Computing homological residue fields in algebra and topology
aa r X i v : . [ m a t h . C T ] J u l COMPUTING HOMOLOGICAL RESIDUE FIELDSIN ALGEBRA AND TOPOLOGY
PAUL BALMER AND JAMES C. CAMERON
Abstract.
We determine the homological residue fields, in the sense of tensor-triangular geometry, in a series of concrete examples ranging from topologicalstable homotopy theory to modular representation theory of finite groups. Introduction
One important question in tensor-triangular geometry is the existence of residuefields. Just as we can reduce some problems in commutative algebra to problemsin linear algebra by passing to residue fields, we would like to study general tensor-triangulated categories via ‘field like’ ones. Indeed, in key examples of tensor-triangulated categories such fields do exist: In the stable homotopy category wehave Morava K-theories, in the stable module category of a finite group schemeover a field we can consider π -points, and of course for the derived category of acommutative ring we have ordinary residue fields.Even though residue fields exist in those examples, at the moment there is notensor-triangular construction of them and it is not known if they always exist,beyond very special cases [Mat17]. It is not even known exactly what properties oneshould expect from a residue field functor F : T → F from our tensor-triangulatedcategory of interest T to such a tensor-triangulated field F . In particular, some ofthe examples above fail to give symmetric tensor functors or tensor functors at all.The recent work [BKS19, Bal20b, Bal20a] introduced and explored homologicalresidue fields as an alternative that exists in broad generality and is always tensor-friendly. They consist of symmetric monoidal homological functors(1.1) ¯ h B : T → ¯ A B from T to ‘simple’ abelian categories ¯ A B . One such functor exists for each so-called homological prime B , as recalled in Section 2. These homological residue fieldscollectively detect the nilpotence of maps [Bal20b] and they give rise to a supporttheory for not necessarily compact objects [Bal20a].Homological residue fields are undeniably useful but they are defined in a ratherabstract manner and it is not clear how they relate to the tensor-triangulated residuefields F : T → F that partially exist in examples. For each homological prime B ,there is a canonical pure-injective object E B of T that completely determines thehomological residue field (1.1). Given a functor F : T → F to a tensor-triangulatedfield satisfying mild assumptions, there is a corresponding homological prime B . Date : July 10, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Tensor-triangular geometry, homological residue field.First named author supported by NSF grant DMS-1901696. (See Lemma 2.2.) In the standard examples of such F : T → F , it is desirableto determine the corresponding pure-injective E B and residue category ¯ A B . Ourfirst contribution is to show that E B is a direct summand of U ( ), where U is rightadjoint to F . (The existence of U is a mild assumption, by Brown Representability.) Theorem (Theorem 3.1) . Let F : T → F be a monoidal exact functor to a tensortriangulated field and suppose that F admits a right adjoint U . Then the pure-injective object E B ∈ T is a direct summand of U ( ) . In the examples of the topological stable homotopy category and of the derivedcategory of a commutative ring, this U ( ) is respectively a Morava K -theory spec-trum and a residue field. These are indecomposable objects. So we conclude thatfor these examples the potentially mysterious E B coincides with the better under-stood ring object U ( ). See Corollary 3.5 and Corollary 3.3. The latter generalizesthe case of a discrete valuation ring obtained in [BKS19, Example 6.23].In the case of the stable module category of a finite group scheme, there is acomplication because the best candidates for tensor-triangulated fields – associatedto π -points – are almost never monoidal functors for the tensor product comingfrom the group Hopf algebra structure. However, in the case of the stable modulecategory of an elementary abelian p -group with the p -restricted Lie algebra tensorproduct structure, π -points do give tensor functors and we show in Corollary 3.11that in this case the pure-injective E B is again determined by the π -point.In addition to understanding the pure-injective object E B we would like to de-scribe the abelian category ¯ A B in which a homological residue field takes its values.When B arises from a tensor-triangular field F : T → F we would like to relate¯ A B and F . Our second contribution is to do just this. The adjunction F ⊣ U gives a comonad F U on F . This comonad F U then gives a comonad d F U on theabelian category of (functor) F c -modules, where F c denotes the compact objectsof F ; see details in Section 2. The Eilenberg-Moore category of comodules (a. k. a.coalgebras) for this comonad d F U is precisely the homological residue field ¯ A B : Theorem (Theorem 4.2) . Let F : T → F be a monoidal functor with right ad-joint U , where F is a tensor triangulated field. The category of comodules for thecomonad d F U on the functor category of F c -modules is equivalent to the homologicalresidue field ¯ A B corresponding to F . In cases where the tensor-triangulated field F in question is semisimple, such asthe examples in topology and commutative algebra, this abelian category of F c -modules is just F itself. So the homological residue field ¯ A B is the category ofcomodules for an explicit comonad on F itself. Let us rephrase this result in heuris-tic terms. When the residue field functor F : T → F takes values in a semi-simpletriangulated category F ( i.e. one that is also abelian), then the abstractly con-structed ¯ A B contains more information than F . There is a faithful exact functor¯ A B → F but objects of ¯ A B ‘remember’ more information than just being an objectof F , they remember that they come from T via F . This additional information isencoded in the comodule structure with respect to F U : F → F .Section 2 sets up the stage. We prove our two theorems in Sections 3 and 4. Acknowledgements.
