My research interests lie in the fields of Algebraic topology, Representation theory, and Arthimetic.
Specifically, I am intersted in interactions between these fields which
are inspired by cohomology theories and triangulated
structures, among others.
For example, I like to think about homotopy theoretic questions in algebraic subjects like the
derived categories of homological algebra and the stable module categories of modular representation theory.
In my thesis (at the University of Washington under the supervision
of John Palmieri) I have studied the refinements of chromatic towers
and Krull-Schmidt type decompositions in various stable homotopy categories including the derived categories of rings,
the stable homotopy category of spectra and the stable modules categories of group schemes; see papers [1,2,3,4] below.
More recently, in several joint projects with my collaborators Benson,
Carlson,
Minac,
Swallow,
and Ido Efrat, I have been working on several problems in
Galois cohomology, modular representation theory (which are inspired
from stable homotopy theory), and related topics which are explained below.
Brief description of my main current research projects:
Freyd's generating hypothesis: This is joint with Minac
and Carlson. A long standing
conjecture of Peter Freyd called the Generating Hypothesis in stable homotopy theory claims that a map between
finite spectra that is zero on stable homotopy groups is null-homotopic. We have formulated and solved the
Generating Hypothesis in the stable module category of a finite group;
see paper 11 below. Now we are working on variations of Freyd's generating hypothesis, some of which seem to yield
more positive results.
Refining the Bloch Kato conjecture: This is joint with
Benson, Minac, and Swallow. TheBloch-Kato
conjecture gives a description of the cohomology of
the absolute Galois group of field via generators and
relations. This has been proved recently by Voevodsky and Rost.
We are working on refinements of this conjecture.
Finite generation of Tate cohomology: This is joint with
Minac and Carlson. We are working on a very fundamental problem
about Tate cohomology; namely, when is Tate cohomology finitely
generated? We made some progress on this question; see paper 14 below.
In our current work which is in progress we hope to give a complete answer to
this question.
Detecting absolute Galois groups: This is joint with Minac
and Efrat. One of the burning problems about absolute Galois groups
is their detection problem. More precisely, given a profinite group G, when is it an absolute
Galois group? or when is it not an absolute Galois group? We are
expoliting the Bloch-Kato conjecture to rule out many pro-p-groups
which cannot be absolute Galois groups.
You can read the
reviews of my papers at the
Mathscinet and also at the Zentralblatt Math.
I will be happy to receive comments/feedback on my papers, and if you are interested in having reprints of the
published papers, please email me.
PUBLICATIONS AND PREPRINTS
All papers below are in PDF format. If you wish to download them in other forms, please visit the Front for the
arXiv
Abstract:
Much has been written on reciprocity laws in number theory and their
connections with group representations. In this paper we explore more on these
connections. We prove a reciprocity Law for certain specific
representations of semidirect products of two cyclic groups which is in
complete analogy with classical reciprocity laws in number theory. In fact,
we show that the celebrated quadratic reciprocity law is a direct consequence
of our main theorem applied to a specific group. As another consequence of
our main theorem we also recover a classical theorem of Sylvester. Our main
focus is on explicit constructions of representations over sufficiently small
fields. These investigations give further evidence that there is still much
unexplored territory in connections between number theory and group
representations, even at an elementary level.
Abstract:
Let G be a finite group and let k be a field of characteristic p. Let M be a finitely generated indecomposable non-projective kG-module. We conjecture that if the Tate cohomology H^*(G, M) of G with coefficients in M is finitely generated over the Tate cohomology ring H^*(G, k) then the support variety V_G(M) of M is equal to the entire maximal ideal spectrum V_G(k). We prove various results all of which support this conjecture. Modules in the connected component of k in the stable Auslander-Reiten quiver for kG are shown to have finitely generated Tate cohomology. It is also shown that all finitely generated kG-modules over a group G have finitely generated Tate cohomology if and only if G has periodic cohomology.
