New York Journal of Mathematics

New York J. Math. 15 (2009) 219–243.

Semisimple ring spectra

Mark Hovey and Keir Lockridge

Abstract. We define global dimension and weak dimension for the structured ring spectra that arise in algebraic topology. We provide a partial classification of ring spectra of global dimension zero, the semisimple ring spectra of the title. These ring spectra are closely related to classical rings whose projective modules admit the structure of a triangulated category. Applications to two analogues of the generating hypothesis in algebraic topology are given.

Contents

1. Introduction 219 2. Semisimplicity and von Neumann regularity 222

2.1. Triangulated categories 222 2.2. Ring spectra 225 2.3. Morita invariance 227

3. Projective modules 231 3.1. Notation and overview 231 3.2. Triangulations of projective modules 232

4. Stable homotopy categories 238 5. The generating hypothesis 240 References 241

1. Introduction

Classical rings are often studied via their categories of modules or their derived categories. Important invariants such as global and weak dimension are defined in terms of resolutions (by either projective or flat modules) in the module category; such resolutions are examples of exact sequences,

Received May 7, 2009. Mathematics Subject Classification. Primary: 55P43; Secondary: 18E30, 16E10. Key words and phrases. Ring spectrum, semisimple, von Neumann regular, triangu-

lated category, quasi-Frobenius ring, IF-ring, generating hypothesis.

219^{ISSN 1076-9803/09}

which make sense in any abelian category. The ring analogues that arise in algebraic topology, on the other hand, do not have abelian module categories. The derived category of modules over a symmetric ring spectrum ([15]) or S-algebra ([7]) is a triangulated category, and it therefore has more in common with the derived category of an ordinary ring. Consequently, we use derived categorical formulations of homological dimension that carry over to more general triangulated categories. These ideas have much in common with the theory of projective classes presented in [4]. In this paper, we will use the term ‘ring spectrum’ to mean either a symmetric ring spectrum or an S-algebra. The categories of symmetric ring spectra and S-algebras are Quillen equivalent ([21, 0.4], [20, 0.2]), and up to homotopy these ring spectra are the A_{∞} ring spectra ([7, 3.4]).

Let E be a ring spectrum, and let D_{E} denote the derived category of right E-module spectra. Recall that thick⟨E⟩, the smallest full subcategory of D_{E} containing E and closed under suspension, retraction, and cofiber sequences, is exactly the collection of compact E-modules. Call an E-module projective if it is a retract of a coproduct of suspensions of E; note that the compact projective modules are exactly the retracts of finite coproducts. Write ⟨E⟩^{0} for the class of projective modules, and inductively define a module M to belong to ⟨E⟩^{i} if there exists a cofiber sequence

\xymatrix{P \ar[r] & Y \ar[r] & M̃ \ar[r] & ΣP}

with P projective, Y ∈ ⟨E⟩^{i−1}, and M a retract of M̃. This defines a filtration of D_{E} that is not in general exhaustive. Define ⟨E⟩_{f}^{i} similarly, where ⟨E⟩_{f}^{0} is the class of compact projective modules; this defines an exhaustive filtration of thick⟨E⟩. The right global dimension of E, written r. gl. dim. E, is the smallest integer n such that ⟨E⟩^{n} = D_{E} (or ∞ if no such n exists), and the right weak dimension of E, written r. w. dim. E, is the smallest integer n such that ⟨E⟩_{f}^{n} = thick⟨E⟩ (or ∞ if no such n exists). Call E semisimple if it has global dimension zero, and call E von Neumann regular if it has weak dimension zero (we prove in §2.1 that semisimplicity and von Neumann regularity are right-left symmetric).

Remark 1.1. Our definitions are related to at least two others. By [4, 3.2], ⟨E⟩^{0} is part of a stable projective class with respect to the ideal of ghosts (maps inducing the trivial map on homotopy groups). Hence, in the terminology of [4], E has right global dimension at most n if and only if every E-module has length at most n + 1 with respect to this projective class. It is shown in [4, §7] that the right (and left) global and weak dimensions of the sphere spectrum S are both ∞. Note that ⟨E⟩_{f}^{0} does not form part of a projective class in thick⟨E⟩ if there is any compact E-module M such that π_{∗}M is not finitely generated over π_{∗}E. But, according to Rouquier ([25, 3.2]), a triangulated category T has dimension zero if there exists any object M in T such that the smallest full subcategory of T containing M and closed under finite coproducts, retracts, and suspensions is T itself. Hence, if a ring spectrum E is von Neumann regular, then thick⟨E⟩ is zero-dimensional in this sense. We do not know whether the converse is true.

In Proposition 2.6 we observe that these definitions are consistent with those in classical ring theory, in the sense that the Eilenberg–Mac Lane spectrum HR is a semisimple (von Neumann regular) ring spectrum if and only if R is a semisimple (von Neumann regular) ring. More generally, the main result of [12, §1] shows that the right global dimension of a ring R is equal to the right global dimension of the ring spectrum HR. The work in [12, §2] shows that the weak dimension of R is at most the right weak dimension of HR, with equality if R is right coherent. We do not know whether the two are equal in general.

The following theorem summarizes the implications of Theorems 3.2, 4.1, and 4.2 for the classification of semisimple and von Neumann regular ring spectra.

Theorem 1.2. Let E be a ring spectrum. If π_{∗}E is graded commutative, then:

(1) If E is semisimple, then π_{∗}E ≅ R_{1} × ⋯ × R_{n}, where R_{i} is either a graded field k or an exterior algebra k[x]/(x^{2}) over a graded field containing a unit in degree 3|x|+1 (i.e., π_{∗}E is a graded commutative Δ^{1}-ring).

(2) If E_{∗} is countable and E is von Neumann regular, then (π_{∗}E)_{𝔭} is a

graded commutative local Δ^{1}-ring for every prime ideal 𝔭 of π_{∗}E. If E is commutative, then:

(3) E is semisimple if and only if π_{∗}E is a graded commutative Δ^{1}-ring and for every factor ring of π_{∗}E of the form k[x]/(x^{2}), x · π_{∗}C ≠ 0, where C is the cofiber of x · E.

(4) If π_{∗}E is local or Noetherian, then E is semisimple if and only if E is von Neumann regular.

Further, any graded commutative Δ^{1}-ring is the homotopy of some (not necessarily commutative) ring spectrum (see Remark 3.12). Hence, the homotopy of a semisimple ring spectrum need not be a semisimple ring.

This paper is organized as follows. In §2 we present a few general facts about semisimplicity and von Neumann regularity, some in a more general context (triangulated categories) and others, such as Morita invariance, in the context of ring spectra. Whenever E is semisimple, the functor π_{∗} induces a triangulation of the projective π_{∗}E-modules, leading to the strictly algebraic problem of classifying the rings for which the associated category of projective modules admits a triangulation. This is addressed in §3 for commutative rings. In §4, we apply the results in §3 to stable homotopy categories; there is slightly more at issue than whether projective modules admit triangulations. Finally, in §5, we consider two forms of the generating hypothesis. In the stable category of spectra, Freyd’s generating hypothesis ([9, §9]) is the conjecture that any map of finite spectra inducing the trivial map of homotopy groups must itself be trivial. The global version of this conjecture — that this is true for all maps of spectra — is easily seen to be false. The present work was motivated in part by the following question: what must be true about a triangulated category in order for it to support a global version of the generating hypothesis?

We would like to thank Sunil Chebolu, who informed us of his work with Benson, Christensen, and Mináč on the generating hypothesis in the stable module category. It is his presentation of the material in [2] that led us to consider certain questions raised in the present paper.

2. Semisimplicity and von Neumann regularity

Before turning to the classification of semisimple and von Neumann regular ring spectra, we record a few general facts. Certain results are more usefully stated in a general setting; this framework is established in §2.1. In §2.2, we compare ring spectra to classical rings and show that, up to homotopy, semisimple ring spectra E with E_{∗} local are wedges of suspensions of a Morava K-theory K(n). We discuss Morita invariance in §2.3.

2.1. Triangulated categories. We assume that the reader is somewhat familiar with triangulated categories. In brief, a triangulated category is an additive category T together with an automorphism Σ of T called suspension and a collection of diagrams called exact triangles of the form

\xymatrix{A \ar[r] & B \ar[r] & C \ar[r] & ΣA}

satisfying several axioms (see [14, A.1.1] or [24] or [28]). Using the suspension functor, one can view the morphism sets in T as graded groups: define [X, Y]_{k} = Hom_{T} (Σ^{k}X, Y) and

[X, Y]_{∗} = ⨁_{k∈ℤ}[X, Y]_{k}.

