# RESEARCH

## A quick overview of my research:

Current Research:
I have broad research interests which spread across the fields of Number Theory, Algebra, Modular Representation Theory and Galois Cohomology. I have been working (since 2013) with Keir Lockridge on some open problems in algebra which were posed by Laszlo Fuchs' around 1960. We solved one of his problems for the class of indecomposable abelian groups (paper 22) and dihedral groups (paper 28) In joint work with Keir Lockridge and Gail Yamskulna we investigated some questions on the structure of units in a group algebra and discovered several new characterizations of Mersenne prime and primes for which 2 is a primitive roots. It also led to new results in combinatorial number theory; see paper 23. In joint work with Carlson and Minac we recently solved the strong generating hypothesis for the stable module category (see paper 27) In work with Claudio Quadrelli, Ido Efrat, Jan Minac and Andy Schultz, we have worked on some problems in Galois cohomology and the Absolute Galois groups. In work with Minac and Efrat, we found a remarkably small quotient of the absolute Galois group which determines the entire Galois cohomology. This led to new example of profinite groups that are not realizable as absolute Galois groups of any field. see paper 16. In work with Claudio and Minac, we found a new characterization of p-rigid fields in terms of quotients of the absolute Galois group. For p-rigid fields F, we have shown that it is possible to solve for the roots of any irreducible polynomials in F[x] whose splitting field over F has a p-power degree via non-nested radicals; see paper 21. Some of this research was funded by a grant from the NSA.

Past research
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. In my postdoctoral research, I worked on some fundamental problem in Tate cohomology including Freyd's generating hypothesis for the stable module category and the problem of finite generation of Tate cohomology. This appeared in several papers written with my coauthors Dave Benson, Jon Carlson, Jan Minac and Dan Christensen. See papers 5, 6, 7, 8, 9, and 13 below.

## PUBLICATIONS AND PREPRINTS

All these papers are available from my author page of the arXiv. 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.

30. How many units can a commutative ring have? joint with Lockridge, Amer. Math. Monthly, to appear.

Abstract: Laszlo Fuchs posed the following problem in 1960, which remains open: classify the abelian groups occurring as the group of all units in a commutative ring. In this note, we provide an elementary solution to a simpler, related problem: find all cardinal numbers occurring as the cardinality of the group of all units in a commutative ring. As a by-product, we obtain a solution to Fuchs' problem for the class of finite abelian $p$-groups when $p$ is an odd prime.

29. Witt's cancellation theorem seen as a cancellation, joint with McQuillan and Minac, to appear in Expositiones Mathematicae

Abstract: Happy birthday to the Witt ring! The year 2012 marks the 75th anniversary of Witt's famous paper containing some key results, including the Witt cancellation theorem, which form the foundation for the algebraic theory of quadratic forms. We pay homage to this paper by presenting a transparent, algebraic proof of the Witt cancellation theorem, which itself is based on a cancellation. We also present an overview of some recent spectacular work which is still building on Witt's original creation of the algebraic theory of quadratic forms

28. Fuchs' problem for dihedral groups joint with Lockridge, Journal of Pure and Applied Algebra 221 (2017) 971-982.

Abstract: More than 50 years ago, Laszlo Fuchs asked which abelian groups can be the group of units of a ring. Though progress has been made, the question remains open. One could equally well pose the question for various classes of nonabelian groups. In this paper, we prove that D_2, D_4, D_6, D_8, and D_12 are the only dihedral groups that appear as the group of units of a ring of positive characteristic (or, equivalently, of a finite ring), and D_2 and D_4k, where k is odd, are the only dihedral groups that appear as the group of units of a ring of characteristic 0.

26. Galois p-groups and Galois modules joint with Minac and Schultz, Rocky Mountain Journal of Mathematics 46 (2016), 1405-1446.

Abstract: The smallest non-abelian p-groups play a fundamental role in the theory of Galois p-extensions. We illustrate this by highlighting their role in the definition of the norm residue map in Galois cohomology. We then determine how often these groups --- as well as other closely related, larger p-groups --- occur as Galois groups over given base fields. We show further how the appearance of some Galois groups forces the appearance of other Galois groups in an interesting way.

