sunil k. chebolu illinois state university joint work...

Progress Report on the Generating Hypothesis

Sunil K. CheboluIllinois State University

Joint work with Jon Carlson and Jan Minac

AMS meeting: University of Kentucky, March 27-28, 2010

1/31 Sunil Chebolu Progress Report on the Generating Hypothesis

Outline

I Freyd’s generating hypothesis for spectra

I The generating hypothesis in triangulated categories

I Derived categories of rings

I Stable module categories of group algebras

I Equivariant generating hypothesis


The Philosophy of the Generating Hypothesis

If an object “generates” a category, then it should “detect”non-zero maps in the category.


Freyd’s generating hypothesis

Generating Hypothesis (Peter Freyd - 1966): If φ : X → Y is amap between finite spectra such that π∗(φ) = 0, then φ is nullhomotopic.

Categorical formulation of the GH: The stable homotopy functoron the category of finite spectra is faithful.

π∗ : category of finite spectra −→ π∗(S0)−modules

X 7→ π∗(X )

is faithful


Yet another formulation of the GH

The generating hypothesis is equivalent to the following statement.

Generating Hypothesis: For any finite torsion spectrum T , π∗(T )is an injective π∗(S

0)–module.

Note that this formulation of the GH does not involve mapsbetween arbitrary finite spectra. It uses only modules over π∗(S

0).

The generating hypothesis is a very deep statement about finitespectra. It is one of the biggest open problems in stable homotopytheory.


Counterexamples to variations of the GH

The GH is false unstably: Consider the natural quotient map fromthe torus to the sphere:

P : S1 × S1 → S1 ∧ S1.

π∗(P) = 0 but P is not null homotopic.

The GH is false for infinite-dimensional spectra: Consider theelement Sq1 in the mod-2 Steenrod algebra.

Sq1 : ΣHF2 → HF2

π∗(Sq1) = 0 but Sq1 is non-zero.


Question: Why are we interested in the generating hypothesis?

Answer: Because the generating hypothesis, if true, leads to someconsequences which are absolutely amazing.

So what seems to be more interesting are the consequences of theGH than the statement of the GH itself!


An unbelievable consequence of the GH

The GH implies that the functor π∗(−) is full.

That is, if X and Y are finite spectra, then any π∗(S0)-module

map between π∗(X ) and π∗(Y ) is induced by a topological mapX → Y .

Therefore, the GH, if true, reduces the study of finite spectra X tothe study of their homotopy groups π∗(X ) as modules over π∗(S

0)!


A believable consequence of the GH

Theorem (Whitehead) The GH implies that there is no finitespectrum X other than a wedge of suspensions of the spherespectrum whose homotopy groups π∗(X ) is finitely generated overπ∗(S

0).

Proof: Suppose π∗(X ) is a finitely generated π∗(S0)-module.

∨i S

ni f // Xg // Y

π∗(f ) is onto, this implies π∗(g) = 0. The GH implies that g = 0.

Therefore X is a retract of a wedge of spheres, and hence X itselfis a wedge of spheres.


Why is the above consequence believable?

It is believable in view of the following theorems.

Theorem (Hovey - 2006) Suppose Y is a finite ring spectrum oftype n > 0. Then π∗(Y ) is not a finitely generated π∗(S

0)-module.

Theorem (Kahn 1969) For any finite spectrum Y , it is possible toattach two cells (one if Y is not torsion) to Y to get a newcomplex Z such that π∗(Z ) is not finitely generated.

Moral: Finite generation of homotopy groups is a very rarephenomenon.


Another consequence of the GH

The GH implies that the ring π∗(S0) is a very bad ring.

It is known that π∗(S0) is not noetherian, and it is not even

coherent. The GH implies that this ring is even worse.

The GH implies that π∗(S0) is totally non-coherent. That is, no

finitely generated proper non-zero ideal is finitely presented.


What is known about the GH?

Answer: Very little

The following theorem of Ethan Devinatz is the ONLY affirmativeresult in favour of the GH.

Theorem (Devinatz 97) The GH with target S0: Let φ : X → S0

be a map such that π∗(φ) = 0, then L1(φ) = 0 for p > 2.

Proof is quite involved. It uses among other things, the machineryof nilpotence and periodicity.