Let S be an object in T , and let π_{∗}(−) = [S, −]_{∗}; we will sometimes write S_{∗} for π_{∗}S. Recall that a π_{∗}S-module N is realizable if there is an object X ∈ T such that π_{∗}X = N. For any X ∈ T , the localizing subcategory generated by X, written loc⟨X⟩, is the smallest full subcategory of T containing X that is thick and closed under arbitrary coproducts. If T = loc⟨X⟩, then we say that X generates T . A cohomology functor H on T is representable if there is an object Y in T such that the functors H and [−, Y]_{0} are naturally isomorphic (see [14] for more details). T is cocomplete if arbitrary coproducts exist in T . Call T a weak stable homotopy category if it is a cocomplete triangulated category where every cohomology functor on T is representable. This definition is ‘weak’ in the sense that it is obtained from the definition of stable homotopy category given in [14, 1.1.4] by deleting conditions (b) and (c) concerning smash products and generators. Note that D_{E} is a weak stable homotopy category with compact generator E. Call any object S in a triangulated category T semisimple if loc⟨S⟩ = ⟨S⟩^{0} and von Neumann regular if thick⟨S⟩ = ⟨S⟩_{f}^{0}.

If T is cocomplete, then idempotents split in T ([3]), so every projective right π_{∗}S-module is realizable as π_{∗}Y, where Y ∈ ⟨S⟩^{0}. Hence, π_{∗} induces an equivalence of categories

Φ : ⟨S⟩^{0} ⟶ P,

where P is the category of projective right π_{∗}S-modules.

Proposition 2.1. Let T be a weak stable homotopy category with compact generator S. The following are equivalent:

(1) S is semisimple.

(2) The functor π_{∗}(−) is full and faithful.

(3) The functor π_{∗}(−) is faithful.

(4) The realizable modules and the projective modules coincide.

(5) The realizable modules and the injective modules coincide. If any one of the above conditions is satisfied, then π_{∗}S is a quasi-Frobenius ring.

A ring is quasi-Frobenius if it is right Noetherian and right self-injective. This condition is right-left symmetric, and the quasi-Frobenius rings are exactly the rings for which the collections of projective and injective modules coincide. In fact, a ring is quasi-Frobenius if and only if every projective module is injective, if and only if every injective module is projective. More information on quasi-Frobenius rings may be found in [18, §15].

Proof. The implications (1) ⟹ (2) ⟹ (3) should be clear; we now prove that (3) ⟹ (1). Assume π_{∗} is faithful. Using a generating set for the π_{∗}S-module π_{∗}Y, construct a map f : X ⟶ Y such that X ∈ ⟨S⟩^{0} and π_{∗}f is surjective. This map fits into an exact triangle

\xymatrix{X \ar[r]^{f} & Y \ar[r]^{g} & Z \ar[r] & ΣX.}

Since π_{∗}f is surjective, π_{∗}g = 0. Since π_{∗} is faithful, g = 0, forcing Y to be a retract of X. Hence, T = ⟨S⟩^{0}.

We next prove that (4) and (5) are equivalent. If I is an injective π_{∗}Smodule, then the functor Hom_{ℤ}^{0}(π_{∗}(−), I) is a cohomology functor on T , hence representable by an object E(I) ∈ T . In this situation, π_{∗}E(I) = I, so every injective module is realizable. As already observed, every projective module is realizable. If every realizable module is projective, then every injective module is projective, and π_{∗}S a quasi-Frobenius ring. If every realizable module is injective, then every projective module is injective, and again π_{∗}S is a quasi-Frobenius ring. In either case, the projective, injective, and realizable modules coincide.

It is clear that (1) ⟹ (4), so it remains to prove that (4) ⟹ (1). For any X ∈ loc⟨S⟩, π_{∗}X = 0 if and only if X is trivial. Consequently, for any map f in loc⟨S⟩, π_{∗}f is an isomorphism if and only if f is an equivalence. Now fix X ∈ loc⟨S⟩. Since π_{∗}X is projective, there is a map f : Y ⟶ X such that Y ∈ ⟨S⟩^{0} ⊆ loc⟨S⟩ and π_{∗}f is an isomorphism. Hence, X is equivalent to Y. Since T = loc⟨S⟩, the implication is established. □

We next prove a similar proposition characterizing von Neumann regularity. As above, π_{∗} induces an equivalence of categories

Φ_{f} : ⟨S⟩_{f}^{0} ⟶ P_{f},

where P_{f} denotes the category of finitely generated projective right π_{∗}Smodules.

Proposition 2.2. Let T be a cocomplete triangulated category, and let S ∈ T be compact. The following are equivalent:

(1) S is von Neumann regular.

(2) For all X ∈ thick⟨S⟩, π_{∗}X is projective. If either condition is satisfied, then π_{∗}S is a left and right IF-ring.

A ring is injective-flat (an IF-ring) if every injective module is flat. According to [6], this condition is not left-right symmetric, and a ring is a left and right IF-ring if and only if it is right and left coherent and right and left FP-injective. More information on IF-rings may be found in [8, §6].

Proof. If S is von Neumann regular, then (2) holds since Φ_{f} is an equivalence of categories; the converse is also clear. Now suppose S is von Neumann regular; to show that π_{∗}S is a right IF-ring, it suffices to show that every finitely presented module M embeds in a finitely generated projective module ([8, 6.8]). Since M is finitely presented, it is the cokernel of a map f : A ⟶ B of finitely generated projective modules. There is a map f̃: Ã ⟶ B̃ in thick⟨S⟩ = ⟨S⟩_{f}^{0} such that f = π_{∗}f̃ and an exact triangle

\xymatrix{Ã \ar[r]^{f̃} & B̃ \ar[r] & C̃ \ar[r] & ΣÃ}

with C̃ ∈ ⟨S⟩_{f}^{0}. Hence, M embeds in the finitely generated projective module π_{∗}C̃, completing the proof. Using a similar argument, one can show that π_{∗}S is also a left IF-ring; π_{∗}(−) is replaced with the functor [−, S]_{∗}, which induces an equivalence of categories from ⟨S⟩_{f}^{0} to the category of projective left π_{∗}S-modules. □

Remark 2.3. Assume T is a monogenic stable homotopy category and Brown category (see [14, 4.1.4]); then, S is von Neumann regular if and only if the flat modules and realizable modules coincide. Indeed, for all X ∈ T , π_{∗}X is the direct limit over a system of modules of the form π_{∗}X_{α}, where X_{α} ∈ thick⟨S⟩. Consequently, if S is von Neumann regular, then every realizable module is flat ([18, 4.4]), and every flat module is realizable (since (−) ⊗_{π_{∗}S} F is representable for any flat module F). On the other hand, representability of cohomology functors implies that arbitrary products of π_{∗}S are realizable. If every realizable module is flat, this forces π_{∗}S to be left coherent ([18, 4.47]). Now, for all X ∈ thick⟨S⟩, π_{∗}X is a finitely presented flat module and therefore projective ([18, 4.30]). By Proposition 2.2, S is von Neumann regular.

We next show that if π_{∗}S is commutative and S is semisimple, then so are its algebraic localizations. In a Brown category, the same is true of von Neumann regularity. For any π_{∗}S-module M, let M_{𝔭} denote the localization of M at the prime ideal 𝔭. Suppose T is a cocomplete triangulated category with compact generator S. The proofs of [14, 2.3.17, 3.3.3, 3.3.7] show that there exists a localization functor L_{𝔭} on T such that π_{∗}L_{𝔭}X ≅ (π_{∗}X)_{𝔭}. We mean ‘localization functor’ in the sense of definition [14, 3.1.1], omitting the condition involving smash products, since T may not have a product struc-

ture. In particular, the natural map \xymatrix{Y \ar[r]^{ι_{Y}} & L_{𝔭}Y} induces an isomorphism

(2.1)  \xymatrix{[L_{𝔭}Y, L_{𝔭}X]_{∗} \ar[r]^{≅} & [Y, L_{𝔭}X]_{∗}.}

Call an object 𝔭-local if it lies in the image of L_{𝔭}, and let T_{𝔭} denote the full subcategory of 𝔭-local objects. Since S is compact, L_{𝔭} commutes with coproducts, and L_{𝔭}S is compact in T_{𝔭} = loc⟨L_{𝔭}S⟩.

Proposition 2.4. Suppose T is a cocomplete triangulated category with compact generator S, and suppose π_{∗}S is commutative. If S is semisimple, then so is L_{𝔭}S for all prime ideals 𝔭 in π_{∗}S. If T is also a Brown category, then L_{𝔭}S is von Neumann regular whenever S is von Neumann regular.

Proof. If S is semisimple, then the fact that L_{𝔭} commutes with coproducts implies that L_{𝔭}S is semisimple.