We thank Jacob Lurie for an interesting discussion, that madeus realize that the monadic adjunction of [BKS19, §
6] also satisfied co monadicity.We also thank Tobias Barthel and Greg Stevenson for useful comments. OMPUTING HOMOLOGICAL RESIDUE FIELDS 3 Background on homological residue fields
In this section we recall some definitions and properties of the category of mod-ules of a tensor triangulated category and of homological residue fields. By a bigtt-category T , we mean a compact-rigidly generated tensor-triangulated category, asin [BKS19]. The (small) tt-subcategory of compact and rigid objects is denoted T c .2.1. Recollection.
For T a big tt-category, the (functor) category of T c -modules A = Mod- T c is the category of additive functors from ( T c ) op to abelian groups.This is a Grothendieck category, hence admits enough injectives. The subcategoryof finitely presented objects of A is denoted A fp = mod- T c . We have a restrictedYoneda functor h : T → A defined by h ( X )( − ) = ˆ X ( − ) := Hom T ( − , X ) | T c forevery X ∈ T . This gives a commutative diagram T c (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) h / / A fp (cid:127) _ (cid:15) (cid:15) T h / / A whose first row is the usual Yoneda embedding for T c .For B a homological prime , i.e. a maximal Serre tensor-ideal of A fp , we obtainthe homological residue field ¯ A B as the Gabriel quotient Q : A ։ ¯ A B := A / Loc( B ).In that quotient ¯ A B we write ¯ X instead of Q ( ˆ X ), for X ∈ T . In ¯ A B we can take theinjective hull of the unit and by [BKS19, §
3] this is of the form ¯ E B for a canonicalpure-injective E B of T . Furthermore we have Loc( B ) = Ker( ˆ E B ⊗ − ) in A .A tt-field is a big tt-category F such that every object of F is a coproduct ofcompact objects and such that tensoring with any object is faithful. Equivalentlyby [BKS19, Theorem 5.21], a big tt-category F is a tt-field if and only if for everynon-zero X ∈ F the internal hom functor hom F ( − , X ) : F op → F is faithful.We now summarize and mildly generalize some results from [BKS19, Bal20a].2.2. Lemma.
Given a big tensor-triangulated category T , a tensor-triangulatedfield F , and a monoidal exact functor F : T → F with right adjoint U , we havethe following diagram: (2.3) T h / / F (cid:15) (cid:15) Mod - T c = A ˆ F (cid:15) (cid:15) Q * * ❚❚❚❚❚❚❚❚❚❚❚❚ Mod - T c / Ker( ˆ F ) = ¯ A B R j j ❚❚❚❚❚❚❚❚❚❚❚❚ ¯ F t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥ F h / / U O O Mod - F c ˆ U O O ¯ U ❥❥❥❥❥❥❥❥❥❥❥❥❥❥ in which ˆ F is the exact cocontinuous functor induced by F , the functor Q is theGabriel quotient with respect to Ker( ˆ F ) and the functor ¯ F is induced by the universalproperty, hence ˆ F = ¯ F Q and ¯ F is exact and faithful. We have adjunctions F ⊣ U , ˆ F ⊣ ˆ U , ¯ F ⊣ ¯ U , and Q ⊣ R , as depicted. Also, ˆ F h = h F and ˆ U h = h U .We have that B := Ker( ˆ F ) ∩ A fp is a homological prime and Ker( ˆ F ) = Loc( B ) .Therefore ¯ h B = Q ◦ h : T → ¯ A B is a homological residue field of T . PAUL BALMER AND JAMES C. CAMERON
Proof.