Devinatz and Hopkins have a program to prove the GH with targetS0 using the telescope conjecture and the chromatic splittingconjecture.

However, there hasn’t been any progress on this program.


What do we do next?

To gain some insight into the GH we study the analogues of theGH in triangulated categories which are formally similar to thestable homotopy category of spectra.

Examples: Derived categories of rings, Localisation of stablehomotopy category, Equivariant stable categories, modulecategories over the Steenrod algebra, stable module categories ofgroup algebras, etc.

This line of research is inspired by the work on Axiomatic StableHomotopy Theory, AMS memoirs, by Hovey, Palmieri, andStrickland.


The GH in a triangulated category

T = Triangulated category with a distinguished object SThick(S) = Thick subcategory generated by S .

The thick subcategory generated by S is the smallest triangulatedsubcategory of T that contains S and is closed under retractions.

Generating Hypothesis in T : The functor

π∗(−) : Thick(S)→ π∗(S)−modules

X 7→ HomT (Σ∗S ,X )

is faithful.


The derived category of a ring

R be a commutative ring.

Ch(R) – category of unbounded chain complexes of R-modules.

D(R) – the derived category of R obtained from Ch(R) byinverting quasi-isomorphisms.

D(R) is a triangulated category with R as a distinguished object.

Thick(R) – the category of perfect complexes

π∗(−) = H∗(−) is the homology functor.


The GH in D(R)

When does the GH holds in the D(R)? That is, for which rings Ris the homology functor on the category of perfect complexesfaithful?

Theorem (Lockridge 2007) The GH holds in D(R) if and only if Ris a von Neumann regular ring.

A ring R is von Neumann regular if for all r in R the equationr = rxr has a solution in R.

Homological characterisation: These are rings R with the propertythat every R-module is flat.

Examples: Direct product of fields and Mn(R).


A closer look

It turns out that the GH holds in D(R), for R commutative,precisely when the category of perfect complexes is trivial in thesense that every perfect complex is a direct sum of suspensions ofR.

If the category of perfect complexes is indeed build from R in thismanner, then it is obvious that the GH holds. However, theconverse is not obvious.

Therefore, commutative von Neumann regular rings are preciselythose rings for which the category of perfect complexes is trivial inthe above sense.


The GH for rings not necessarily commutative is more involved andhas been studied by Hovey, Lockridge and Puninski.

Theorem (Hovey-Lockridge-Puninski 2007) The GH holds forD(R) if and only if R satisfies the following two conditions:

1. submodules of flat R-modules are flat

2. Short exact sequences of finitely presented R-modules split.

These 3 authors have produced examples of non-commutativerings for which

1. H∗(−) faithful does not imply H∗(−) full.

2. H∗(−) faithful does not imply that the category of perfectcomplexes is a collection of wedges of suspensions of R.


Theorem (Lockridge 2007). Let T be any triangulated categorywith a distinguished object S for which π∗(S) is gradedcommutative and concentrated in even degrees. Then thefollowing are equivalent.

1. The GH holds in thick(S).

2. π∗(S) is a graded von Neumann regular ring.

3. Thick(S) is the collection of wedges of suspensions of S .

Note: This theorem does not apply for the stable homotopycategory of spectra because π∗(S

0) is not concentrated in evendegrees.

Therefore, to better understand GH for spectra, we have toconsider a triangulated category with unit object S whosehomotopy groups are non-zero in both even and odd degrees.


The stable module category

G finite groupk field of characteristic p > 0 (p | |G |).

f , g : M → N between finite dimensional kG -modules arehomotopic if their difference (f − g) factors through a projective.f : M → N is null-homotopic if it factors through a projective.

The stable module category – stmod(kG )

I Objects: finite-dimensional kG -modules.

I Morphisms: homotopy classes of kG -linear maps.


Tate Cohomology

stmod(kG ) is a triangulated category:

1. Ω(M) := ker(PM M)

2. exact triangles are given by the short exact sequences ofkG -modules.

3. The trivial representation k is the distinguished object

Fact: Tate cohomology is well encoded in this category.

Hi(G ,M) ∼= HomkG (Ωik ,M) for all i .


