DG algebras with exterior homology

We study differential graded algebras whose homology is an exterior algebra over a commutative ring R on a generator of degree n, and also certain types of differential modules over these DGAs. We obtain a complete classification when R is the integers, or the prime field of characteristic p>0, and n is greater than or equal to -1. The examples are unexpectedly interesting.


Introduction
We mainly study differential graded algebras (DGAs) whose homology is an exterior algebra over either F p or Z on a class of degree −1, as well as (left) modules over these DGAs whose homology is Z/p in degree 0. In the case of an exterior algebra over F p , we find many DGAs, each having one such module; in the case of an exterior algebra over Z, one DGA having many such modules. In both cases, the enumeration of possibilities involves complete discrete valuation rings with residue field F p . There are remarks about other types of DGAs in Section 6.
In more detail, a DGA is a chain complex A of abelian groups together with a multiplication map A ⊗ A → A that is both unital and associative. A morphism f : A → B of DGAs is a map of chain complexes which respects multiplication and unit; f is said to be an equivalence if it induces isomorphisms H i A ∼ = H i B, i ∈ Z. The homology H * A is a graded ring, and A is said to be of type β S , for S a commutative ring, if H * A is an exterior algebra over S on a class of degree −1. The notation β S is meant to suggest that A captures a type of Bockstein operation. In topology, the 'Bockstein' is an operation β on mod p homology, of degree −1 and square 0, which arises from a generator of Ext 1 Z (Z/p, Z/p). A module over A is a chain complex X together with an action map A ⊗ X → X with the usual unital and associativity properties. Morphisms and equivalences between modules are defined in the evident way. The module X is of type (M, 0) for an abelian group M if there are isomorphisms of abelian groups We are particularly interested in DGAs of type β S for S = F p or S = Z, and modules over these DGAs of type (Z/p, 0). 3 is the profusion of modules, since there are many pairs (O, π) as above. For instance, if π ∈ Zp is divisible by p but not by p 2 , then (Zp, π) is such a pair. Since Zp has no nontrivial automorphisms, these pairs are distinct for different choices of π.
The object A of Theorem 1.3 can be taken to be the DGA F that contains copies of Z in degrees 0 and −1, is trivial elsewhere, and has zero differential. In Remark 5.5, there is an explicit description of how to pass from a pair (O, π) to a module over F. Remark 1.5. If A is a connective DGA, that is, H i A = 0 for i < 0, and X is a module over A such that H i X = 0 for i = 0, then X is determined up to equivalence by the isomorphism class of H 0 X as an (ordinary) module over H 0 A; see, for instance, [5, 3.9]. This is in strong contrast to what happens in the nonconnective setting of Theorem 1.3.

Generalizations.
We have some remarks in Section 6 about DGAs with other types of exterior algebra homology.
The arguments below can be interpreted in the setting of stable homotopy theory, and they lead to a classification of associative ring spectra of type β Z or β Fp (appropriately interpreted) and of module spectra over these ring spectra of type (Z/p, 0). No new examples come up; all of these ring spectra and module spectra are obtained in a standard way [9] from DGAs.

DG-objects, equivalences, and formality
As suggested above, a map between differential graded (DG) objects of any kind is said to be an equivalence if it induces an isomorphism on homology groups. Two objects are equivalent if they are related by a zig-zag ←→← · · · → of equivalences. A DG-object X is formal if it is equivalent to a DG-object Y of the same kind that has zero differentials. Of course, in this case the graded constituent Y i of Y must be isomorphic to H i X. For instance (cf. Remark 1.5), if A is a connective DGA and X is a module over A such that H i X vanishes except for a single value of i, then X is formal as an A-module.
If A is a DGA such that H i A = 0 for i = 0, then A is formal as a DGA. To see this, let A ⊂ A be the subcomplex given by Then A is a DGA, and there is a zig-zag of DGA-equivalences where the object on the right is a ring, treated as a DGA concentrated in degree 0. If A is a DGA such that H * A is a polynomial algebra Z[x] on a class x of (arbitrary) degree n, then A is also formal: choose a cycle χ ∈ A n representing x and construct an equivalence On the other hand, if S is a commutative ring other than Z and , then it is not necessarily the case that A is formal, even for S = F p (see Section 6 for examples); the above argument applies only if A itself is an algebra over S, or at least equivalent to an algebra over S.