27. Ghosts and strong ghosts in the stable category joint with Carlson and Minac, Canadian Mathematical Bulletin 59 (2016), 682-692

Abstract: Suppose that G is a finite group and k is a field of characteristic p>0. A ghost map is a map in the stable category of finitely generated kG-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow p-subgroup of G is cyclic of order 2 or 3. In this paper we introduce and study some variations of ghosts maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We find all groups satisfying a strong form of Freyd's generating hypothesis and show that ghost can be detected on a finite range of Tate cohomology. We also consider maps which mimic ghosts in high degrees.

25. When is a subgroup of a ring an idea? joint with Henry, Involve 9 (2016) no. 3, 503-516.

Abstract: Let $R$ be a commutative ring. When is a subgroup of $(R, +)$ an ideal of $R$? We investigate this problem for the rings $\mathbb{Z}^{d}$ and $\prod_{i=1}^{d} \mathbb{Z}_{n_{i}}$. For various subgroups of these rings we obtain necessary and sufficient conditions under which the above question has an affirmative answer. In the case of $\mathbb{Z} \times \mathbb{Z}$ and $\mathbb{Z}_n \times \mathbb{Z}_m$, our results give, for any given subgroup of these rings, a computable criterion for the problem under consideration. We also compute the probability that a randomly chosen subgroup from $\mathbb{Z}_n \times \mathbb{Z}_m$ is an ideal.

24. Fields with indecomposable multiplicative groups joint with Lockridge, Expositiones Mathematicae 34, (2016) no 2. 237-242.

Abstract: The multiplicative group of a finite field is well known to be cyclic; in this note, we determine the finite fields whose multiplicative groups are direct sum indecomposable. We obtain our classification using a direct argument and also as a corollary to Catalan's Conjecture. Turning to infinite fields, we prove that any infinite field whose characteristic is not equal to 2 must have a decomposable multiplicative group. We conjecture that this is also true for infinite fields of characteristic 2.

23. Characterizations of Mersenne and 2-rooted primes joint with Lockridge and Yamskulna, Finite Fields and Their Applications 35 (2015) 330-351.

Abstract: We give several characterizations of Mersenne primes and of primes for which 2 is a primitive root. These characterizations are based on group algebras, circulant matrices, binomial coefficients, and bipartite graphs.

22. Fuchs' problem for indecomposable abelian groups joint with Lockridge, Journal of Algebra 438 (2015) 325-336.

Abstract: More than 50 years ago, Laszlo Fuchs asked which abelian groups can be the group of units of a commutative ring. Though progress has been made, the question remains open. We provide an answer for indecomposable abelian groups by classifying the indecomposable abelian groups that are realizable as the group of units of a ring of any given characteristic. This led to new characterizations of Mersenne primes and Fermat primes.

21. Detecting Fast solvability of equations via small powerful Galois groups joint with Minac and Quadrelli, Transactions of the AMS 367 (2015), no. 12, 8439-8464.

Abstract: Fix an odd prime $p$, and let $F$ be a field containing a primitive $p$th root of unity. It is known that a $p$-rigid field $F$ is characterized by the property that the Galois group $G_F(p)$ of the maximal $p$-extension $F(p)/F$ is a solvable group. We give a new characterization of $p$-rigidity which says that a field $F$ is $p$-rigid precisely when two fundamental canonical quotients of the absolute Galois groups coincide. This condition is further related to analytic $p$-adic groups and to some Galois modules. When $F$ is $p$-rigid, we also show that it is possible to solve for the roots of any irreducible polynomials in $F[X]$ whose splitting field over $F$ has a $p$-power degree via non-nested radicals. We provide new direct proofs for hereditary $p$-rigidity, together with some characterizations for $G_F(p)$ -- including a complete description for such a group and for the action of it on $F(p)$ -- in the case $F$ is $p$-rigid.

20. What is special about the divisors of 12?, with Micahel Mayers Math. Mag. Vol. 86, No. 2 (2013).

Abstract: Let R be the ring of polynomials in finitely many commuting varibles with coefficients in Z_n. It is shown that every unit of R has order 2 if and only of n is a divisor of 12.

19. Representations of the Miraculous Klein group, joint with Minac, Ramanujan Mathematics Society Newsletter 22 (2012) No. 1. 135-145.

