Guozhen WANG

Abstract The author introduces the notion of a minimal resolution for BP*BP-comodules,and gives an effective algorithm to produce minimal resolutions.This produces the data needed in the work[3]for studying motivic stable stems up to stem 90.

Keywords Stable homotopy,Adams-Novikov spectral sequence,Hopf algebroid

1 Introduction

The Adams-Novikov spectral sequence,constructed in[5],is an important tool for the study of stable homotpy groups.See[6]for a detailed treatment of its construction and computational techniques.One of the difficulty for this method is that the computations of the E2term is very hard.This paper gives an efficient algorithm for stemwise computations of the Adams-Novikov E2term using computers.Together with techniques introduced in[2],this has made breakthroughs in the stemwise computations of classical and motivic stable homotopy groups(see[3]and forthcoming works).

Usually people compute the Adams-Novikov E2term by the following three methods:The algebraic Novikov spectral sequence,the Bockstein spectral sequence,and the chromatic spectral sequence.The chromatic spectral sequence,introduced in[4],separates informations of different heights(or periods),which are computed individually.This gives many global structure theorems for the Adams-Novikov spectral sequence(see[6]for details).However,for stem-wise computations,especially for small primes,it is hard to get complete informations using the chromatic methods alone.For example,to get to the stem 126 at prime 2,informations up to chromatic level 6 are involved,which are beyond our current knowledge.

The Bockstein spectral sequence is often used for stem-wise computations.For example,this method is illustrated in[6]for the computation of the first 25 stems.The method used in[6]to the computation the differentials in the Bockstein spectral sequence is to do the computation in the cobar complex.As the cobar complex grows very fast,direct computations in the cobar complex become impossible very quickly,even for computers.

The algebraic Novikov spectral sequence is also often used in stem-wise computations.This spectral sequence is very important from a theoretic as well as a practical point of view.It is proved in[2]that the algebraic Novikov spectral sequence for the sphere spectrum is isomorphic to the motivic Adams spectral sequence for the cofiber of tau,giving us a very powerful method for stemwise computations provided we know the structure of the algebraic Novikov spectral sequence.As in the case of the Bockstein spectral sequence,the method of computing the algebraic Novikov differentials using the cobar complex becomes too complicated very quickly.

The main technique of this paper is the observation that,the Bockstein and the algebraic Novikov filtrations are defined on any resolution of BP*BP-comodules.So we can replace the cobar complex with any cofree resolution.In particular we introduce the notion of a minimal resolution which is smallest among cofree resolutions.

Recall that the maximal ideal I of BP*is an invariant ideal,with the quotient BP*BP/I isomorphic to the even part of the dual Steenrod algebra.We define a cofree resolution to be minimal,if its modulo I reduction is a minimal resolution as BP*BP/I-comodules.Then we use the Bockstein and the algebraic Novikov filtrations on the minimal resolution to compute the Bockstein and the algebraic Novikov spectral sequences respectively.

To construct the minimal resolution,we first construct a minimal resolution for the modulo I reduction.Then an arbitrary lift of this resolution almost gives us what we want,except that the compositions of consecutive maps are only zero modulo I.Then we do adjustments using some kind of Gaussian elimination.One such algorithm is given in Section 4.The naive method can be optimized by observing that,for maps between cofree comodules,we only need to know the projection to the cogenerators.This reduces the sizes of the matrices to be computed by one order.This optimization is given in Section 5.

Implementations

The algorithm is implemented with C++code(using C++11 standard),using the library GMP(GNU Multiple Precision Arithmetic Library)to deal with large integers,and the library OpenMP(Open Multi-Processing)to deal with parallelism.The source code is available at:https://github.com/pouiyter/MinimalResolution.

The program has been performed on the MiG(minimum intrusion grid)at University of Copenhagen,the Wayne State University Grid high performance computing cluster,and the high performance computer at Shanghai Center for Mathematical Sciences.The(current)outputs are available at:https://github.com/pouiyter/morestablestems/raw/master/algNovikovmachine.csv.

