``First steps in brave new commutative algebra.''
[28/4/06]
Click here (.dvi) or
Click here (.pdf)
for
M.Ando and J.P.C.Greenlees ``Circle-equivariant classifying spaces and
the rational equivariant sigma genus.'' 69pp
We analyze the circle-equivariant spectrum MString_C, which is the
equivariant analogue of the cobordism spectrum MU<6> of stably almost complex
manifolds with c_1=c_2=0. The second author showed
how to construct the ring T-spectrum EJ representing the T-equivariant
elliptic cohomology associated to a rational elliptic curve J. In the
case that J is a complex elliptic curve, we construct a map of ring
T-spectra MString_C---->EJ which is the rational equivariant analogue of
the sigma orientation of Ando-Hopkins-Strickland. Our method gives a proof
of a conjecture of the first author. (Added 17/5/07)
Click here
for
D.J.Benson and J.P.C.Greenlees ``Commutative algebra for the cohomology of
classifying spaces of compact Lie groups.'' 12pp
Establishes duality
properties (such as the local cohomology theorem) for the cohomology ring of the
classifying space of compact Lie group. Companion to the paper by Elmendorf and
May below.
Click here (.dvi)
or here (.ps)
for
D.J.Benson and J.P.C.Greenlees ``Commutative algebra for the cohomology
rings of virtual duality groups.'' 16pp
Establishes duality properties
(such as the local cohomology theorem) for the cohomology ring of the
classifying space of virtual duality groups. We give a purely algebraic proof
which applies to all such groups, and a topological proof which applies to
groups which are virtual duality groups for geometric reasons. (added 28/2/96)
Click here (.dvi) or here (.ps) for
D.J.Benson and J.P.C.Greenlees ``Localization and duality in topology and
modular representation theory'' 41pp
Establishes Gorenstein duality
properties for localizations of the module category of kG for a finite group G,
proving a conjecture of Benson's. The method is to use an S-algebra version of
local duality in the setting of Dwyer-Greenlees-Iyengar,
and then to translate this into purely algebraic categories. (added 20/5/04)
Click here (.dvi)
or here (.ps)
for
R.Bruner and J.P.C.Greenlees ``The algebraic Bredon-Loffler conjecture.''
13pp
The conjecture is an Ext-level analogue of the conjecture better known
as `the-root-invariant-never-more-than-triples-the-stem conjecture'; this is
verified out to the 30-stem.
Click here (.tar.gz) (212K)
or here (.ps)
(1.8Mb) for
R.Bruner and J.P.C.Greenlees ``Connective K theory of finite groups.'' 129pp
We give a systematic study of the connective K theory of finite groups.
This includes consideration of the variety of the ku cohomology ring, and a
large number of calculations in particular cases. It also includes calculations
of the ku homology by the local cohomology spectral sequence and a discussion of
duality. Finally we give an extensive study of the case of an elementary abelian
2-group, where the commutative algebra is remarkably intricate. [The .tar.gz
format is used since we use 22 .eps files in addition to the .dvi file] (added
3/1/01)
Click here (.dvi) or here (.ps) for
M.Cole, J.P.C.Greenlees and I.Kriz ``Equivariant formal group laws.'' 28
pp
We define the algebraic notion of an A-equivariant formal group law for
an abelian compact Lie group A. We give tools for calculation and show that
there is a universal ring representing A-formal group laws. Any complex oriented
A-equivariant cohomology theory gives rise to an A-formal group law by taking
cohomology of the representing A-space for A-line bundles. (Added 19/11/97)
Click here (.dvi) or here (.ps) for
M.Cole, J.P.C.Greenlees and I.Kriz ``The universality of equivariant complex
bordism.'' 16 pp
For any abelian group A, and any complex oriented
cohomology theory all complex vector bundles are orientable. It follows that
complex equivariant bordism is universal for complex oriented cohomology
theories. In the course of the proof we calculate the complex oriented
cohomology of all complex grassmannians. (Added 20/11/98; slightly revised
4/8/00)
Click here (.dvi) or here (.ps) for
W.G.Dwyer and J.P.C.Greenlees ``Complete modules and torsion modules'' 20
pp
For a ring R and an ideal I we discuss categories of torsion and complete
modules (subcategories of the derived category of dg R-modules), and in the
commutative case give an algebraic characterization in terms of their homology.