Suppose T is a Brown category. By [14, 4.2.2, 4.2.3], every object X in T is the minimal weak colimit (see [14, 2.2.1]) of the diagram of compact objects mapping to X. Now, let L_{𝔭}Z be a compact object in T_{𝔭}. Since localization preserves minimal weak colimits ([14, 3.5.1]), L_{𝔭}Z is a minimal weak colimit of objects in L_{𝔭}(thick⟨S⟩). By compactness, the identity map of L_{𝔭}Z must factor through an object in L_{𝔭}(thick⟨S⟩) ([14, 4.2.1]). Consequently, every small object in T_{𝔭} is the retract of an object in L_{𝔭}(thick⟨S⟩). If S is von Neumann regular, then L_{𝔭}(thick⟨S⟩) ⊆ ⟨L_{𝔭}S⟩_{f}^{0}, so ⟨L_{𝔭}S⟩_{f}^{0} = thick⟨L_{𝔭}S⟩. This completes the proof. □

Note that the proof of Proposition 2.4 is in fact valid for any smashing localization functor (i.e., one that preserves coproducts; see [14, 3.3.2]), not just the algebraic localizations.

2.2. Ring spectra. In this section we establish a few basic facts specific to ring spectra. First, we address left-right symmetry.

Proposition 2.5. Let E be a ring spectrum. E is right semisimple (von Neumann regular) if and only if E is left semisimple (von Neumann regular).

Proof. Since the left E-modules are the same as the right E^{op}-modules, we need to show that E is right semisimple (von Neumann regular) if and only if E^{op} is right semisimple (von Neumann regular). As in [7, §III], we will use the notation M_{E}(−, −) to denote morphisms in the category M_{E} of right E-modules and F_{E}(−, −) for function objects in M_{E}. Further, we have two duality functors

D(−) = F_{E}(−, E) : M_{E} ⟶ M_{E^{op}}

and

D^{op}(−) = F_{E^{op}}(−, E) : M_{E^{op}} ⟶ M_{E}.

Now, let M be a right E-module. We have the following sequence of natural bijections:

M_{E}(M, D^{op}DM) ≅M_{E∧_{S}E^{op}}(M ∧_{S}DM, E)
≅M_{E^{op}∧_{S}E}(DM ∧_{S}M, E)
≅M_{E^{op}}(DM, DM).

The first and third bijections are applications of [7, III.6.5(i)]. The identity map of DM therefore provides a natural transformation from the identity functor to the functor

M ↦ D^{op}DM.

Since this natural transformation induces an isomorphism when M = E, it is an isomorphism for all M in thick⟨E⟩.

Note that both D and D^{op} preserve compact projective modules. Hence, if every compact right E^{op}-module is projective, then so is every compact right E-module. This proves that E is von Neumann regular if and only if E^{op} is. Now, if E is right semisimple, then it is right von Neumann regular, and so E^{op} is right von Neumann regular. As in Remark 2.3, this means that every realizable right π_{∗}E^{op}-module is flat. However, since E is right semisimple, π_{∗}E is quasi-Frobenius by Proposition 2.1; hence, π_{∗}E^{op} is quasi-Frobenius and every flat module is projective. Therefore D(E^{op}) = ⟨E^{op}⟩^{0} and E^{op} is semisimple. □

The following proposition gives the correspondence between our definitions and the ones in classical ring theory, where the semisimple rings are those of global dimension zero and the von Neumann regular rings are those of weak dimension zero.

Proposition 2.6. Let HR denote the Eilenberg–Mac Lane spectrum associated to the ring R. HR is semisimple if and only if R is semisimple, and HR is von Neumann regular if and only if R is von Neumann regular.

Proof. It is well-known (see, for example, [7, IV.2.4]) that the derived category of HR-module spectra is equivalent to the derived category of Rmodules, D(R). In [19, §4] it is shown that π_{∗} is faithful on D(R) if and only if R is semisimple, and in [13, 1.3, 1.5] it is shown that thick⟨R⟩ = ⟨R⟩_{f}^{0} if and only if R is von Neumann regular. □

Semisimple ring spectra are also related to field spectra, as defined in [11, 1.7]. A noncontractible ring spectrum E is a field if E_{∗}X = π_{∗}E∧X is a free E_{∗}-module for all spectra X. According to [11, 1.9], every field has the homotopy type of a wedge of suspensions of the Morava K-theory K(n), for some fixed n (0 ≤ n ≤ ∞) and prime p. If E is semisimple and E_{∗} is a ring over which every projective module is free, then E is a field. Note, however, that since E_{∗} is quasi-Frobenius by Proposition 2.1, E_{∗} must in fact be local in this case (see [17, 23.6]). We now have

Proposition 2.7. If E is a semisimple ring spectrum and E_{∗} is local, then E has the homotopy type of a wedge of suspensions of K(n) for some fixed n and prime p.

Even if E_{∗} is not local, some K(n) is a summand of E. Note that a field spectrum is not necessarily semisimple: for example, consider E = Hℚ[x], the polynomial algebra over Hℚ generated by x in degree 2. Since E is flat over Hℚ, E_{∗}X ≅ E_{∗} ⊗_{ℚ} Hℚ_{∗}X is always a free E_{∗}-module, hence E is a field. But, the homotopy of the cofiber of multiplication by x on E is ℚ, a neither free nor projective E_{∗}-module, so E is not semisimple. We obtain a better correspondence between semisimplicity and the notion of a skew field object given in [14, 3.7.1(d)]. A skew field object in a homotopy category is a ring object R such that every R-module is isomorphic to a coproduct of suspensions of R. Consequently, a ring spectrum E that is a skew field object in D_{E} is semisimple if and only if E is a semisimple ring spectrum and E_{∗} is local.

2.3. Morita invariance. We should begin by emphasizing that our general definitions of semisimplicity and von Neumann regularity depend on the ambient triangulated category, not just on the object S. For example, the Morava K-theory ring spectrum K(n) is semisimple as an object in the derived category D_{K(n)}, but it is not von Neumann regular as an object in the stable category of spectra S. For example, the spectrum L_{K(n)}F, where F is a finite type n spectrum, is in thick⟨K(n)⟩ ⊆ S ([16, 8.12]), but it is not a wedge of suspensions of K(n). The following proposition summarizes how the general definitions of semisimplicity and von Neumann regularity relate to the particular definitions for ring spectra.

Proposition 2.8. A right E-module M is von Neumann regular as an object in D_{E} if and only if the endomorphism ring spectrum End_{E}(M) is von Neumann regular as a ring spectrum. If M is compact in D_{E}, then the same is true of semisimplicity.

This proposition follows easily from the following more general one. Recall that for any object S in a triangulated category T , thick⟨S⟩ is filtered by the subcategories ⟨S⟩_{f}^{i} and loc⟨S⟩ is filtered by the subcategories ⟨S⟩^{i}. If T is a triangulated category with filtration F_{i} and U is a triangulated category with filtration G_{i}, then call an equivalence of triangulated categories Φ : T ⟶ U a filtered equivalence if the restriction of Φ to F_{i} induces an equivalence

\xymatrix{F_{i} \ar[r]^{≅} & G_{i}} for all i.

Proposition 2.9. Let E be a ring spectrum, let M be a right E-module, and let K = End_{E}(M). There is a filtered equivalence of thick⟨M⟩ in D_{E} with thick⟨K⟩ in D_{K}. If M is compact, then there is a filtered equivalence of loc⟨M⟩ with loc⟨K⟩ = D_{K}.

Proof. We continue to use the notation in the proof of Proposition 2.5. Let

K = End_{E}(M) = F_{E}(M, M).

Note that M is a K-E-bimodule. Consider the adjoint functor pair

\xymatrix{D_{E} \ar[r]^{F_{E}(M,−)} & D_{K}. \ar[l]^{(−)∧_{K}M}}

Adjointness induces two natural transformations

\xymatrix{η : 1_{D_{E}} \ar[r] & F_{E}(M, −)∧_{K}M}

and

\xymatrix{ζ : F_{E}(M, (−)∧_{K}M) \ar[r] & 1_{D_{K}}.}

Since η_{M} is an isomorphism, η induces an isomorphism on thick⟨M⟩. Similarly, ζ induces an isomorphism on thick⟨K⟩. Our adjoint functors therefore induce an equivalence between thick⟨M⟩ and thick⟨K⟩. If M is compact in D_{E}, then F_{E}(M, −) commutes with coproducts, so η induces an isomorphism on loc⟨M⟩ and ζ induces an isomorphism on loc⟨K⟩ = D_{K}. In this situation, we have an equivalence of loc⟨M⟩ with D_{K}.