2 Preliminaries

Let BP be the Brown-Peterson spectrum.It is a complex oriented ring spectrum whose associated formal group law is the universal p-typical formal group law over Z(p).We have

Define

The pair(BP*,BP*BP)forms a Hopf algebroid which represents the moduli stack of formal groups over Z(p).See[6]for a detailed treatment of Brown-Peterson theory and the theory of Hopf algebroids.

Reduction modulo I gives the Hopf algebra P,which is a sub-Hopf algebra of the dual Steenrod algebra.For p=2,P is isomorphic to the dual Steenrod algebra with degrees doubled.

A BP*BP-comodule F is cofree if F is a direct product of copies of BP*BP(and its degree shifts).

For a BP*BP-comodule M,let Prim(M)denote the primitive elements of M.

We carry out all constructions in the graded sense.In particular,a direct product of objects which are finite in each degree is also a direct sum.

Definition 2.1A graded Z(p)-module is locally finite if it is bounded below and finitely generated in each degree.

For example,BP*,BP*BP and P are all locally finite.In this paper,we implicitly assume that all modules are locally finite.

3 Minimal Resolutions of BP*BP-Comodules

In this section we introduce the notion of a minimal resolution for BP*BP-comodules,which are lifts of minimal resolutions of P-comodules.

Let M be a BP*BP-comodule which is free as a BP*-module.Then M/I becomes a Pcomodule.

Proposition 3.1Let M and N be BP*BP-comodules which are locally finite and free as BP*-modules.Then a comodule map

f:M→N

is strongly injective(resp.,strongly surjective)if and only if the reduction

modulo I is injective(resp.,surjective).

Recall that the data of a long exact sequence 0→M→F0→F1→···is equivalent to the data of a collection

of short exact sequences.

Definition 3.2A long exact sequence 0→M→F0→F1→···of BP*BP-comodules is a cofree resolution of M if each Fiis cofree,and each short exact sequence 0→Mi→Fi→Mi+1→0 is strongly exact.

Recall that we have the following notion of minimal resolutions of P-comodules.

Definition 3.3A cofree resolution

is bijective.(By convention,M0=M.)

Remark 3.2Since Prim is a left exact functor from P-comodules to Fp-modules,it follows that the induced map

is trivial.

For BP*BP-comodules,we define minimal resolutions in terms of reductions modulo I.

Definition 3.4Let M be a BP*BP-comodule which is free over BP*.A cofree resolution of M is called a minimal resolution if its reduction modulo I is a minimal resolution of M/I.

One can see that a minimal resolution has the smallest size among all cofree resolutions.

4 Construction of Minimal Resolutions

We will construct minimal resolutions of BP*BP-comodules by lifting minimal resolutions of reductions modulo I.

First we introduce the notion of cogenerators dual to the notion of generators.Let M be a locally finite BP*BP-comodule which is free as BP*-module,and let X be a locally finite free BP*-module.Let f:M→X be a BP*-module map.Recall that the adjoint map of f is the composite

We say that f exhibits X as cogenerators of M if the adjoint map of f is strongly injective.We will also abbreviate to say that X is the cogenerators of M.

In this case,if g:N→M is a comodule map,then its corestriction to the cogenerators is defined to be the composite mapOne finds that the adjoint of the above map factors through g,and we have a commutative diagram

It follows that g is determined by its corestriction to the cogenerators of M.

Proposition 4.1A map f:M→X exhibits X as cogenerators of M if and only if its reduction

modulo I exhibits X/I as cogenerators of M/I as a P-comodule.

ProofThe adjoint map

Now we construct minimal resolutions as follows.Suppose that M is a locally finite BP*BPcomodule with free underlying BP*-module.We will construct strongly exact sequences

such that each Fiis cofree.