The main point is to give a version of Morita theory showing that I-tors-R-mod =
mod-E = I-comp-R-mod under rather general circumstances where E is the
endomorphism dga of R/I. The case when R is the integers and I = (p) is already
interesting, and more general cases are related to the algebraicization of
rational stable homotopy theory. (Added 21/12/99; slightly revised 30/3/01)
Click here (.dvi) or here (.ps) for
W.G.Dwyer, J.P.C.Greenlees and S.B.Iyengar ``Duality in algebra and
topology.'' 39 pp
This paper allows us to view (1) Poincare duality for
manifolds, (2) Gorenstein duality for commutative rings, (3) Benson-Carlson
duality for cohomology of finite groups (4) Poincare duality for groups and (4)
Gross-Hopkins duality in chromatic stable homotopy theory as instances of a
single phenomenon. One new consequence is the local cohomology theorem for the
cohomology of a p-compact group. (Added 11/3/02; complete expository revision
28/10/05)
Click here (.dvi) or here (.ps) for
W.G.Dwyer, J.P.C.Greenlees and S.B.Iyengar ``Finiteness in derived
categories of local rings.'' 39 pp
New homotopy invariant finiteness
conditions on modules over commutative rings are introduced, and their
properties are studied systematically. A number of finiteness results for
classical homological invariants like flat dimension, injective dimension, and
Gorenstein dimension, are established. It is proved that these specialize to
give results concerning modules over complete intersection local rings. A
noteworthy feature is the use of techniques based on thick subcategories of
derived categories. (Added 4/6/04)
Click here
for
A.D.Elmendorf and J.P.May ``Algebras over equivariant sphere spectra.''
12pp
A corollary of the main result is that Borel cohomology is represented
by a highly structured ring spectrum for any compact Lie group. Companion to the
paper by Benson and Greenlees above.
Click here
(.dvi) or here
(.ps) for
J.P.C.Greenlees ``Commutative algebra in group cohomology.'' 11 pp
The
cohomology ring of a finite group is shown to enjoy certain global duality
properties in the sense that the local cohomology theorem holds: in particular
this gives a new proof of the result of Benson and Carlson that if the
cohomology ring of a finite group is Cohen-Macaulay it is also Gorenstein.
Various observations are made about the Tate cohomology: in particular it is
essentially the cohomology of the projective space of the cohomology ring.
(added 15/9/95)
Click here (.dvi) or here (.ps) for
J.P.C.Greenlees ``Rational Mackey functors for compact Lie groups I.'' 32pp
The category of rational Mackey functors is shown to be equivalent to the
category of equivariant sheaves on the Weyl-toral category. The advantage is
that the latter category only has morphisms in one direction, and is thus easily
seen to have global dimension equal to the rank of the group. This version is a
simplified and corrected version of an earlier preprint. (Added 9/9/96)
J.P.C.Greenlees ``Rational S^1 equivariant stable homotopy
theory.'' viii + 287 pp
For reasons of size, this is available in pieces as
indicated.
For Introduction and contents click here (.ps 212kb)
For Introduction and contents and Part I click here (.ps
686kb)
For Part II click here (.ps
331kb)
For Part III click here (.ps
421kb)
For Part IV click here (.ps
527kb)
For Appendices, Index and Bibliography click here (.ps
295kb)
For the whole thing here (.dvi 1120kb), here (.ps 1762kb) or here (.ps 2 pages to a
sheet 1836kb)
A complete algebraic model is given for the category of rational S^1 spectra
is given in Part I. In Part II the algebraic counterparts of various change of
groups functors are described. Part III gives numerous applications. Part IV
gives algebraic models for the smash product and function spectra.
This
version is substantially reorganized, and somewhat improved, from the 28/3/96
version, and the two chapters on smash products and function spectra have now
expanded to form Part IV. (added 28/3/96; revised version 19/11/97)
Click here for
J.P.C.Greenlees ``Augmentation ideals of equivariant cohomology rings.''
10pp
It is shown that if a group acts freely on a product of spheres then
for a complex oriented Noetherian cohomology theory the radical of the ideal
generated by Euler classes is the augmentation ideal; there is an analagous
result for general finite groups. This implies good behaviour of the ideals
under restriction. (added 25/4/96; updated 27/2/97)
Click here (.dvi) or here (.ps) for
J.P.C.Greenlees ``Rational O(2)-equivariant cohomology theories.'' 8
pp
A complete algebraic model is given for rational O(2)-spectra. The main
input is the model for SO(2)-spectra described in ``Rational
S^1-equivariant stable homotopy theory'', and the special case of ``Rational
Mackey functors for compact Lie groups I'' applying to O(2). It turns out that
any O(2)-spectrum is described by an SO(2)-spectrum with O(2)/SO(2)-action,
together with an equivariant sheaf over the space of dihedral subgroups of O(2).