We next show that these inverse equivalences are filtration preserving. The functor Φ = (−)∧_{K}M preserves coproducts, so Φ(⟨K⟩_{f}^{0}) ⊆ ⟨M⟩_{f}^{0} and Φ(⟨K⟩^{0}) ⊆ ⟨M⟩^{0}. Since Φ preserves cofiber sequences, Φ(⟨K⟩_{f}^{i}) ⊆ ⟨M⟩_{f}^{i} and Φ(⟨K⟩^{i}) ⊆ ⟨M⟩^{i} for all i ≥ 0, by induction on i. The same is true of the functor F_{E}(M, −) for the filtration on thick⟨M⟩, and also for the filtration on loc⟨M⟩ when M is compact. This completes the proof. □

In the Morita theory of classical rings, two rings R and S are Morita equivalent if and only if S ≅ End_{R}(P) for some progenerator P in the category of R-modules Mod(R) ([18, 18.33]). A right R-module P is a progenerator if and only if it is a finitely generated projective module such that R is a retract of a finite sum of copies of P ([18, 18.8]). If E is a ring spectrum, then P_{∗} is a progenerator of Mod(E_{∗}) if and only if it is realizable as the homotopy of an E-module P such that ⟨P ⟩_{f}^{0} = ⟨E⟩_{f}^{0}. As a consequence of the last proposition, we have the following result on the ‘classical’ Morita invariance of global and weak dimension.

Proposition 2.10. Let E be a ring spectrum, and let P be a right E-module such that ⟨P ⟩_{f}^{0} = ⟨E⟩_{f}^{0}. Then

r. gl. dim. E = r. gl. dim. End_{E}(P) and r. w. dim. E = r. w. dim. End_{E}(P).

Proof. This follows from Proposition 2.9 with M = P. □

If P is an arbitrary compact generator of D_{E}, then we do not know whether global and weak dimension are preserved in the sense of this proposition. For example, suppose r. gl. dim. E = n. If P is a compact generator, then E ∈ ⟨P ⟩_{f}^{k} for some k, and the global dimension of End_{E}(P) is at most k+n, but we do not know of an example where the dimension of End_{E}(P) is actually larger than n. Put another way, we do not know whether concepts such as semisimplicity and von Neumann regularity are Morita invariant in the more general sense of [26]. Following [26, 4.20], call a pair of ring spectra E and F Morita equivalent if there is a chain of spectral Quillen equivalences between the model categories M_{E} and M_{F} of right modules. Semisimplicity is Morita invariant in the following sense.

Proposition 2.11. Suppose E and K are Morita equivalent symmetric ring spectra with π_{∗}E commutative. If E is semisimple, then K is semisimple.

Our proof requires an understanding of the relationship between generators in the derived category D_{E} and generators in the category of right E_{∗}-modules. The next two propositions address this relationship, but our analysis is far from complete when E_{∗} is noncommutative. We do not know whether von Neumann regularity is also Morita invariant.

Proposition 2.12. Let E be a ring spectrum, and let M be a compact projective E-module. Then, M is a generator of D_{E} if and only if

Hom_{E_{∗}}(M_{∗}, N) ≠ 0

for every nontrivial realizable E_{∗}-module N.

Proof. Throughout the proof, we use the fact that

D_{E}(X, Y)_{∗} ≅ Hom_{E_{∗}}(X_{∗}, Y_{∗})

whenever X is a projective E-module.

Suppose M is a generator of D_{E}, and let N = X_{∗} ≠ 0 be a realizable E_{∗}-module. The functor D_{E}(−, X)_{∗} vanishes on a localizing subcategory of D_{E}. If it vanishes on M, then it also vanishes on X, forcing X to be trivial. Consequently, since X is nontrival, D_{E}(M, X)_{∗} = Hom_{E_{∗}}(M_{∗}, N) ≠ 0.

Conversely, suppose Hom_{E_{∗}}(M_{∗}, N) ≠ 0 for all nontrivial realizable modules N. Then, D_{E}(M, X)_{∗} must be nontrivial whenever X is nontrivial.

There exists a localization functor L whose kernel is the localizing subcategory generated by the compact object M. Now, since D_{E}(M, LX)_{∗} = 0 for all X, the only local object is the trivial object. Hence, the kernel of L must be all of D_{E}; i.e., M generates D_{E}. □

We do not know whether the commutativity assumption in the following proposition may be dropped. If semisimplicity is indeed a Morita invariant notion, then the proposition must be true without it (provided E is semisimple). Note that any right (or left) Artinian ring — and hence any quasi-Frobenius ring — is semiperfect ([1, p. 303]).

Proposition 2.13. Let E be a ring spectrum, and let M be a compact projective E-module. Then:

(1) If M_{∗} is a generator of Mod(E_{∗}), then M is a generator of D_{E}.

(2) Suppose π_{∗}E is commutative and semiperfect. If M is a generator of D_{E}, then M_{∗} is a generator of Mod(E_{∗}).

Proof. Suppose M_{∗} is a generator of Mod(E_{∗}). By definition of generator, Hom_{E_{∗}}(M_{∗}, −) acts faithfully on Mod(E_{∗}); by Proposition 2.12, M is a generator of D_{E}. This establishes (1).

Assume E_{∗} is commutative and semiperfect, and suppose M is a generator of D_{E}. Since E_{∗} is semiperfect, it admits a decomposition as a product of rings E_{∗} = (e_{1}E_{∗}) × ⋯ × (e_{n}E_{∗}), where the e_{i} form a complete set of orthogonal local idempotents ([1, 27.6]). Since M_{∗} is projective, it is a product of copies of these factors ([1, 27.11]); we will show that each factor appears at least once, implying that M_{∗} is a generator of Mod(E_{∗}) ([18, 18.8]). Since M is a generator of D(E), Hom_{E_{∗}}(M_{∗}, e_{i}E_{∗}) ≠ 0 for all i by Proposition 2.12. This forces e_{i}E_{∗} to be a factor of M_{∗}, as there are no maps from e_{j}E_{∗} to e_{i}E_{∗} when i ≠ j (this uses commutativity). This establishes (2). □

We now give the proof of Proposition 2.11.

Proof of Proposition 2.11. Assume E is semisimple. By [26, 4.20], there exists a K-E-bimodule M such that M is a compact generator of D_{E} and K ≅ F_{E}(M, M). Consider the E-K-bimodule Q = F_{E}(M, E). Since E is semisimple, M is projective, and Q_{∗} = Hom_{E_{∗}}(M_{∗}, E_{∗}). By Proposition 2.13 (2), M_{∗} is a finitely generated projective generator of Mod(E_{∗}), so Q_{∗} is a finitely generated projective generator of Mod(K_{∗}) by [18, 18.22]. By Proposition 2.13 (1), Q must be a compact projective generator of D_{K}.

The map

\xymatrix{E \ar[r] & F_{K}(Q, Q)},

adjoint to the identity map of Q as a K-module ([7, III.6.2]), is an equivalence since

F_{K}(Q, Q)_{∗} ≅ Hom_{K_{∗}}(Q_{∗}, Q_{∗}) ≅ E_{∗}

(the first isomorphism by projectivity of Q, the second by [18, 18.17(c)]). If either E or K is flat (a ring spectrum E is flat if E∧− preserves stable equivalences of symmetric spectra), then by the proof of [26, 4.20], there exists an E-K-bimodule Q′ derived equivalent to Q such that the derived smash product

−∧_{E}^{L}Q′ : D_{E} ⟶ D_{K}

is an equivalence of categories. Now every object of D_{K} is a retract of a coproduct of suspensions of the projective K-module Q′, and K is semisimple.

If neither E nor F is flat, take a cofibrant replacement QE of E. Since QE is flat and Morita equivalent to both E and F, two applications of the above argument complete the proof. □

3. Projective modules

In section §2.1, we proved that if E is semisimple, then the derived category of E-module spectra is equivalent to the category P of projective right E_{∗}-modules. This equivalence endows P with a triangulation. Similarly, if E is von Neumann regular, then the category P_{f} of finitely generated projective right E_{∗}-modules admits a triangulation. For any graded module M, write M[n] for the shifted module with M[n]_{i} = M_{i−n}. In the induced triangulation on either P or P_{f}, the shift functor is determined by the suspension of spectra: π_{∗}ΣX = π_{∗−1}X = π_{∗}X[1]. We have proved:

Proposition 3.1. Let T be a cocomplete triangulated category with compact generator S. If S is semisimple (von Neumann regular), then the category P (or P_{f}) admits a triangulation with shift functor Σ = (−)[1].

This proposition implies that if S is semisimple, then π_{∗}S must be a product of fields and exterior algebras on one generator by Theorem 3.2 (see §3.1). However, the converse of this statement if false; the triangulation on P may not be induced by π_{∗} (a counter-example is provided in Remark 5.2). The relevant condition is included in Theorem 4.1, presented in §4.

We now characterize the local rings and the commutative rings for which the associated category of projective modules admits a triangulation. Our two main results are Theorems 3.2 and 3.3, stated in the next section.