Abstract: The Klein group contains only four elements. Nevertheless this little group contains a number of remarkable entry points to current highways of modern representation theory of groups. In this paper, we shall describe all possible ways in which the Klein group can act on vector spaces over a ﬁeld of two elements. These are called representations of the Klein group. This description involves some powerful visual methods of representation theory which builds on the work of generations of mathematicians starting roughly with the work of K. Weiestrass. We also discuss some applications to properties of duality and Heller shifts of the representations of the Klein group.

18. What is special about the divisors of 24?, Math. Mag. to appear.

Abstract: What is an interesting number theoretic or a combinatorial characterisation of the divisors of 24 amongst all positive integers? In this paper I will provide one characterisation in terms of modular multiplication tables. This idea evolved interestingly from a question raised by a student in my elementary number theory class. I will give the characterisation and then provide 5 different proofs using various techniques: Chinese remainder theorem, structure theory of units, Dirichlet's theorem on primes in an arithmetic progression, Bertrand-Chebyshev theorem, and results of Erdos and Ramanujan on the pi(x) function.

17. Freyd's generating hypothesis for group with periodic cohomology, with Dan Christensen and Ján Minác, Canadian Mathematical Bulletin 55 (2012), 48-59

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.

16. Quotients of absolute Galois groups which determine the entire Galois cohomology, with Ido Efrat and Ján Minác, Math. Annalen.(2011) 352 (2012), 205-221.

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.

15. Counting irreducible polynomials over finite fields using the inclusion-exclusion principle, with Ján Minác, Math. Mag 84 (2011), No. 5, 369-371.

Abstract: C. F. Gauss discovered a beautiful formula for the number of irreducible polynomials of a given degree over a finite field. Assuming just a few elementary facts in field theory and the exclusion-inclusion formula, we show how one see the shape of this formula and its proof instantly.

14. On a small quotient of the big absolute Galois group (joint work with Ido Efrat and Jan Minac), Oberwolfach Reports, 32 2010 14-17.

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

13. Finite generation of Tate cohomology, with Jon Carlson and Ján Minác, Representation theory (Journal of the AMS), 15 (2011) 244-257.

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.

12. Reciprocity laws for representations of finite groups, with Ján Minác, and Clive Reis, Annales des sciences math´ematiques du Qu´ebec. 34 (2010), no 1, 37-61.

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.

11. Classifying subcategories of modules over a PID, JP Journal of Algebra, Number theory and its Applications, 2 (2009) 211 - 220.

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).

10. Absolute Galois groups viewed from small quotients and the Bloch-Kato conjecture, with Ján Minác, Geometry and Topology Monographs 16 (2009) 31-47.

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.

9. Freyd's generating hypothesis with almost split sequences, with Jon Carlson and Ján Minác, Proc. Amer. Math. Soc., 137 (2009) 2575-2580.

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.

8. Auslander-Reiten sequences for homotopists and arithmeticians, with Ján Minác, Annales des sciences math´ematiques du Qu´ebec 32 (2008), no 2. 19 pages.

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.

7. Ghosts in modular representation theory, with Dan Christensen and Ján Minác, Advances in Mathematics, 217 (2008), 2782-2799.

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.

6. Groups which do not admit ghosts, with Dan Christensen and Ján Minác, Proc. Amer. Math. Soc., 136 (2008), 1171-1179.

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.

5. The generating hypothesis for the stable module category of a p-group, with Dave Benson, Dan Christensen and Ján Minác, Journal of Algebra 310 (2007) 428-433.

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.

4. Abelian subcategories closed under extensions: K-theory and decompositions, Communications in Algebra 35 (2007) 807-819.

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.

3. Krull-Schmidt decompositions for thick subcategories, Journal of Pure and Applied Algebra 210 (2007) 11-27.

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.

2. Thick subcategories in stable homotopy theory, Oberwolfach Report 8 (2006), 12-20,

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.

1. Refining thick subcategory theorems, Fundamenta Mathematicae 189 (2006), 61-97.

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.

0. Refinements of chromatic towers and Krull-Schmidt decompositions in stable homotopy categories, Ph.D. thesis (2005), University of Washington.

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.

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