Set M0=M.Suppose that we have constructed a locally finite BP*BP-comodule Mnwhich is free as a BP*-module.We do the following to construct Mn+1:

3.Take Fnto be BP*BP⊗BP*Xn,and Mn+1to be the cokernel of the adjoint map gn:Mn→Fnof fn,so we have the quotient map hn:Fn→Mn+1.

In this way,we construct a minimal resolution inductively.

Remark 4.1The first step is the standard one for computing Adams E2terms using minimal resolutions.In the second step,in addition to constructing the map fn,we also need to do a Gaussian elimination for its adjoint map gn,in order to find the quotient matrix needed in Step 3.And in Step 3,we need to compose the quotient map with the coaction map on Fnto get the coaction map on Mn+1.

Remark 4.2The formulas for coactions of cofree comodules can be computed beforehand,e.g.save a comultiplication table for BP*BP in the disk.

5 Optimization of the Process

The complexity of computing a minimal resolution of M/I has smaller order than computing a minimal resolution of M.So the process can be optimized by computing a minimal resolution of M/I first,and then using it as a model for a resolution of M.

Once we know the structure of the minimal resolution of M/I,the structures of the cofree comodules Fiare already known.The problem with an arbitrary lift of the minimal resolution of M/I is that the compositions of consecutive maps are not guaranteed to be zero.

Once we know the structure of Fn,we already know a set of cogenerators for it.Moreover,these also make a set of cogenerators for Mn.So the data for gnand hn(in Step 3 of Section 4)are determined by their corestrictions to cogenerators.This reduces the order of the size of the matrices for the data of fnand gn.So the optimized process is as follows:

1.This is the same as Step 1 of Section 4,but it is computed beforehand.

2.This is the same as Step 2 of Section 4,but it is computed beforehand.

3.Compute the matrix for the composite map

instead of the full formula for the coaction of Mn.We do this by composing the coaction of Fnwith the composite

Remark 5.1In Step 3,instead of solving the linear equations hn◦gn=0,we solve the equationsgn=0,which only involve the cogenerators.This reduces the complexity of Step 3 by one order.

Similarly,in Step 4,only the coactions of cogenerators are computed.This reduces the complexity of Step 4 by one order.

6 Computations of Homology

With the minimal resolution constructed,our next step is to compute the homology of the chain complex of its primitives.We will modify the algorithm of[1].

Suppose X0→X1→···is a complex of locally finite Z(p)-modules.

Suppose each Xiis given a maximal filtration.This means that,there is an ordinal α0such that for each ordinal number α<α0,there is a submodule Fil≥αXiof Xi,decreasing with respect to α.Moreover,the maximality means that,Fil≥αXi/Fil≥α+1Xi～=Fp,and when α is a limit ordinal,

For each i,we take a set Aiconsisting of an Fpbasis for the graded pieces of Xi.Then each Aihas a canonical order.For any x∈Xi,we say a∈Aiis the leading term of x,if the projection of x to the graded pieces of Xiis a nontrivial multiple of a.

Remark 6.1Here we allow the generality that the filtration is indexed by any ordinal number.In actual computations we will use appropriate truncations to make the filtration finite.

Remark 6.2We do not require that the differentials in the complex respect the filtrations.

A Curtis table is a list with entries of the form:

•a,where a∈Aifor some i,or

•a→b,where a∈Aiand b∈Ai+1for some i;

such that

1.An entry a is in the table if and only if a is the leading term of a cycle,and no boundary has leading term a.

2.An entry a→b is in the table if and only if

(a)there is an element x∈Xiwith leading term a,and d(x)has leading term b,and

(b)for any element x′such that d(x′)has leading term b,the leading term of x′is at least a(i.e.,x′has filtration at most that of a).

The following proposition is proved in[7].

Proposition 6.1Suppose that X0→X1→···is a complex of locally finite Z(p)-modules such that each Xiis maximally filtered,and let Aibe an Fpbasis for the graded pieces of Xi.Then a Curtis table exists and is unique.And in addition,all the elements of Aiappears in the table exactly once.