Abstract:For prime power q=p^d and a field F containing a root of unity of order q we show that the Galois cohomology ring H^*(G_F_q) is determined by a quotient G_F^{[3]} of the absolute Galois group G_F related to its descending q-central sequence. Conversely, we show that G_F^{[3]} is determined by the lower cohomology of G_F. This is used to give new examples of pro-p groups which do not occur as absolute Galois groups of fields.
Abstract:
Let G be a finite group and k be a field whose characteristic p divides the order of G.
Freyd's generating hypothesis for the stable module category of a
non-trivial finite group G is the statement that a map between finite-dimensional
kG-modules in the thick subcategory generated by k factors through a
projective if the induced map on Tate cohomology is trivial. We show that if G
has periodic cohomology then the generating hypothesis holds if and only if the Sylow
p-subgroup of G is C_2 or C_3. We also give some other conditions that are equivalent to the GH
for groups with periodic cohomology.
Abstract:
We introduce Auslander-Reiten sequences for group algebras and give several
recent applications. The first part of the paper is devoted to some fundamental
problems in Tate cohomology which are motivated by homotopy theory. In the
second part of the paper we interpret Auslander-Reiten sequences in the context
of Galois theory and connect them to some important arithmetic objects.
Abstract:
Let R be a commutative ring. A full additive subcategory C of R-modules is
triangulated if whenever two terms of a short exact
sequence belong to C, then so does the third term. In this note
we give a classification of triangulated subcategories of finitely
generated modules over a principal ideal domain. As a corollary we
show that in the category of finitely generated modules over a PID,
thick subcategories (triangulated subcategories closed under direct
summands), wide subcategories (abelian subcategories closed under
extensions) and Serre subcategories (wide subcategories closed under
kernels) coincide and correspond to specialisation closed subsets of
Spec(R).
Abstract:
In this paper we concentrate on the relations between the structure of small Galois groups, arithmetic of fields, Bloch-Kato conjecture, and Galois groups of maximal pro-$p$-quotients of absolute Galois groups.
Abstract:
Freyd's generating hypothesis for the stable module category of a non-trivial finite group G is the statement that a map between finitely generated kG-modules that belongs to the thick subcategory generated by k factors through a projective if the induced map on Tate cohomology is trivial. In this paper we show that Freyd's generating hypothesis fails for kG when the Sylow p-subgroup of G has order at least 4 using almost split sequences. By combining this with our earlier work, we obtain a complete answer to Freyd's generating hypothesis for the stable module category of a finite group. We also derive some consequences of the generating hypothesis.
Abstract:
A ghost over a finite group Gis a map between modular
representations of G which is invisible in Tate cohomology. Motivated by the
failure of the generating hypothesis ---the statement that ghosts
between finite-dimensional G-representations factor through a
projective---we define the compact ghost number of kG to be the smallest integer h
such that the composition of any h ghosts between finite-dimensional
G-representations factors through a projective. In this paper we study ghosts
and the compact ghost numbers of p-groups. We begin by showing that a weaker version
of the generating hypothesis, where the target of the ghost is fixed to be the
trivial representation k, holds for all p-groups.
We do this by proving that a map between finite-dimensional
G-representations is a ghost if and only if it is a dual ghost.
We then compute the compact ghost
numbers of all cyclic p-groups and all abelian 2-groups with C_2 as a
summand. We obtain bounds on the compact ghost numbers for abelian p-groups and for
all 2-groups which have a cyclic subgroup of index 2. Using these bounds
we determine the finite abelian groups which have compact ghost number at most 2.
Our methods involve techniques from group theory, representation theory,
triangulated category theory, and constructions motivated from homotopy
theory.
Abstract:
A ghost in the stable module category of a group G is a map between
representations of G that is invisible to Tate cohomology. We show that the
only non-trivial finite p-groups whose stable module categories have no non-trivial
ghosts are cyclic groups of order 2 and 3.
We compare this to the situation in the derived category of a
commutative ring. We also determine for which groups G the second power of the Jacobson radical is stably isomorphic to a suspension of k.
Abstract:
Freyd's generating hypothesis for a finite p-group G is the statement that
a map between finite-dimensional kG-modules factors through a projective if
the induced map on Tate cohomology is trivial. We show that Freyd's generating
hypothesis holds for a non-trivial p-group G if and only if G is a cyclic group of order 2 or 3.