(Added 13/9/96; slightly revised 26/9/97)
Click here (.dvi) or here (.ps) for
J.P.C.Greenlees ``Equivariant forms of connective K-theory'' 18 pp
A
good equivariant version of connective complex K-theory is constructed for the
group of prime order. This has the following properties (i) it is
non-equivariantly ku, (ii) it is a split ring spectrum (iii) it becomes
equivariant periodic K-theory if the Bott element is inverted (iv) it is complex
orientable (v) its coefficient ring is Noetherian and in even degrees. (Added
19/11/97) [Comment (August 2000): A construction is now available for all
compact Lie groups. It is known to have properties (i), (ii) and (iii) in
general, and (iv) if the groupo is abelian. Its coefficient ring is Noetherian
in all known cases, but not generally in even degrees.]
Click here (.dvi) or
here (.ps) for
J.P.C.Greenlees ``Tate cohomology in axiomatic stable homotopy theory.'' 18
pp
It is observed that any smashing localization gives in an axiomatic
stable homotopy theory in the sense of Hovey-Palmieri-Strickland gives rise to a
Tate theory. Various known versions of Tate cohomology (for example in
commutative algebra, in the cohomology of groups, in equivariant homotopy theory
and in chromatic stable homotopy theory) are considered from this point of view.
The revised version is better organized and more accurate. It also contains a
new section on homotopically Gorenstein rings. (Added 20/11/98; Revised version
15/11/99)
Click here (.dvi) or here (.ps) for
J.P.C.Greenlees ``Multiplicative equivariant formal group laws.'' 10
pp
It is shown that there is a universal ring for multiplicative equivariant
formal group laws, and the ring is identified explicitly. It is closely related
to the Rees ring of the representation ring at the augmentation ideal, but only
equal to it if the group is topologically cyclic. If the group is of prime order
the universal ring is the coefficient ring of equivariant connective K-theory,
as constructed in ``Equivariant forms of
connective K-theory.'' (Added 13/1/99; significantly improved version
15/11/99; minor revisions 3/8/00)
Click here (.dvi) or here (.ps) for
J.P.C.Greenlees ``Rational SO(3)-equivariant cohomology theories.'' 20
pp
A complete algebraic model is given for rational SO(3)-spectra. The main
input is the model for SO(2)-spectra described in ``Rational
S^1-equivariant stable homotopy theory'',, but special cases of results from
``Rational Mackey functors for compact Lie groups I'' applying to subgroups of
SO(3) are used, as are results of ``Rational
O(2)-equivariant cohomology theories.'' . Some of the methods for reducing
to subgroups are interesting in themselves. (Added 3/8/00.)
Click here (.dvi) or
here (.ps)
for
J.P.C.Greenlees ``Local cohomology in equivariant topology.'' 30 pp
The
article (based on talks at the Guanajuato Workshop on Local Cohomology, December
1999) describes the role of local homology and cohomology in understanding the
equivariant cohomology and homology of universal spaces. This brings to light an
interesting duality property related to the Gorenstein condition. The phenomena
are studied and illustrated in several rather different families of examples.
Both topology and commutative algebra benefit from the connection, and many
interesting questions remain open. AMS classification numbers: 13D45, 19L41,
20Jxx, 55N91, 55N22, 55P43 (Posted 4/10/00; minor amendments 23/2/01)
Click here (.dvi) or here (.ps) for
J.P.C.Greenlees ``Equivariant connective K theory for compact Lie groups''
16 pp
An equivariant version of connective K theory is constructed for all
compact Lie groups. It is shown to be ring valued, Noetherian, non-equivariantly
ku, v-periodically K and complex orientable. This is sufficient justification
for the name. The coefficient ring is shown to be related to the representing
ring of multiplicative equivariant formal group laws as in ``Multiplicative
equivariant formal group laws.'' and equal to it for the product of two
topologically cyclic groups, and to the modified Rees ring. Explicit
calculations in special cases may be obtained from those of ``Connective K theory of
finite groups'' (with R.R. Bruner) . (Posted 3/1/01; revised version
22/11/01)
Click here (.dvi) or here (.ps) for
J.P.C.Greenlees ``Equivariant formal group laws and complex oriented
cohomology theories.'' 32 pp
This is a survey of what I knew about
equivariant formal group laws at the time, based on talks at the 2000 Stanford
workshop. It includes summaries of various other papers on this page (those
present at the time with ``equivariant formal group'' in the title, together
with those on connective K theory), together with a sketch of the definition for
non-abelian groups. (Posted 23/2/01, slightly revised 3/4/01)
Click here (.dvi)
or here
(.ps) for
J.P.C.Greenlees ``The coefficient ring of equivariant bordism classifies
equivariant formal groups over Noetherian rings.'' 19 pp
This paper gives
the proof of the theorem of the title. Unfortunately this does not determine the
coefficient ring of equivariant bordism since it is not itself Noetherian.
