element of ExtA(Z/2, Z/2) can be detected by restricting to an exterior sub-Hopf algebra of A. We prove that certain spectra have self maps easily described in terms of Ext over the Steenrod algebra. We apply this to prove Jeff Smith's theorem that for each n, there is a finite spectrum admitting a vn-map. Other applications are Margolis' killing construction for spectra and a spectral sequence for computing the homotopy of some vn-periodic telescopes. Using Steenrod algebra tools (Pst-homology, in particular), we develop machinery analogous to the chromatic spectral sequence (and associated paraphernalia) of Miller, Ravenel, and Wilson in their Annals paper; and for any p-local spectrum X we use the Steenrod algebra to construct a variant of the chromatic tower over X. Let B be a sub-Hopf algebra of the mod 2 Steenrod algebra, and let M be a finite dimensional B-module. We prove that one can detect nilpotence of elements in ExtB (M,M) by restricting to elementary sub-Hopf algebras, and we prove a similar result at odd primes. We imitate much of the work of Hopkins and Smith, in the context of modules over the mod 2 Steenrod algebra A. Our main result is an analog of their periodicity theorem, implying, for example, the existence of infinitely many non-nilpotent elements in ExtA(Z/2, Z/2). In the context of finite dimensional cocommutative Hopf algebras, we prove versions of various group cohomology results: the Quillen-Venkov theorem on detecting nilpotence in group cohomology, Chouinard's theorem on determining whether a kG-module is projective by restricting to elementary abelian p-subgroups of G, and Quillen's theorem which identifies the cohomology of G, ``modulo nilpotent elements.'' This last result is only proved for graded connected Hopf algebras. We define and investigate a class of categories with formal properties similar to those of the homotopy category of spectra. This class includes suitable versions of the derived category of modules over a commutative ring, or of comodules over a commutative Hopf algebra, and is closed under Bousfield localization. We study various notions of smallness, questions about representability of (co)homology functors, and various kinds of localization. We prove some theorems analogous to those of Hopkins and Smith about detection of nilpotence and classification of thick subcategories. We define the class of Noetherian stable homotopy categories, and investigate their special properties. Finally, we prove that a number of categories occurring in nature (including those mentioned above) satisfy our axioms. Go to the article (at the Springer-Verlag website). In this paper the authors study the cohomological varieties associated to the finite-dimensional sub-Hopf algebras of the Steenrod algebra. A stratification theorem like the Quillen stratification theorem for finite groups is proven. With this stratification one can then invoke results from restricted Lie algebra cohomology to study these cohomological varieties. Several results and conjectures of Adams and Margolis are obtained through this approach. 10. Quillen stratification for the Steenrod algebra, Annals of Mathematics 149 (1999), 421-449. Here is a link to the paper at the Mathematics ArXiv. Let A be the mod 2 Steenrod algebra, and let Q denote the category of exterior sub-Hopf algebras of A, where the morphisms are given by inclusions. The restriction maps
ExtA (Z/2,Z/2) --> ExtE (Z/2,Z/2),
for E in Q can be assembled into a map
i : ExtA (Z/2, Z/2) --> limQ ExtE (Z/2,Z/2).
There is an action of A on this inverse limit, and i factors through
the invariants under this action, giving a map
g : ExtA (Z/2, Z/2) --> (limQ ExtE (Z/2,Z/2))A.
We show that g is an F-isomorphism.
I have found one mistake in the paper, described in this dvi file or pdf file. In this paper, we study general questions about the Bousfield lattice of spectra. Our starting point is Ohkawa's theorem that the collection B of Bousfield classes is a set. Using this result, we perform a number of constructions with Bousfield classes. For example, we describe a greatest lower bound operator; this gives B the structure of a lattice. We examine the sub-poset DL of B. Since the greatest lower bound in DL distributes over arbitrary least upper bounds, then DL is a frame; we use this observation to construct a complete Boolean algebra cBA, which is related to the Boolean algebra of spectra, BA, which was first constructed by Bousfield. We also discuss a number of other conjectures regarding Bousfield classes, and we prove that various of these are equivalent to various others. Our hope is that these conjectures, as well as the lattice-theoretic approach that Ohkawa's theorem provides, will lead to new structural understanding of the stable homotopy category. Here is a link to the abstract and the paper at the Geometry and Topology web site. Let E be a nice ring spectrum, let X be a spectrum, and consider the E-based Adams spectral sequence converging to the homotopy groups of X. We prove that, for any number m, the property that the spectral sequence has a vanishing line of slope m at some term of the spectral sequence is generic. Let A be the mod p Steenrod algebra. We study A by means of the category Stable(A), in which the objects are cochain complexes of injective comodules over the dual of A. This is a stable homotopy category, in the sense of Hovey-Palmieri-Strickland, so we study this category using the tools of stable homotopy theory. In particular, we set up basic homotopy theoretic tools (like Postnikov towers), and we also generalize some well-known results about A-modules to the category Stable(A) (like the vanishing line theorems of Anderson-Davis and Miller-Wilkerson). We then use the methods and philosophy of modern stable homotopy theory to study Stable(A). We end up with generalizations of deeper results about A-modules (such as the main theorem of Nilpotence for modules over the mod 2 Steenrod algebra I), as well as new results (such as a disproof of one version of the telescope conjecture in the category Stable(A)). I have found one mistake in the paper, described in this dvi file or pdf file. Go to the issue of the Proceedings containing this article (at the AMS website). We give a short elementary proof of Ohkawa's theorem: there is a set of Bousfield classes (we are working stably, by the way). There are now classifications of