This is a more general version of [BKS19, Diagram 6.19], whose proof workswithout assuming F symmetric monoidal. We do not need Condition (3) in [BKS19,Hypothesis 6.1] either, as this was later used only to guarantee faithfulness of ¯ U ,which we do not use here. (cid:3) Remark.
It is important that F is only assumed to be monoidal but not sym-metric monoidal. In other words F ( X ⊗ Y ) is naturally isomorphic to F ( X ) ⊗ F ( Y ),but this isomorphism is not necessarily compatible with the braiding. In one of thekey examples coming from Morava K -theory the functor F cannot be made sym-metric monoidal but it is monoidal. The reason we ask that F is monoidal is sothat the kernel of ˆ F is a tensor-ideal. The monoidality of F is a reasonable condi-tion that guarantees this but because F is not monoidal in other examples ( e.g. inmodular representation theory) there may be better hypotheses to be discovered.3. The pure-injective E B in examples We now would like to study the pure-injective object E B that determines thehomological residue field corresponding to F : T → F as in Section 2.3.1. Theorem.
With the hypotheses as in Lemma 2.2, the pure-injective object U ( ) in T admits E B as a direct summand.Proof. That U ( ) is pure-injective is [BKS19, Lemma 6.12]. It corresponds to theinjective ¯ U ( ) = U ( ) in ¯ A B .By the ¯ F ⊣ ¯ U adjunction we have that ¯ F ( ) → ¯ F ¯ U ¯ F ( ) is a monomorphism.As ¯ F is faithful, we deduce that the unit → ¯ U ¯ F ( ) = ¯ U ( ) is a monomorphismin ¯ A B . But → ¯ E B is the injective hull in ¯ A B . Hence ¯ U ( ) = ¯ E B ⊕ ¯ X for somepure-injective X ∈ T .Recall by [BKS19, Corollary 2.18 (c)] that for any injective ¯ I in ¯ A B and any X ∈ T we have that Hom T ( X, I ) ∼ = Hom ¯ A B ( ¯ X, ¯ I ). So, since ¯ U ( ) has ¯ E B as asummand in ¯ A B , it follows that U ( ) also has E B as a summand in T . (cid:3) Now, in several examples U ( ) is a familiar object and it does not have anynon-trivial summands.3.2. Lemma.
Suppose that T is a tensor-triangulated category that is generated byits ⊗ -unit . For E ∈ T if π ∗ E := Hom(Σ ∗ , E ) is indecomposable as a π ∗ -modulethen E is indecomposable in T .Proof. We have π ∗ ( E ′ ⊕ E ′′ ) ∼ = π ∗ E ′ ⊕ π ∗ E ′′ as π ∗ -modules. So the result followsimmediately from π ∗ being a conservative functor when T is generated by . (cid:3) Corollary.