We also give various equivalent characterisations of the generating hypothesis.
Abstract: A full subcategory of modules over a commutative ring R
is wide if it is abelian and closed under extensions. Hovey gave a
classification of the wide subcategories of finitely presented modules over regular coherent rings in terms of
certain specialisation closed subsets of Spec(R) . We use this classification theorem to study K-theory and
Krull-Schmidt type decompositions for wide subcategories. It is shown that the K-group, in the sense of Grothendieck, of a wide
subcategory W of finitely presented modules over a regular coherent ring is isomorphic to that of the thick subcategory of
perfect complexes whose homology groups belong to W. We also show that the wide subcategories of finitely generated modules
over a noetherian regular ring can be decomposed uniquely into indecomposable ones. This result is then applied to obtain
a decomposition for the K-groups of wide subcategories.
Abstract:
Following Krause, we prove Krull-Schmidt decompositions for thick subcategories of various triangulated categories: derived categories, noetherian
stable homotopy categories, stable module categories over Hopf algebras, and the stable homotopy category of spectra. In all these categories, it
is shown that the thick ideals of small objects decompose uniquely into indecomposable thick ideals. Some consequences of these decompositions are
also discussed. In particular, it is shown that all these decompositions respect K-theory.
Abstract:
This a very "long abstract" of three lectures given at a Mini-Workshop at Oberwolfach on Thick Subcategories: Classifications and Application (Feb 19 - 25, 2006).
In these lectures we give an exposition of the seminal work of Devinatz, Hopkins and Smith which is
surrounding the classification of the thick subcategories of finite
spectra in stable homotopy theory. The lectures are expository and are aimed primarily at
non-homotopy theorists. We begin with an introduction to the stable
homotopy category of spectra, and then talk about the celebrated
thick subcategory theorem and discuss a few applications to the
structure of the Bousfield lattice. Most of the results that we
discuss here were conjectured by Ravenel and were proved
by Devinatz, Hopkins, and Smith.
Abstract:
We use a K-theory recipe of Thomason to obtain classifications of triangulated subcategories via
refining some standard thick subcategory theorems. We apply this recipe to the full subcategories of finite objects in the
derived categories of rings and the stable homotopy category of spectra. This gives, in the derived categories, a complete classification
of the triangulated subcategories of perfect complexes over some noetherian rings. In the homotopy category of spectra we obtain only a partial classification of
the triangulated subcategories of the finite p-local spectra. We use this partial classification to study the lattice of triangulated subcategories.
This study gives some new evidence to a conjecture of Adams that the thick subcategory C_2 can be generated by iterated cofiberings of the Smith-Toda complex
We also discuss several consequences of these classifications theorems.
Abstract:
We study the triangulated subcategories of compact objects in stable homotopy categories such as the homotopy category of spectra, the derived categories
of rings, and the stable module categories of Hopf algebras. In the first part of this thesis we use a K-theory recipe of Thomason to classify these
subcategories. This recipe when applied to the category of finite p-local spectra gives a refinement of the ``chromatic tower''. This refinement has some interesting
consequences. In particular, it gives new evidence to a conjecture of Frank Adams that the thick subcategory C_2 can be generated
by iterated cofiberings of the Smith-Toda complex V(1).
Similarly by applying this K-theory recipe to derived categories, we obtain a complete classification of the triangulated subcategories
of perfect complexes over some noetherian rings. Motivated by these classifications, in the second part of the thesis, we study Krull-Schmidt decompositions for thick
subcategories. More precisely, we show that the thick subcategories of compact objects in the aforementioned stable homotopy categories decompose uniquely into
indecomposable thick subcategories. Some
consequences of these decompositions are also discussed. In particular, it is shown that all these decompositions respect K-theory. Finally in the last chapter we mimic
some of these ideas in the category of R-modules. Here we consider abelian subcategories of R-modules that are closed under extensions and study their K-theory and
decompositions.
Sunil Kumar Chebolu
Department of Mathematics
Illinois State Univeristy
Normal, IL 61790 USA.