thick subcategories of modules over several different sorts of Hopf algebras: group algebras of p-groups and sub-Hopf algebras of the mod 2 Steenrod algebra. These results require one to work over an algebraically closed field. In this paper, we prove that such a classification over an algebraically closed field yields a similar classification over any subfield, and in particular over finite fields. Extending work of Benson, Carlson, and Rickard on varieties for infinitely generated modules over p-groups, we prove a Quillen stratification theorem for cohomological varieties of modules over a finite-dimensional graded cocommutative Hopf algebra B. The stratification is in terms of quasi-elementary subHopf algebras of B. We show that this reduces the classification of thick subcategories of stable B-modules to the classification of thick subcategories over these quasi-elementary subHopf algebras. In the particular case where B is a finite-dimensional sub-Hopf algebra of the mod 2 Steenrod algebra, we are able to complete the program, giving a classification of thick subcategories, as well as a description of the Bousfield lattice in this case. 17. There is no paper 17 at the moment. An A-infinity algebra is a natural generalization of an associative algebra. We provide some examples and applications of A-infinity algebras and show that A-infinity algebras can be used to solve questions in ring theory. We construct four families of Artin-Schelter regular algebras of global dimension four. Under some generic conditions, this is a complete list of Artin-Schelter regular algebras of global dimension four that are generated by two elements of degree 1. These algebras are also strongly noetherian, Auslander regular and Cohen-Macaulay. One of the main tools is Keller's higher-multiplication theorem on A-infinity Ext-algebras. Download: dvi file or pdf file. Fix a prime p, and let A be the polynomial part of the dual Steenrod algebra. The Frobenius map on A induces the Steenrod operation P0 on cohomology, and in this paper, we investigate this operation. We point out that if p=2, then for any element in the cohomology of A, if one applies P0 enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and that ``enough times'' should be ``once.'' The bulk of the paper is a study of some quotients of A in which the Frobenius is an isomorphism of order n. We show that these quotients are dual to group algebras, the resulting groups are torsion-free, and hence every element in Ext over these quotients is nilpotent. We also try to relate these results to the questions about P0. The dual complete Steenrod algebra makes an appearance. 21. The Lambda algebra and Sq0, in Geometry and Topology Monographs 11 (2007), 201-216. Go to the article at the Geometry and Topology website. We study the action of Sq0 on the Adams E2-term using the Lambda algebra. In particular, there is a Bockstein spectral sequence whose E1-term is the cohomology of a quotient of the Lambda algebra, which converges to the Adams E2-term, and whose differentials reflect the action of Sq0. Using this, we show that Sq0 is injective on Exts for s < 4, and the kernel of Sq0 on Ext4 is the vector space spanned by h04. 22. A user's guide to the Bockstein spectral sequence, preprint (2005), 7 pages. Download: dvi file or pdf file. We consider a non-traditional presentation of the Bockstein spectral sequence. This is an expository paper — there is nothing new in it. Let A be a connected graded algebra and let E denote its Ext-algebra ExtA(kA, kA). There is a natural A-infinity structure on E, and we prove that this structure is mainly determined by the relations of A. In particular, the coefficients of the A-infinity products mn restricted to the tensor powers of E1 give the coefficients of the relations of A. We also relate the mn's to Massey products. Go to the article at the New York Journal website. We prove a version of Koszul duality and the induced derived equivalence for Adams connected A-infinity algebras that generalizes the classical Beilinson-Ginzburg-Soergel Koszul duality. As an immediate consequence, we give a version of the Bernstein-Gel'fand-Gel'fand correspondence for Adams connected A-infinity algebras. We give various applications. For example, a connected graded algebra A is Artin-Schelter regular if and only if its Ext-algebra ExtA(k,k) is Frobenius. This generalizes a result of Smith in the Koszul case. If A is Koszul and if both A and its Koszul dual A! are noetherian satisfying a polynomial identity, then A is Gorenstein if and only if A! is. The last statement implies that a certain Calabi-Yau property is preserved under Koszul duality. Go to the article at the Geometry and Topology website. Go to the article at the HHA website. The global structure of the unbounded derived category of a truncated polynomial ring on countably many generators is investigated, via its Bousfield lattice. In particular, the following results are proved: (1) the Bousfield lattice is shown to have at least 22aleph0 elements. (2) The Bousfield lattice has a unique nonzero minimal element. (3) For each positive integer n, there is an object X in this derived category so that the n-fold derived tensor product of X with itself is nonzero, while the (n+1)-fold derived tensor product is zero. 27. Commutators in the Steenrod algebra, joint with J. J. Zhang, New York J. Math. 19 (2013), 23-37. Go to the article at the New York Journal website. We construct a family of bases for the mod p Steenrod algebra as products of iterated commutators of the algebra generators. Go to the preprint at arXiv 1401.0793 (31 pages). We use the discriminant to determine the automorphism groups of some noncommutative algebras, and we prove that a family of noncommutative algebras has tractable automorphism groups. We compute the automorphism groups of some quantized algebras, including tensor products of quantum Weyl algebras and some skew polynomial rings. 30. Invariant theory for quantum Weyl algebras under finite group action, joint with S. Ceken, Y. Wang, J. J. Zhang, preprint arXiv 1501.07881 (15 pages). We study the invariant theory of a class of quantum Weyl algebras under group actions and prove that the fixed subrings are always Gorenstein. We also verify the Tits alternative for the automorphism groups of these quantum Weyl algebras. |
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