Now we suppose that there is an additional filtration Fion each Xiwhich is preserved by the differentials.Moreover,we suppose that the maximal filtration associated to Aiis a refinement of Fi.Then the Curtis table describes the structure of the spectral sequence defined by Fi(see[7]for details).

Proposition 6.2The entries of the form a in the Curtis table correspond bijectively to the surviving permanent cycles in the spectral sequence defined by the filtration Fi.

The entries of the form a→b correspond bijectively to the differentials in the spectral sequence,where the length of the differential is the difference of the F-filtration degrees of a and b.(We include all the differentials from d0.)

7 The Algebraic Adams-Novikov Spectral Sequence

Let

be a cofree resolution of a BP*BP-comodule M,which is free over BP*.

Then Ext(M)is computed by the complex

We order Aiby the following rules.We first order them by using the Adams-Novikov filtration.Then,amongst elements with the same Adams-Novikov filtration,we use a lexicographic order.

By Proposition 6.2,we have the following propositon.

Proposition 7.1The Curtis table associated to the above Aigives the structure of the algebraic Adams-Novikov spectral sequence for M.

Remark 7.1We can also introduce an order,by taking the lexicographic order directly.This gives results in the Bockstein spectral sequence.

8 The Atiyah-Hirzebruch Spectral Sequence and Multiplicative Structure

We fix a cofree resolution

for BP*.

Let M be a BP*BP comodule.Recall that the tensor product of two BP*BP-comodules(over BP*)has the structure of a BP*BP-comodule(see[6,A1.1.2]).BP*is the unit for this tensor product,i.e.,

Since Fiis cofree,M⊗Fiis relatively injective,and

So Ext(M)can be computed by the complex M⊗BP*Prim(Fi).

Remark 8.1The formula for the above isomorphism involves the coaction on M,because we used the diagonal coaction.

Now suppose that M is filtered as a BP*BP-comodule.Then there is an associated algebraic Atiyah-Hirzebruch spectral sequence computing Ext(M).We have the following standard fact for Atiyah-Hirzebruch differentials.

Proposition 8.1Let M have a two-step filtration as shown in the short exact sequence

Let h∈Extj,1(BP*)be the element corresponding to this extension.Then the Atiyah-Hirzebruch differentials for M corresponds to multiplication by h.

So we can compute multiplication by elements in homological degree 1 by computing the Atiyah-Hirzebruch differentials.To do this,we do the following.

Suppose in general that M is a filtered BP*BP-comodule.We select a set K of BP*-generators for M,with order refining the filtration on M.Then an element in

is a linear combination of expressions of the form

with k∈K and a∈Gi.Here Giis a set of BP*-generators for Prim(Fi),as in Section 7.

We order these elements as follows.We first consider the filtration of k,then the Adams-Novikov filtration ofand finally the lexicographic order.By Proposition 6.2,the Curtis table for this order gives the structure of the Atiyah-Hirzebruch spectral sequence for M.

Remark 8.2More generally,if the comodule M has a three step filtration,then the Atiyah-Hirzebruch spectral sequence gives data for the corresponding Massy products.The same generalizes to higher Massy products.One needs to be careful that the choice of M fixes some part of the indeterminacies of the Massey products.

Remark 8.3Once we have a cofree resolution of BP*,we always get a relative injective resolution for any M,which needs not to be free over BP*.In particular,we can take M to be an extension of BP*/p.This allows us to compute multiplications and Massey products involving p as well as the β families,the latter living in homological degree 1 on the top cell of BP*/p.

AcknowledgementsThe author thanks Dan Isaksen who has read through the paper and made many modifications.The author also thanks Mark Behrens,Robert Bruner,Jesper Grodal,Lars Hesselholt,Haynes Miller,Doug Ravenel,John Rognes and Zhouli Xu for many useful discussions.