However a pullback square for A-equivariant bordism is given for any finite
abelian group A, generalizing that of tom Dieck and Kriz for groups of prime
order. There are a number of useful facts about the universal ring for
equivariant formal groups. (Posted 23/2/01)
Click here (.dvi) , here (.ps) or here (.pdf) for
J.P.C.Greenlees ``Rational S^1-equivariant elliptic cohomology'' 51
pp
For each elliptic curve A over the rational numbers we construct a
2-periodic S^1-equivariant cohomology theory E whose cohomology ring is the
sheaf cohomology of A; the homology of the sphere of the representation z^n is
the cohomology of the divisor A(n) of points with order dividing n. The
construction proceeds by using the algebraic models of ``Rational S^1
equivariant homotopy theory.'' and is natural and explicit in terms of
sheaves of functions on A.
The following additional topics were first added
in the Fourth Edition: (a) periodicity and differentials treated (b) dependence
on coordinate divisor (c) relationship with Grojnowksi's construction and, most
importantly, (d) equivalence between the derived category of
O_{A}-modules and EA-modules. Minor changes for Version 4.2. Version 4.3
includes minor changes and adds treatment of non-split torus.
The Fifth
Edition included (a) the Hasse square and (b) explanation of how to calculate
maps of EA-module spectra. Version 5.2 makes numerous small improvements, partly
thanks to the fact that ``An algebraic model for
rational torus-equivariant stable homotopy.'' now shows that EA may be taken
to be a strictly commutative ring spectrum.
(Added 2/8/99; revised version
added 21/12/99; first readable version posted 12/9/00; addendum on naturality
8/11/00; Fourth edition, 27/7/01; Version 4.2 13/2/02; Version 4.3 6/9/02;
Version 5.2 (expected to be the final form) 19/4/05)
Click
here (.dvi) or
here (.pdf) for
J.P.C.Greenlees ``Rational torus-equivariant stable homotopy I: calculating
groups of stable maps.'' (22pp)
This is the first in a series giving a
complete algebraic model for rational G equivariant cohomology theories for an
r-torus G. It constructs an abelian category A(G) of injective dimension r
designed to capture localization and inflation information about cohomology
theories. The category A(G) is the target of a homology theory on G-spectra, and
there is a finite Adams spectral sequence for calculating maps of rational
T-spectra for a torus T, with E2 term an Ext in the abelian category.
(27/7/01; reorganized version 31/8/01; version with characterization of basic
cells 31/8/04; revised version 15/2/07 with expanded exposition, flatness)
Click
here (.dvi) or
here (.pdf) for
J.P.C.Greenlees ``Rational torus-equivariant stable homotopy II: the algebra
of localization and inflation.''
This is the second in a series giving a
complete algebraic model for rational torus equivariant cohomology theories. It
gives an algebraic study of the category A(G) for an r-torus G. The first main
result is one which explains how global information can be reassembled from
small sets of isotropy groups, and the second is to explain how A(G) can be
viewed as a category of modules over a ring R with many objects.
(27/7/01; reorganized version 31/8/01; revised version
15/2/07 with torsion functor, and proof of exact injective dimension)
Click
here (.dvi) or
here (.ps) for
J.P.C.Greenlees and B.E.Shipley ``An algebraic model for rational
torus-equivariant stable homotopy.''
This is the final paper in a series
giving a complete algebraic model for rational torus equivariant cohomology
theories. The paper combines the Schwede-Shipley Morita theory with an intrinsic
formality statement to give a Quillen equivalence between the category of
rational torus-equivariant spectra and the derived category of the category A(G)
constructed in ``Rational torus-equivariant stable homotopy I,II.'' The functors
in the Quillen equivalence are lax symmetric monoidal, so rings and commutative
rings correspond in the two settings. (31/8/04; expository revision
29/11/04) [[Substantially revised and expanded version in preparation.]]
Click here (.dvi) or here (.ps) for
J.P.C.Greenlees ``Algebraic groups and equivariant cohomology theories.''