Let R be a noetherian ring with derived category T = D( R ) and let B be a homological prime corresponding to the prime p of Spec( R ) ∼ = Spc(D perf ( R )) .Then the pure-injective object E B is isomorphic to the residue field κ ( p )[0] .Proof. The homological prime B is obtained as in the setting of Theorem 3.1 fromthe residue field functor F : D( R ) → D( κ ( p )). By the above discussion, we justneed to show that U ( ) = κ ( p )[0] does not split in D( R ). By Lemma 3.2 thisfollows from the fact that κ ( p ) does not split as an R -module. (cid:3) Morava K -theories furnish all of the tt-primes in the stable homotopy cat-egory SH by [HS98]. Indeed, each tt-prime of SH c is the kernel Ker(SH c → K ( p, n ) ∗ - GrMod) of a functor given by X K ( p, n ) ∗ X where p is a prime and OMPUTING HOMOLOGICAL RESIDUE FIELDS 5 n ≥ ∞ . We write K ( p, ∞ ) for H Z /p and K ( p,
0) for H Q .When 0 < n < ∞ , we have K ( p, n ) ∗ = F p [ v ± n ] with v n in degree 2( p n − K ( p, n ) ∗ - GrMod the category of graded K ( p, n ) ∗ -modules; this is asemisimple triangulated category.Each spectrum K ( p, n ) is a ring spectrum but they are not E except for n = 0and n = ∞ ; see [ACB19, Corollary 5.4] for a proof of this folklore result. The K (2 , n ) are not even commutative rings in the homotopy category except for n = 0and n = ∞ [W¨ur86]. Nevertheless the homotopy category of K ( p, n )-modules isequivalent via taking homotopy groups to the category K ( p, n ) ∗ - GrMod.The K ( p, n )-homology has a K¨unneth isomorphism which shows that the functorSH → K ( p, n ) ∗ - GrMod is monoidal, although not necessarily symmetric monoidal.Because K ( p, n ) ∗ is a graded field, the category K ( p, n ) ∗ - GrMod is a tensortriangulated field. So for each Morava K -theory we have a homological residuefield ¯ A B and a pure-injective E B that we want to identify.3.4. Lemma.
Each Morava K -theory spectrum K ( p, n ) is indecomposable in thestable homotopy category.Proof. This follows from [HS98, Proposition 1.10], which states that any retract inSH of a K ( p, n )-module is a wedge of shifts of K ( p, n ). Because K ( p, n ) ∗ is either F p or 0 in each dimension, it follows that K ( p, n ) itself is indecomposable in SH. (cid:3) Corollary.
For B the homological prime of Spc h (SH c ) corresponding to K ( p, n ) ,we have an isomorphism E B ≃ K ( p, n ) in SH .Proof. Because E B is constructed as in Lemma 2.2 from the monoidal functor F : SH → Ho( K ( p, n )-Mod), by Theorem 3.1 we have that E B is a summandof K ( p, n ). Therefore by Lemma 3.4 we have that E B is isomorphic to K ( p, n ). (cid:3) Remark.
The tt-primes of the G -equivariant stable homotopy category SH( G )have been determined for G finite in [BS17] and for general compact Lie groups G in [BGH20]. They are all pulled-back from the chromatic ones in SH via geometricfixed-point functors Φ H : SH( G ) → SH for (closed) subgroups H ≤ G . The func-tor Φ H admits a right adjoint that we denote U H : SH → SH( G ). It follows that thepure-injective E B corresponding to H and K ( p, n ) is a summand of U H ( K ( p, n )).When H = G it is easy to show that these objects remain indecomposable in SH( G )but we do not know whether this holds in general.3.7. Remark.
We have shown that in two of the main examples, the derived categoryof a noetherian ring and the stable homotopy category, the E B are ring objects. Itis natural to wonder if this is always the case. By [BKS19], there is always a map E B ⊗ E B → E B retracting E B ⊗ η where η : → E B comes from the definitionof ¯ E B as the injective hull of ¯ . Those ‘weak rings’ E B would be actual rings ifwe always had E B = U ( ) but the latter is not necessarily true. Indeed, one can‘overshoot’ the mark as explained in the following example.3.8. Example.