3.1. Notation and overview. Let R be a graded ring. By a module we mean a graded right R-module. Let Mod(R) be the category of all Rmodules, let P be the category of projective R-modules, and let P_{f} denote the category of finitely generated projective R-modules. For any R-module M, write M[n] for the shifted module with M[n]_{i} = M_{i−n}. For any element m ∈ M, denote by |m| the degree of m. Write Hom_{k}(M, N) for degree k module maps from M to N; observe that Hom_{k}(M, N) = Hom_{0}(M[k], N). If M is an R-R-bimodule (or, if R is graded commutative), then, for any element x ∈ R of degree i, let x · M denote the right module map from M[i] to M induced by left multiplication by x.

In considering triangulations of P or P_{f}, it seems natural to consider one of two possibilities for Σ. If Σ(−) = (−)[1], then [M, N]_{∗} is the graded group of graded module maps from M to N; this shift functor already appeared in Proposition 3.1. Alternatively, we could consider Σ = 1, the identity functor. When R is concentrated in degree zero, one then obtains a triangulation of the category of ungraded projective (or finitely generated projective) Rmodules by identifying this category with the thick subcategory of P (or P_{f}) generated by the modules concentrated in degree zero. Usually, we will assume that the suspension functor is of the form Σ = (−)[n] for some n.

For convenience, we will use the term Δ^{n}-ring to refer to any ring for which P admits a triangulation with suspension Σ = (−)[n] and the term Δ_{f}^{n}-ring for any ring for which P_{f} admits a triangulation with suspension Σ = (−)[n]. Our goal is to characterize the graded commutative rings R for which the categories P and P_{f} admit triangulations. In §3.2, we prove:

Theorem 3.2. Let P be the category of projective modules over a graded commutative ring R. P admits a triangulation with suspension Σ = (−)[1] if and only if R ≅ R_{1} ×⋯×R_{n}, where each factor ring R_{i} is either a graded field k or an exterior algebra k[x]/(x^{2}) over a graded field k containing a unit of degree 3|x| + 1.

In the ungraded case, we have:

Theorem 3.3. Let P be the category of projective modules over a commutative ring R. P admits a triangulation with suspension Σ = 1 if and only if R ≅ R_{1} × ⋯ × R_{n}, where each factor ring R_{i} is either a field k, an exterior algebra k[x]/(x^{2}) over a field k of characteristic 2, or T/(4), where T is a complete 2-ring.

A complete 2-ring is a complete local discrete valuation ring of characteristic zero whose maximal ideal is generated by 2 (see [22, p. 223]). In [23], it is shown that the category of finitely generated projective T/(4)-modules admits a unique triangulation, and we use their methods to construct triangulations for the rings appearing in the above two theorems.

3.2. Triangulations of projective modules. It is in this section that we prove Theorems 3.2 and 3.3. Before we begin, we make a simple observation that we will use frequently. Suppose P (or P_{f}) is triangulated. If we apply the functor [R, −]_{0} to an exact triangle

\xymatrix{A \ar[r]^{f} & B \ar[r]^{g} & C \ar[r]^{h} & ΣA,}

then we must obtain a long exact sequence of R-modules. Hence, any exact triangle is exact as a sequence of R-modules. (‘Exact’ at ΣA means ker(−Σf) = im h.)

First we show that if P admits a triangulation, then R must be a quasi-Frobenius ring, and if P_{f} admits a triangulation, then R must be an IF-ring (cf. Propositions 2.1 and 2.2). Note that IF-rings are coherent ([8, 6.9]), so all Δ^{n}-rings and Δ_{f}^{n}-rings are coherent.

Proposition 3.4. If P admits a triangulation, then R is quasi-Frobenius.

Proof. Let M be an R-module. There is a map f : A ⟶ B of projective modules whose cokernel is M. This map must lie in an exact triangle in P,

\xymatrix{A \ar[r]^{f} & B \ar[r] & C \ar[r] & ΣA.}

Exactness at B implies that M is isomorphic to a submodule of the projective module C. If M is injective, then it must be a summand of C and therefore projective. Hence, every injective R-module is projective, and R is quasi-Frobenius. □

Proposition 3.5. If P_{f} admits a triangulation, then R is a left and right IF-ring.

Proof. In light of the proof of Proposition 3.4, we see that every finitely presented right R-module embeds in a finitely generated projective module. By [8, 6.8], R is a right IF-ring. To show that R is also a left IF-ring, use the functor [−, R]_{0} to show that every finitely presented left R-module embeds in a finitely generated projective module. □

Remark 3.6. When R is quasi-Frobenius, one can form StMod(R), the stable module category of R. The objects of StMod(R) are R-modules and the morphisms are R-module maps modulo an equivalence relation: two maps are equivalent if their difference factors through a projective module. StMod(R) is a triangulated category. If M is an R module, then the Heller shift of M, written ΩM, is the kernel of a projective cover \xymatrix{P (M) \ar[r] & M}. This descends to a well-defined, invertible endomorphism of the stable module category, and Ω^{−1} is the suspension functor for the triangulation of StMod(R). In Proposition 3.4, the exact triangle gives rise to three short exact sequences:

\xymatrix{0 \ar[r] & M \ar[r] & C \ar[r] & Σ ker f \ar[r] & 0 \\ 0 \ar[r] & ker f \ar[r] & A \ar[r] & im f \ar[r] & 0 \\ 0 \ar[r] & im f \ar[r] & B \ar[r] & M \ar[r] & 0.}

Together they imply that, if P admits a triangulation, then Ω^{3}ΣM ≅ M. This observation was made by Heller in [10].

The following proposition shows that a triangulation imposes a severe restriction on the local rings that may occur.

Proposition 3.7. Suppose P admits a triangulation. If R is a local ring with maximal right ideal 𝔪, then 𝔪 is principal and contains no nontrivial proper ideals.

Proof. Let 𝔪 be the maximal right ideal of R; since the proposition is clearly true if 𝔪 = (0), assume 𝔪 is nontrivial. Let S be the socle of R. Note that R must be quasi-Frobenius by Proposition 3.4; consequently, S is simple and is the minimal nontrivial right (and left) ideal of R ([18, 15.7]). We will argue that S and 𝔪 are equal by showing that they have the same composition length, an analogue of dimension for Artinian rings.

Since R is local, we must in fact have S ≅ R/𝔪; hence, there is a map f : R ⟶ R whose image is S and whose kernel is 𝔪. Since f must lie in an exact triangle, there is a free (since R is local) module F and a short exact sequence

(3.1)  \xymatrix{0 \ar[r] & R/S \ar[r] & F \ar[r] & Σ𝔪 \ar[r] & 0.}

For any R-module M, let c(M) denote its composition length (see [1, pp. 134–138]). Note that c(M) = 0 if and only if M = 0 and c(M) = 1 if and only if M is simple. By [1, 11.4], if

\xymatrix{0 \ar[r] & A \ar[r] & B \ar[r] & C \ar[r] & 0}

is exact, then c(B) = c(C)+c(A). Applied in the current context, we obtain

c(F) = c(R/S) + c(𝔪)
= (c(R) − c(S)) + (c(R) − c(S))
< 2c(R).

Hence, we must have c(F) = c(R) and c(S) = c(𝔪). Since 𝔪 and S have identical composition lengths, they must be equal, and 𝔪 is the only proper nontrivial right ideal of R; 𝔪 is principal by uniqueness. □

If R is a graded commutative local ring, then we need only assume that P_{f} is triangulated, as the next proposition demonstrates. Note that if P admits a triangulation, then thick⟨R⟩ admits a triangulation, and since R must be Noetherian, thick⟨R⟩ = P_{f}.

Proposition 3.8. Suppose P_{f} admits a triangulation. If R is a graded commutative local ring with maximal ideal 𝔪, then 𝔪 is principal and contains no nontrivial proper ideals.

Proof. First, we observe that, even if R is neither graded commutative nor local, it must satisfy the double annihilator condition ann_{l} ann_{r} Rx = Rx. Let x ∈ R, and let i = |x|. We have an exact triangle in P

\xymatrix{R[i] \ar[r]^{x} & R \ar[r]^{ψ} & P \ar[r]^{ϕ} & ΣR[i].}

Observe that im ϕ = ann_{r} Rx, and if y ∈ ann_{l} ann_{r} Rx, then (y · R) ∘ ϕ = 0. Exactness at R[i] therefore implies that ann_{l} ann_{r} Rx = Rx.

For the remainder of the proof, we assume R is graded commutative and local. Since P_{f} is triangulated, R is coherent; this implies that ann(x) is finitely generated for all x ∈ R.