John Tate wins 2010 Abel prize


The GH in the stable module category

Question: For which finite groups G is the Tate cohomologyfunctor H

∗(G ,−) faithful on the thick subcategory generated by k?

In other words, when does the GH hold in ThickG (k)?

Main Theorem: [Carlson-C.-Minac (2009)]The generating hypothesis holds for the stable module category ofkG if and only if the Sylow p-subgroup of G is either C2 or C3.

This builds on some previous joint work with Dave Benson andDan Christensen.

Proof uses Block theory, structure of the principal block for cyclicSylow p-subgroups, Auslander-Reiten sequences, varieties ofmodules, some general constructions which make use of thetriangulated structure of the stable module category.


Why only C2 and C3?

Question: Which representation theoretic property distinguishesfinite groups whose Sylow p-subgroup is either C2 or C3?

Answer: These are the only groups with the property that everykG -module in the thick subcategory generated by k is a direct sumof suspensions of k .

Note that the situation here is similar to that of the derivedcategory of a commutative ring.


Faithfull implies Full

Corollary Let G be a finite group. On the category Thick(k), if

H∗(G ,−) is faithful, then it is also full.

Proof By our main theorem, if H∗(G ,−) is faithful, then the Sylow

p-subgroup is C2 or C3.

As noted above, in these cases, every finitely generated kG -modulein Thick(k) is a direct sum of suspension of k .

The Tate cohomology of such modules is clearly free, and thereforethe corollary follows.


Sketch proof of the main theorem

Assume G is a p-group.

Note that for a p-group stmod(kG ) = Thick(k).

1. It is an exercise to show that for a cyclic group G , the GHholds if and only if G is C2 or C3

2. If H is a subgroup of G and if φ : M → N is a counterexampleto the GH in stmod(kH), then the inductionφ↑G : M↑G → N↑G is a counterexample to the GH instmod(kG ).

3. The GH fails when G = Cp ⊕ Cp.

4. If G 6= C2 or C3, then G contains either a copy of Cp ⊕ Cp orCpr (pr > 3).

When G is not a p-group, the above strategy fails. So we use adifferent technique to produce counterexamples to the GH.


Auslander-Reiten sequences

A short exact sequence of finitely generated kG -modules

ε : 0 −→ Mα−→ E

β−→ N −→ 0

is an Auslander-Reiten (AR) sequence if:

I ε is a non-split sequence.

I α is left almost split,

I β is right almost split

N ′

ψ

~~0 // M

α //

φ

Eβ //

~~

N // 0

M ′


Theorem (Auslander-Reiten-Smalø) Given any indecomposablenon-projective finitely generated kG -module N, there exists aunique Auslander-Reiten sequence ending in N. Moreover, the firstterm of the AR-sequence is isomorphic to Ω2N.

0→ Ω2N → E → N → 0.

This short exact sequences gives a triangle in the stable modulecategory

Ω2N → E → Nf−→ ΩN

f : N → ΩN is a counter-example to the GH in stmod(kG )provided N is module in thick(k) and is not isomorphic to Ωnk forany n.


New invariants for group algebras

As we see from our main theorem, the GH is seldom true.

We therefore introduce an integer which measures the degree offailure of the GH for a given group algebra kG .

A ghost is a map between kG -modules which induces the zero mapon Tate cohomology.

The ghost number of kG is the smallest positive integer t suchthat the composition of any t ghosts in Thick(k) is zero in thestable category.

Main Theorem: The ghost number of kG is one if and only if theSylow p-subgroup of G is C2 or C3.

This definition opens the door to a whole new project which will bea subject of another talk.


Rational S1- Equivariant Generating Hypothesis.

Very recently Anna Marie Bohmann has worked on the generatinghypothesis in the equivariant world.

Theorem (Bohmann - 2009) The generating hypothesis fails in thecategory of rational S1-equivarant spectra.

In other words, there exist finite rational S1-spectra X and Y anda map f : X → Y such that

I f∗ : πH∗ (X )→ πH

∗ (Y ) is zero for all closed subgroups H ⊂ S1

I f is not stably nullhomotopic.

A key ingredient in her proof is Greenlees’s description of thecategory of S1-equivariant spectra.


Thank You


sunil k. chebolu illinois state university joint work...

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