For a field extension K → L , we have a tt-functor on derived cat-egories T := D( K ) → D( L ) =: F . Note that T is already a tt-field itself. Thepure-injective E B associated to the only homological prime B = 0 of T is the ⊗ -unit T = K [0] itself, whereas U ( F ) is L [0] and of course L ≃ K dim K ( L ) .Our final example is that of π -points in modular representation theory. Let k bea field of characteristic p dividing the order of a finite group G . Recall [FP05] that PAUL BALMER AND JAMES C. CAMERON a π -point of the group ring kG is a flat algebra homomorphism π : KC p → KG which factors through KA for A an elementary abelian p -subgroup of G where K isa field extension of k . There is a corresponding functor π ∗ : Stab kG → Stab KC p composed of extension-of-scalars to K followed by restriction along π . This functoracts like a residue field functor, but frustratingly this functor is not monoidal, andso Lemma 2.2 cannot be used to construct a homological residue field.However, every π -point is equivalent to one where the homomorphism KC p ∼ = K [ t ] /t p → K [ x , . . . , x n ] / ( x pi ) ∼ = KA is of the form t P ni =1 α i x i for α =( α , . . . , α n ) ∈ K n r { } . We will denote such a map by π α . Up to equivalence, oneindexes the π -points π α by the closed points of P nK ≃ Spc(Stab( KA ) c ). Allowingall field extensions K/k , the π α parameterize all points of Spc(Stab( kA ) c ).The map π α is a Hopf algebra map when the source and target have the Hopfalgebra structure coming from thinking of KC p and KA as the restricted envelopingalgebra of a p -restricted Lie algebra. Consequently the functor π ∗ α : Stab( KA ) → Stab( KC p ) is a monoidal functor with respect to this different tensor. In formulas,this corresponds to the comultiplication ∆ on KA = K [ x i , . . . , x n ] / ( x pi ) given by∆( x i ) = x i ⊗ ⊗ x i . See [CI17] for a discussion of these structures.3.9. Convention.
From now on we use the Hopf algebra structure on KA comingfrom the p -restricted Lie algebra structure to put a tensor structure on Stab( KA ).The π ∗ α : Stab( kA ) → Stab( KC p ) are now tensor-triangular residue field func-tors. So, there are corresponding homological residue fields Stab( kA ) → ¯ A B andcorresponding pure-injectives E B in Stab( kA ). We would like to relate the π -pointto the pure-injective object. First, we note what π -points do on cohomology.3.10. Lemma.
Let π α : kC p → kA be a k -rational π -point. For p odd, the inducedmap on cohomology π ∗ α : H ∗ ( A, k ) ∼ = k [ η , . . . , η n ] ⊗ Λ( ξ , . . . , ξ n ) → k [ ζ ] ⊗ Λ( ω ) ∼ =H ∗ ( C p , k ) is given by π ∗ α ( η i ) = α pi ζ and π ∗ α ( ξ i ) = α i ω for all i = 1 , . . . , n . For p = 2 , there is no exterior power and π ∗ α ( η i ) = α i ζ for all i = 1 , . . . , n .Proof. For the polynomial classes this is [Car83, Propositions 2.20 and 2.22] andthe result for the exterior classes follows from the proof of the first proposition. (cid:3)
Corollary.
Let A be an elementary abelian p -group. See Convention 3.9. (a) The pure-injective E B of Stab( kA ) associated to a (closed) k -rational π -point π α : kC p → kA is given by coind kAkC p k = Hom kC p ( kA, k ) . (b) Let
K/k be an extension. The pure-injective E B of Stab( kA ) associated to a π -point π α : KC p → KA is a summand of res KAkA coind
KAKC p K = Hom KC p ( KA, K ) .Proof. The second claim follows immediately from Theorem 3.1. The first claimwill follow from the second (with K = k ) if we can show that coind kAkC p k is indecom-posable in Stab( kA ). From Lemma 3.2 it is enough to show that π ∗ (coind kAkC p k ) isindecomposable as a π ∗ k = ˆH −∗ ( A, k ) module. But π ∗ (coind kAkC p k ) = ˆH −∗ ( C p , k )with the module structure induced by the restriction map π ∗ α : ˆH ∗ ( A, k ) → ˆH ∗ ( C p , k )and this module is indecomposable by Lemma 3.10. (cid:3) Remark.
In (b), there is no reason for the object res
KAkA coind
KAKC p K to beindecomposable, for one can ‘overshoot’ the right residue field as in Example 3.8.3.13. Remark.