Consider x ∈ 𝔪. We now show that x^{2} = 0. Since (x·P)∘ψ = ψ∘(x·R) = 0, we obtain a factorization x · P = f ∘ ϕ, where f : im ϕ ⟶ P. Since im ϕ = ann(x), (x · P) ∘ f = 0. Hence, x^{2} · P = 0. Since R is local, P is free (P is nontrivial since x is not a unit). Therefore x^{2} = 0.

Fix a nonzero element x ∈ 𝔪. We next show that ann(x) = (x). Note that P is an extension of finitely generated modules and is therefore finitely generated. Let {e_{1}, …, e_{n}} be a basis for the free module P, let ϕ_{i} = ϕ(e_{i}), and let ψ(1) = Σe_{i}t_{i}. Since x is nonzero, im ϕ = ann(x) ⊆ 𝔪; hence, ϕ(e_{i}ϕ_{i}) = ϕ_{i}^{2} = 0, and exactness at P provides an element z_{i} ∈ R such that ψ(z_{i}) = e_{i}ϕ_{i}. This implies that t_{j}z_{i} = 0 when i ≠ j, and t_{i}z_{i} = ϕ_{i}. Since im ϕ = ann(x), the elements ϕ_{i} generate ann(x). Let t = t_{1} + ⋯ + t_{n} and consider q = Σϕ_{i}a_{i} ∈ ann(x). We have t(Σz_{i}a_{i}) = q, so ann(x) ⊆ (t). Since ψ ∘ (x · R) = 0, we obtain t_{i}x = 0 for all i. Hence, (t) = ann(x). Since x ≠ 0, t is not a unit, so (t) ⊆ ann(t) = ann ann(x) = (x). This proves that ann(x) = (t) = (x). Further, it is worth remarking that P must be free of rank 1. Suppose n ≥ 2. It is clear that e_{1}x and e_{2}x are in the kernel of ϕ. Hence, for some a, b ∈ R, e_{1}x = ψ(a) and e_{2}x = ψ(b). This means that e_{1}xb − e_{2}xa = 0, forcing a, b ∈ ann(x) = (x). But since t_{i}x = 0 for all i, it must be the case that e_{1}x = ψ(a) = Σe_{i}t_{i}a = 0, a contradiction.

Finally, we show that, if 𝔪 ≠ (0), it must be the unique, proper, nontrivial, principal ideal of R. Suppose a ∈ 𝔪 and b ∈ R have the property that ab ≠ 0. Then, a ∈ ann(ab) = (ab). Hence, there is an element k ∈ R such that a(1−bk) = 0. Since R is local, this forces a to be zero or b to be a unit. Now consider two nonzero elements x, y ∈ 𝔪. We have x ∈ ann(y) = (y) and y ∈ ann(x) = (x), so (x) = (y). This forces 𝔪 to be the unique, proper, nontrivial, principal ideal of R. □

Proposition 3.9. Let R be a graded commutative local ring with residue field k. If P (or P_{f}) admits a triangulation with suspension Σ = (−)[n], then R is either

(1) the graded field k,

(2) an exterior algebra k[x]/(x^{2}) with a unit in degree 3|x| + n, where char k = 2 if n = 0, or

(3) n is even and R ≅ T/(4), where T is the unique (up to isomorphism) complete 2-ring with residue field k (of characteristic 2) containing a unit in degree n.

Proof. In the proof of Proposition 3.8, we showed that the cofiber of x · R is free of rank 1 for any nontrivial element x in the maximal ideal 𝔪. Hence, any such x fits into an exact triangle of the form

(3.2)  \xymatrix{R[i] \ar[r]^{x} & R \ar[r]^{vx} & R[j] \ar[r]^{wx} & R[i + n],}

where i = |x| and v and w are units. One can check that |(vw)^{−1}| = 3i + n.

Suppose R contains a field. As such, R is a ring of equal characteristic; since it is complete, it contains a field isomorphic to k = R/𝔪 ([22, 28.3]). So either 𝔪 = 0 and R is a graded field, or 𝔪 = (x) for some x ≠ 0 (by Proposition 3.8) and R is isomorphic to the exterior algebra k[x]/(x^{2}). In the latter case, we just observed that k must contain a unit of degree 3i + n.

If p is a unit for all primes p, then R contains the field ℚ. If p is not a unit for some prime, then either R has characteristic p and contains a field, or p is nonzero and is in the maximal ideal. If p is in the maximal ideal 𝔪, then by Proposition 3.8, we must have 𝔪 = (p). If n is odd, this is not possible: since p has degree zero, we saw above that there must be a unit of degree 3 · 0 + n, which is odd. But if s is an odd unit, then 2s^{2} = 0 by graded commutativity; this forces 2 = 0, contradicting 𝔪 = (p).

It remains to consider the case where n is even and 𝔪 = (p) for some prime p. Let T be the unique (up to isomorphism) complete p-ring with residue field k (see [22, §29]). By the discussion on p. 225 of [22], since R is a complete local ring of unequal characteristic, it contains a coefficient ring A ≅ T/(p^{2}). Further, it is observed in the proof of [22, 29.4] that every element can be expanded as a power series in the generators of the maximal ideal with coefficients in A. Since the maximal ideal is generated by p ∈ A, we obtain R ≅ A. To see why p = 2, consider the rotation of triangle (3.2),

\xymatrix{R \ar[r]^{vp} & R[j] \ar[r]^{wp} & R[n] \ar[r]^{−p} & R[n].}

The map κ in the diagram

[figure: R; p; R; vp; v; R[j]; wp; κ; R[n]; R; vp; R[j]; wp; R[n]; −p; R[n]]

must exist. Hence, κvp = wpv = −pκv, so 2p = 0. This forces p = 2.

When n = 0 and R = k[x]/(x^{2}), an argument similar to the one just given shows that char k = 2. □

We next show that, for n = 0 and n = 1, the rings in the conclusion of Proposition 3.9 are Δ^{n}-rings.

Proposition 3.10 ([23]). If T is a complete 2-ring, then T/(4) is a Δ^{0}-ring.

Proof. In [23], a triangulation of P_{f} is constructed for any commutative local ring of characteristic 4 with maximal ideal (2), but the proof in fact shows that P admits a triangulation. These rings are exactly the rings of the form T/(4) (where T is a complete 2-ring). Certainly, any ring of the form T/(4) is of the type discussed in [23], and in Proposition 3.9 it is shown that any commutative local ring of characteristic 4 with maximal ideal (2) is of the form T/(4). □

Proposition 3.11. Every graded field k is a Δ^{n}-ring for all n. Every exterior algebra k[x]/(x^{2}) with a unit in degree 3|x|+n is a Δ^{n}-ring, provided n and |x| are not both even when char k ≠ 2.

Proof. We will use a graded version of the construction presented in [23]. Fix n ≥ 0. We will work in the category of differential graded modules over a differential graded algebra A, where the degree of all derivations is −n. So if x, y ∈ A,

d(xy) = d(x)y + (−1)^{n|x|}xd(y).

This category admits a triangulation with suspension functor (−)[n]. For example, if u ∈ A has degree i, then there is an exact triangle

\xymatrix{A[i] \ar[r]^{u} & A \ar[r] & A ⊕ A[i + n] \ar[r] & A[i + n],}

where the differential on A[j] is (−1)^{j}d and the differential D on A⊕A[i+n] is

D(a, b) = (da + ub, (−1)^{i+n}db).

Let A = k with zero differential. Trivially, the homology of any differential graded A-module is projective, so homology induces an equivalence of categories from D(A) (the derived category of A) to P = Mod(k) by Proposition 2.1.

For the exterior algebra k[x]/(x^{2}) with a unit v in degree 3|x| + n, we use the differential graded algebra constructed in [23]. Let i = |x|, and let a and u be symbols with degrees |a| = 2i + n and |u| = i. Let A = k⟨a, u⟩/I, where I is the two sided ideal generated by the homogeneous elements a^{2} and au + ua + v (here we see why the existence of the unit v is necessary). Define the differential on A by da = u^{2} and du = 0. One can check that this differential is well-defined; if i and n have opposite parity, then v has odd degree, and so char k = 2 by graded commutativity and signs do not matter. If i and n are both odd, then the signs work out independent of the characteristic of k. The differential is not well-defined if i and n are both even and char k ≠ 2; fortunately, if n = 0, char k = 2 is forced (see Proposition 3.9).

As in [23], it is straightforward to check that H_{∗}A ≅ k[x]/(x^{2}), where x is the homology class of u, and for any differential graded A-module M, H_{∗}M is projective. Again we see that homology induces an equivalence of categories from D(A) to P. □

Remark 3.12. According to [27], every simplicial, cofibrantly generated, proper, stable model category with a compact generator P is Quillen equivalent to the module category of a certain endomorphism ring spectrum End(P). This is true in particular for the derived category of a differential graded algebra. In light of the proof of Proposition 3.11, we see that every graded commutative Δ^{1}-ring arises as π_{∗}E for some (not necessarily commutative) semisimple ring spectrum E. Because of Theorem 1.2 (3), however, π_{∗}E being a graded commutative Δ^{1}-ring is not sufficient to conclude that E is semisimple (see Remark 5.2).