We discuss a number of examples in which an equivariant cohomology theory
is associated to an algebraic group. The archetype is that S1-equivariant K
theory is associated to the multiplicative group. Slightly less familiar is the
fact that one may associate an S1-equivariant cohomology to an elliptic curve
over a field of characteristic zero (see Rational S1-equivariant
elliptic cohomology). The features of this association are described and we
speculate about higher dimensional examples. [Added 25/11/03]
Click here (.dvi) or here (.ps) for
J.P.C.Greenlees ``Spectra for commutative algebraists.''
This paper is
an expanded version of lectures given at MSRI during the commutative algebra
emphasis year 2002-03. The idea is to give commutative algebraists an idea about
what (strictly) commutative ring spectra are, how to construct a good category
of them, and what use they might be to algebraists. [Added 25/11/03]
Click
here (.dvi) or
here (.pdf) for
J.P.C.Greenlees ``Spectra for commutative algebraists.'' 23pp
The article is based on a series of lectures given at the 2002 MSRI emphasis year
on commutative algebra. It is designed to explain to commutative algebraists what
spectra are, why they were originally defined, and how they can be
useful for commutative algebra.
[Added 28/4/06]
Click
here (.dvi) or
here (.pdf) for
J.P.C.Greenlees ``First steps in brave new commutative algebra.'' 37pp
The article is based on lectures given at the 2004 Chicago Summer School.
It introduces completion, cellularization and localization constructions
in derived categories first from an elementwise point of view and then from
a more conceptual point of view, and discusses Gorenstein ring spectra. This
summarizes joint work with Benson, Dwyer, Iyengar and May.
[Added 28/4/06]
Click
here (.dvi) or
here (.pdf) for
J.P.C.Greenlees ``Triangulated categories from rational equivariant topology.''
7pp
This is a thumbnail sketch introduction to equivariant cohomology
theories and a a very brief summary of the cases where an algebraic model of
rational G-spectra is known.
Click
here (.dvi) for
J.P.C.Greenlees ``Equivariant forms of real and complex connective K-theory.''
24pp
This is a survey of results and calculations as stated in the title; further
details in the real case will appear in a book with R.R. Bruner.
[Added 28/4/06]
Click here (.dvi) or here (.ps) for
J.P.C.Greenlees and G.Lyubeznik ``Rings with a local cohomology theorem, and
applications to group cohomology.'' 17 pp
We study connected graded algebra
over a field which satisfy a local cohomology theorem (such as the cohomology
ring of a group which is a discrete or profinite virtual duality group or a
compact Lie group). It is obvious that if such a ring is Cohen-Macaulay it is
Gorenstein. We show that if it has depth one less than its dimension (ie is
almost Cohen-Macaulay) then it is in fact almost Gorenstein, and its Hilbert
series satisfies a pair of functional equations. This proves a conjecture of
Greenlees and Benson for the cohomology of classifying spaces of compact Lie
groups and virtual duality groups. Minimal primes of local cohomology groups of
cohomology rings of groups are shown to be Quillen strata. (Added 20/11/98)
Click here
for
J.P.C.Greenlees and J.P.May ``Completions in algebra and topology.''
22pp
Click here
for
J.P.C.Greenlees and J.P.May ``Equivariant stable homotopy theory.''
48pp
Click here for
J.P.C.Greenlees and J.P.May ``Localization and completion theorems for MU
module spectra.'' 32pp
The completion theorem and its dual for equivariant
bordism, Morava K-theory, ... and all finite or toral groups. An essential
ingredient is a multiplicative norm map, which is constructed for equivariant
bordism and similar theories arising form global equivariant FSP's. Added
4/8/95.
Click here
for
J.P.C.Greenlees and J.A.Perez ``Connected Lie groups that act freely on
products of linear spheres.'' 11pp
A complete classification of such groups
is given.
Click here for
J.P.C.Greenlees and H.Sadofsky ``Tate cohomology of theories with one
dimensional coefficient ring.'' 14pp
We calculate the Tate cohomology of
vn-periodic complex oriented theories whose coefficient ring is one dimensional.
For p-groups it is self contained, but in general it relies on ``Augmentation
ideals of equivariant cohomology rings'', above. (updated 25/4/96)
Click here (.dvi) or
here (.ps) for
J.P.C.Greenlees and N.P.Strickland ``Varieties and local cohomology for
chromatic group cohomology rings'' 45pp
Following Quillen we use the methods
of algebraic geometry to study the ring E^*(BG) where E is a suitable complete
periodic complex oriented theory and G is a finite group: we describe its
variety in terms of the formal group associated to E and the category of abelian
p-subgroups of G. This also gives information about the homology of BG (`local
cohomology is trivial over pure chromatic strata') (10/12/96) Revised version
(15/5/98).
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