Because K = Stab( kA ) c is generated by the unit, all thick subcat-egories are tensor-ideals, and hence the two different tensor structures on K give OMPUTING HOMOLOGICAL RESIDUE FIELDS 7 the same tensor-triangular spectrum. However, this does not imply that the homo-logical spectra are in canonical bijection. In other words it is not necessarily thecase that if I is a Serre tensor-ideal of mod- K in the restricted Lie algebra tensorstructure then it is also a Serre tensor-ideal in the group tensor algebra structure.Therefore these results do not give classifications of the objects E B of Stab( kA )with the group tensor structure, a priori . Further research might tell.4. Homological residue fields as comodules
Recall Diagram (2.3). It is proved in [BKS19] that ¯ U is monadic under theassumption that U is faithful, i.e. that F is surjective-up-to-summands (an as-sumption we do not make here). In that case, one can describe Mod- F c as theEilenberg-Moore category of ¯ U ( )-modules in ¯ A B . This was the logic of [BKS19]:How to recover F from the abstract ¯ A B ?However, in many of the examples, the category F is more familiar than ¯ A B andwe would rather like a description of ¯ A B in terms of F . Towards this end, we willshow that ¯ F is co monadic, so ¯ A B is the Eilenberg-Moore category of comodules forthe comonad ¯ F ¯ U over Mod- F c .For this, we use the dual version of the Beck monadicity theorem. Denote theEilenberg-Moore category of comodules for a comonad H on D by H - CoMod D .4.1. Lemma.
Suppose we have an adjunction F : C ⇄ D : U and that C and D haveequalizers and F is conservative and preserves equalizers. Then F is comonadic, i.e. the comparison functor f : C → F U - CoMod D is an equivalence of categories.Proof. This follows from the usual Beck Monadicity theorem [ML98, § VI.7] appliedto the opposite category. (cid:3)
Theorem. In (2.3) the functor ¯ F is comonadic, so ¯ A B is equivalent to theEilenberg-Moore category of comodules for the comonad ¯ F ¯ U .Proof. The conditions of Lemma 4.1 hold since ¯ F is conservative and exact and thecategories involved are abelian and hence have all equalizers. (cid:3) Remark.
We have d F U ∼ = ˆ F ˆ U ∼ = ¯ F Q R ¯ U ∼ = ¯ F ¯ U . The functor ˆ F is not conser-vative in general, so ˆ F itself is not comonadic.In two of the examples at hand, the stable homotopy category and the derivedcategory of a ring, the tensor triangulated fields F in the picture are semisimpletriangulated categories and hence are already abelian. So, the abelian category of(functor) modules on these tensor triangulated fields is just the tensor triangulatedfield F itself, i.e. restricted Yoneda h : F → F c -Mod is an equivalence.4.4. Corollary.
In the case of SH , the homological residue category ¯ A B for the ho-mological prime B corresponding to a Morava K -theory spectrum K ( p, n ) is equiv-alent to the Eilenberg-Moore category of comodules for the comonad F U on thecategory of graded K ( p, n ) ∗ -modules, where F U is associated to the free/forgetfuladjunction F : SH ⇄ Ho( K ( p, n ) - Mod) ∼ = K ( p, n ) ∗ - GrMod : U . Corollary.
In the case of D( R ) for R a commutative noetherian ring, thehomological residue category ¯ A B for the homological prime B corresponding to aZariski point i : Spec( κ ( p )) → Spec R is equivalent to the Eilenberg-Moore categoryof comodules for the comonad L i ∗ ◦ i ∗ on graded κ ( p ) -modules, where L i ∗ ◦ i ∗ isassociated to the usual adjunction L i ∗ : D( R ) ⇄ D( κ ( p )) ∼ = κ ( p ) - GrMod : i ∗ . PAUL BALMER AND JAMES C. CAMERON
Proof of Corollaries 4.4 and 4.5.
By Theorem 4.2 we have that in these two casesthe category ¯ A B is equivalent to F U -comodules on F , where F is respectively graded K ( p, n ) ∗ -modules and graded κ ( p )-modules.So, the claim follows from the fact that if F is a semisimple abelian category then F c -Mod ∼ = F and, under this equivalence, the comonad d F U boils down to
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Paul Balmer, UCLA Mathematics Department, Los Angeles CA 90095-1555, USA
E-mail address : [email protected] URL : ∼ balmer James C. Cameron, UCLA Mathematics Department, Los Angeles CA 90095-1555, USA
E-mail address : [email protected] URL : ∼∼