Observe that, for any integer n, every Δ^{n}-ring is a Δ_{f}^{n}-ring. For if P admits a triangulation, then thick⟨R⟩ admits a triangulation. Since R must be Noetherian, thick⟨R⟩ = P_{f}. Propositions 3.10 and 3.11 now imply:

Proposition 3.13. Let n ∈ {0, 1}. Every ring R in the conclusion of Proposition 3.9 is a Δ^{n}-ring. Hence, the classes of graded commutative local Δ^{n}- rings and graded commutative local Δ_{f}^{n}-rings coincide.

According to [18, 15.27], a commutative ring is quasi-Frobenius if and only if it is a finite product of local Artinian rings with simple socle. The following proposition allows us to restrict our attention to the local case. Combined with the above characterization of commutative local Δ^{n}-rings for n ∈ {0, 1}, this completes the proofs of Theorems 3.2 and 3.3.

Proposition 3.14. Suppose R ≅ A × B. The category of projective Rmodules admits a triangulation if and only if the categories of projective A-modules and projective B-modules each admit a triangulation. The same is true for finitely generated projective modules.

Proof. This is a consequence of the fact that Mod(R) ≅ Mod(A) × Mod(B). □

As a final note, we broaden the scope of the second statement of Proposition 3.13.

Proposition 3.15. Let n ∈ {0, 1}. The classes of graded commutative Noetherian Δ^{n}-rings and graded commutative Noetherian Δ_{f}^{n}-rings coincide.

Proof. It suffices to check that every commutative Noetherian Δ_{f}^{n}-ring R is a Δ^{n}-ring. Since R must be an IF-ring (Proposition 3.5), it must be a commutative Noetherian self-injective ring ([8, 6.9]). This makes R quasi-Frobenius by definition, and therefore a product of local rings, each of which must be a Δ^{n}-ring by Proposition 3.13 (since each local ring is a Δ_{f}^{n}-ring). Hence, R is a Δ^{n}-ring. □

4. Stable homotopy categories

In this section, we assume that T is a monogenic stable homotopy category with sphere object S (see [14]). Certain conclusions can be drawn with weaker hypotheses; this should be clear in the proofs. In particular, Theorems 4.1 and 4.2 also apply to the derived category D_{E} of right E-module spectra, provided E_{∗} is commutative. We have the following corollary to Theorem 3.2.

Theorem 4.1. Let T be a monogenic stable homotopy category with unit object S. S is semisimple if and only if the following two conditions hold:

(1) π_{∗}S ≅ R_{1} ×⋯×R_{n}, where R_{i} is either a graded field k or an exterior algebra k[x]/(x^{2}) over a graded field containing a unit in degree 3|x|+1 (π_{∗}S is a Δ^{1}-ring).

(2) For every factor ring of π_{∗}S of the form k[x]/(x^{2}), x · π_{∗}C ≠ 0, where C is the cofiber of x · S.

Proof. First, note that every stable homotopy category has a symmetric monoidal structure (for which S is the unit) compatible with the triangulation. This forces R = π_{∗}S to be graded commutative ([14, A.2.1]). Further, we may use the monoidal product to take any element x ∈ π_{∗}S and obtain a map x · X for any X ∈ T . For example, if e ∈ π_{∗}S is idempotent, then e · X is an idempotent endomorphism of X. Since idempotents split, there is a decomposition X ≃ Y ∨Z such that π_{∗}Y ≅ eπ_{∗}X and π_{∗}Z = (1 − e)π_{∗}X (see [14, 1.4.8]).

Suppose S is semisimple. By Proposition 3.1, the category P of projective modules over π_{∗}S admits a triangulation with suspension functor (−)[1]. By Theorem 3.2, condition (1) is satisfied, and π_{∗}S ≅ R_{1} × ⋯ × R_{n}, where R_{i} is either a graded field k or an exterior algebra k[x]/(x^{2}). Since idempotents split in T , S ≅ A_{1}∨ ⋯ ∨A_{n}, where π_{∗}A_{i} = R_{i}. The cofiber C_{i} of x · A_{i} is a summand of C, the cofiber of x · S. Since π_{∗}C_{i} must be a nontrivial free R_{i}-module, x · C_{i} ≠ 0. This proves (2).

Conversely, if condition (1) holds, then R and S admit decompositions as above. We now show that it suffices to assume R is local. Let T_{i} = loc⟨A_{i}⟩; loc⟨A_{i}⟩ is an ideal by [14, 1.4.6]. Every X ∈ T admits a decomposition X ≅ X_{1}∨ ⋯ ∨X_{n}, where X_{i} ∈ T_{i} (take X_{i} = X∧A_{i}). We claim that this decomposition is orthogonal, in that Hom_{T} (T_{i}, T_{j}) = 0 whenever i ≠ j (i.e., every map from an object in T_{i} to an object in T_{j} is trivial). First, observe that if i ≠ j, then any map f : A_{i} ⟶ A_{j} is trivial, as follows. It suffices to show that the induced map f_{∗} : R_{i} ⟶ R_{j} is zero. Let e_{k} ∈ π_{∗}S be the idempotent corresponding to R_{k}. We now have f_{∗}(x) = f_{∗}(e_{i}x) = e_{i}f_{∗}(x) = 0, for all x ∈ R_{i}. So indeed, [A_{i}, A_{j}]_{∗} = 0. Now, [A_{i}, −]_{∗} vanishes on loc⟨A_{j}⟩ since A_{i} is compact; therefore [−, X]_{∗} vanishes on loc⟨A_{i}⟩ for any X ∈ loc⟨A_{j}⟩. In summary, there is an orthogonal decomposition

T = loc⟨S⟩ ≅ T_{1}∨ ⋯ ∨T_{n}.

We now see that π_{∗} is faithful on T if and only if the functors [A_{i}, −]_{∗} are faithful on T_{i} for i = 1, …, n. It therefore suffices to assume that R is a local ring (see Proposition 2.1 (3)).

We must now show that if R = π_{∗}S is local and satisfies conditions (1) and (2), then S is semisimple. By Proposition 2.1 (4), we need only check that π_{∗}X is always projective (or free, since R is local). If R is a graded field, then this is trivially true. If R ≅ k[x]/(x^{2}), then condition (2) tells us that π_{∗}C is free, where C is the cofiber of x · S. Since every map of free k[x]/(x^{2})-modules is the coproduct of a trivial map, an isomorphism, and multiplication by x, it is easy to check that π_{∗}X is free for all X ∈ thick⟨S⟩.

For arbitrary X ∈ T , π_{∗}X is the direct limit of a system of modules of the form π_{∗}X_{α}, where X_{α} ∈ thick⟨S⟩ ([14, 2.3.11]). Since any direct limit of flat modules is flat, π_{∗}X is flat. Over k[x]/(x^{2}), flat implies free. This completes the proof. □

Using localization, we also have the following corollary to Theorem 3.2.

Theorem 4.2. Let T be a monogenic stable homotopy category with unit object S. If π_{∗}S is local or Noetherian, then S is semisimple if and only if it is von Neumann regular. If T is also a Brown category, and S is von Neumann regular, then for any prime ideal 𝔭, (π_{∗}S)_{𝔭} is either a graded field k or an exterior algebra k[x]/(x^{2}) with a unit in degree 3|x| + 1.

The derived category D_{E} is a Brown category if E_{∗} is countable, but there are ring spectra E where D_{E} is not Brown (see [5]).

Proof. Since T is a monogenic stable homotopy category, the unit object S is a compact generator and R = π_{∗}S is commutative.

Certainly, semisimplicity implies von Neumann regularity. Assume S is von Neumann regular. By Proposition 3.1, R is a Δ_{f}^{1}-ring. If R is local or Noetherian, then it is also a Δ^{1}-ring by Proposition 3.13 or 3.15. We now wish to invoke Theorem 4.1. To do so, we must verify condition (2). Arguing as in the proof of Theorem 4.1, it suffices to assume that R ≅ k[x]/(x^{2}). For this ring, condition (2) is implied by the fact that π_{∗}C must be free since S is von Neumann regular. This proves the first implication.

For the second, assume T is a Brown category and S is von Neumann regular. By Theorem 3.2 and Propositions 2.4 and 3.1, R_{𝔭} is a Δ_{f}^{1}-ring. Since it is a graded commutative local ring, it must be Δ^{1}-ring by Proposition 3.13. By Proposition 3.9, R is either a graded field k or an exterior algebra k[x]/(x^{2}) over a field with a unit in degree 3|x| + 1. □

5. The generating hypothesis

Note that D_{E} is almost an example of a stable homotopy category in the sense of [14] (it may not admit a symmetric monoidal product), where E plays the role of the sphere. In any stable homotopy category C with sphere object S, the functor π_{∗}(−) = [S, −]_{∗} is of particular interest, and it is natural to ask how different the homotopy category C is from the algebraic category of graded modules over π_{∗}S. For example, one might like to know whether π_{∗}(−) acts faithfully. More concretely, let T be a triangulated category, and let S ∈ T be a distinguished object. Write π_{∗}(−) for the functor [S, −]_{∗}. We say that T satisfies the global generating hypothesis if π_{∗} is a faithful functor from T to the category of graded right modules over π_{∗}S. We say that T satisfies the strong generating hypothesis if, for any map f : X ⟶ Y with X ∈ thick⟨S⟩ and Y ∈ T , π_{∗}f = 0 implies f = 0. The following proposition follows from Propositions 2.1 and 2.2.

Semisimple ring spectra 241

Proposition 5.1. Let T be a weak stable homotopy category with compact generator S. T satisfies the global generating hypothesis if and only if S is semisimple, and T satisfies the strong generating hypothesis if and only if S is von Neumann regular.

Proof. The first assertion is true by definition of the global generating hypothesis and Proposition 2.1. If S is von Neumann regular, then T trivially satisfies the strong generating hypothesis since thick⟨S⟩ = ⟨S⟩_{f}^{0}. Conversely, if T satisfies the strong generating hypothesis, then one can argue that thick⟨S⟩ ⊆ ⟨S⟩_{f}^{0} (cf. the proof of (4) ⟹ (1) in Proposition 2.1). □

Remark 5.2. As promised in §3, we give an example of a stable homotopy category where the sphere S is not semisimple even though π_{∗}S is a Δ^{1}-ring. Condition (2) in Theorem 4.1 is therefore necessary. Consider the stable module category StMod(k[G]) associated to a finite p-group G and field k of characteristic p. In StMod(k[G]), π_{∗}S = Ĥ(G; k), the Tate cohomology of G. For n ≥ 1, Ĥ(ℤ/(3^{n}); 𝔽_{3}) ≅ 𝔽_{3}[y, y^{−1}][x]/(x^{2}), where |x| = 1 and |y| = 2. However, it is shown in [2] that StMod(𝔽_{3}[ℤ/(3)]) satisfies the global generating hypothesis, though StMod(𝔽_{3}[ℤ/(3^{n})]) does not for n ≥ 2.

References

[1] Anderson, Frank W.; Fuller, Kent R. Rings and categories of modules. Second edition. Graduate Texts in Mathematics, 13. Springer-Verlag, New York, 1992. MR1245487 (94i:16001), Zbl 0765.16001.

[2] Benson, David J.; Chebolu, Sunil K.; Christensen, J. Daniel; Mináč, Ján. The generating hypothesis for the stable module category of a p-group. J. Algebra 310 (2007), 428–433. MR2307802 (2007k:16011), Zbl 1120.20002.

[3] Bökstedt, Marcel; Neeman, Amnon. Homotopy limits in triangulated categories. Compositio Math. 86 (1993), 209–234. MR1214458 (94f:18008), Zbl 0802.18008.

[4] Christensen, J. Daniel. Ideals in triangulated categories: phantoms, ghosts and skeleta. Adv. Math. 136 (1998), 284–339. MR1626856 (99g:18007), Zbl 0928.55010.

[5] Christensen, J. Daniel; Keller, Bernhard; Neeman, Amnon. Failure of Brown representability in derived categories. Topology 40 (2001), 1339–1361. MR1867248 (2003i:16013), Zbl 0997.18007.

[6] Colby, R. R. Rings which have flat injective modules. J. Algebra 35 (1975), 239–252. MR0376763 (51 #12938),Zbl 0306.16015.

[7] Elmendorf, A. D.; Kriz, I.; Mandell, M. A.; May, J. P. Rings, modules, and algebras in stable homotopy theory. With an appendix by M.Cole. Mathematical Surveys and Monographs, 47. American Mathematical Society, Providence, RI, 1997. MR1417719 (97h:55006), Zbl 0894.55001.

[8] Faith, Carl. Rings and things and a fine array of twentieth century associative algebra. Mathematical Surveys and Monographs, 65. American Mathematical Society, Providence, RI, 1999. MR1657671 (99j:01015), Zbl 0917.16001.

[9] Freyd, Peter. Stable homotopy. 1966 Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), pp. 121–172. Springer, New York, 1966. MR0211399 (35 #2280), Zbl 0195.52901.

242 Mark Hovey and Keir Lockridge

[10] Heller, Alex. Stable homotopy categories. Bull. Amer. Math. Soc. 74 (1968), 28– 63. MR0224090 (36 #7137), Zbl 0177.25605.

[11] Hopkins, Michael J.; Smith, Jeffrey H. Nilpotence and stable homotopy theory. II. Ann. of Math. (2) 148 (1998), 1–49. MR1652975 (99h:55009), Zbl 0924.55010.

[12] Hovey, Mark; Lockridge, Keir. The ghost dimension of a ring. Proc. Amer. Math. Soc. 137 (2009), 1907–1913. MR2480270.

[13] Hovey, Mark; Lockridge, Keir; Puninski, Gena. The generating hypothesis in the derived category of a ring. Math. Z. 256 (2007), 789–800. MR2308891 (2008b:18017), Zbl 1137.16016.

[14] Hovey, Mark; Palmieri, John H.; Strickland, Neil P. Axiomatic stable homotopy theory. Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114 pp. MR1388895 (98a:55017), Zbl 0881.55001.

[15] Hovey, Mark; Shipley, Brooke; Smith, Jeffrey H. Symmetric spectra. J. Amer. Math. Soc. (2000) 13 (2000), 149–208. MR1695653 (2000h:55016), Zbl 0931.55006.

[16] Hovey, Mark; Strickland, Neil P. Morava K-theories and localisation. Mem. Amer. Math. Soc. 139 (1999), no. 666, viii+100 pp. MR1601906 (99b:55017), Zbl 0929.55010.

[17] Lam, T. Y. A first course in noncommutative rings. Second edition. Graduate Texts in Mathematics, 131. Springer-Verlag, New York, 2001. MR1838439 (2002c:16001), Zbl 0980.16001.

[18] Lam, T. Y. Lectures on modules and rings, Graduate Texts in Mathematics, 181. Springer-Verlag, New York, 1999. MR1653294 (99i:16001), Zbl 0911.16001.

[19] Lockridge, Keir H. The generating hypothesis in the derived category of Rmodules. J. Pure Appl. Algebra 208 (2007), 485–495. MR2277690 (2007i:55009), Zbl 1112.55011.

[20] Mandell, M. A.; May, J. P. Equivariant orthogonal spectra and S-modules. Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108 pp. MR1922205 (2003i:55012), Zbl 1025.55002.

[21] Mandell, M. A.; May, J. P.; Schwede, S.; Shipley, B. Model categories of diagram spectra. Proc. London Math. Soc. (3) 82 (2001), 441–512. MR1806878 (2001k:55025), Zbl 1017.55004.

[22] Matsumura, Hideyuki. Commutative algebra. Second edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR0575344 (82i:13003), Zbl 0441.13001.

[23] Muro, Fernando; Schwede, Stefan; Strickland, Neil. Triangulated categories without models. Invent. Math. 170 (2007), 231–241.MR2342636 (2008g:18016), Zbl 1125.18009.

[24] Neeman, Amnon. Triangulated categories. Annals of Mathematics Studies, 148. Princeton University Press, Princeton, NJ, 2001. MR1812507 (2001k:18010), Zbl 0974.18008.

[25] Rouquier, Raphaël. Dimensions of triangulated categories. J. K-Theory 1 (2008), 193–256. MR2434186, Zbl pre05348081.

[26] Schwede, Stefan. Morita theory in abelian, derived and stable model categories. Structured ring spectra, 33–86, London Math. Soc. Lecture Note Ser., 315. Cambridge Univ. Press, Cambridge, 2004. MR2122154 (2005m:18015), Zbl 1088.16010.

[27] Schwede, Stefan; Shipley, Brooke. Stable model categories are categories of modules. Topology 42 (2003), 103–153. MR1928647 (2003g:55034), Zbl 1013.55005.

Semisimple ring spectra 243

[28] Weibel, Charles A. An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994. MR1269324 (95f:18001), Zbl 0797.18001.

Department of Mathematics, Wesleyan University, Middletown, CT 06459 hovey@member.ams.org

Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109 keir@alumni.rice.edu

This paper is available via http://nyjm.albany.edu/j/2009/15-12.html.