Relationship to Moore-Koszul duality
Suppose that k is some chosen field. Say that a DGA A over k is admissible if H i A is finitedimensional over k for all i and H 0 A ∼ = k. Applying duality over k to the bar constructions in [6] produces bijections between equivalence classes of the admissible DGAs indicated below.
In topology, for instance, this gives the relationship between the cochain algebra A of a simply connected space X and the chain algebra B of the loop space ΩX. Our technique is to push the boundaries of the above Moore-Koszul duality construction. In proving Theorem 1.1 (Sections 2 and 3), we start with a DGA A of type β Fp , more or less construct by hand an action of A on (something equivalent to) Z/p, and show that the DGA B = End A (Z/p) has its homology concentrated in degree 0, and so is essentially an ordinary ring O (Subsection 1.1). It turns out that O is a complete discrete valuation ring, and that it determines A up to equivalence via the formula Note that it is not necessarily the case that A is a DGA over F p . In Section 4, the same technique succeeds in classifying DGAs of type β Z . The proof in Section 5 of the second part of Theorem 1.3 relies on ideas from [4] which fit modules into a type of Moore-Koszul duality setting.

Notation and terminology
We work in the context of [4,5]. Rings are tacitly identified with DGAs concentrated in degree 0 (Subsection 1.1). An ordinary module M over a ring R is similarly tacitly identified with the chain complex X over R with X 0 = M and X i = 0 for i = 0. Hom is the derived homomorphism complex and ⊗ is the derived tensor product. If X is a module over the DGA (or ring) A, then End A (X) denotes the DGA obtained by taking a projective model for X and forming the usual DGA of endomorphisms of this model. See [10, 2.7.4], but re-index to conform to our convention that differentials always reduce degree by 1. (Up to equivalence, the DGA End A (X) depends only on the equivalence types of A and of X; this can be proved for instance with the bimodule argument of [7, 3.7.6].) If R is a ring and M is an ordinary R-module, then there are isomorphisms for the ordinary endomorphism ring of M over R. If X is a chain complex or graded abelian group, then we write Σ i X for its i-fold shift: . For the sake of clarity, we attempt as much as reasonably possible to make a notational distinction between the field F p and the abelian group Z/p = Z/pZ.

A module of type (Z/p, 0)
In this section, A is a DGA of type β Fp . We construct a module X over A of type (Z/p, 0) and show that for any module Y over A of type (Z/p, 0) there is an equivalence e Y : X → Y . However, the equivalence e Y is not unique in any sense, even up to homotopy. See [5, 3.3, 3.9] for more general constructions of this type.
The construction is inductive. Suppose that Z is a module over A with and let JZ be the mapping cone of κ. There is a natural map Z → JZ that is an isomorphism on H 0 and trivial on the other homology groups (in particular H −1 Z → H −1 JZ is zero), and so the homology exact sequence of the triangle shows that H * JZ again vanishes except for copies of Z/p in degrees 0 and −1. Starting with Z = A, iterate the process to obtain a sequence of chain complexes and chain maps, and let X = hocolim k J k A. (In this case, the homotopy colimit can be taken to be a colimit, that is, the ascending union.) It is immediate that X has type (Z/p, 0). Suppose that Y is an arbitrary module of type (Z/p, 0). There is certainly a map A → Y that induces an isomorphism on H 0 , so to construct an equivalence X → Y it is enough to show that if Z satisfies (2.1) and f : Z → Y induces an isomorphism on H 0 , then f extends to f : JZ → Y . (By induction, this will guarantee that the map A → Y extends to a map X = hocolim k J k A → Y .) The map f extends to f if and only if the composite

Exterior algebras over F p
In this section, we prove Theorem 1.1. The proof depends on two lemmas. Obtaining a ring from a DGA. Suppose that A is a DGA of type β Fp . According to Section 2, up to equivalence there is a unique module X over A of type (Z/p, 0). Let E = End A (X) be the derived endomorphism algebra of X. Lemma 3.1. Let A, X, and E be as above.
(1) The homology group H i E vanishes for i = 0, and the ring H 0 E is a complete discrete valuation ring with residue field F p .
(2) The natural map A → End E (X) is an equivalence.
Obtaining a DGA from a ring. Suppose that O is a complete discrete valuation ring with residue class field F p . Let X denote Z/p with the unique possible O-module structure, and let A = End O (X) be the derived endomorphism algebra of X.  For minor efficiency reasons, we first prove Lemma 3.2 and then Lemma 3.1.

Proof of Lemma 3.2
For part (1), observe that as usual there are isomorphisms By inspection, then, Ext i O (Z/p, Z/p) vanishes unless i = 0 or i = 1, and in these two exceptional cases the group is isomorphic to Z/p.
Since O/m ∼ = F p is a field and hence a regular ring, Lemma 3.2(2) is [5, 4.20].

Remark 3.3.
If O is as in Lemma 3.2, an explicit model for End O (Z/p) can be derived from (3.1) as follows: This is a DGA which is nonzero only in degrees 1, 0, −1; the notation -denotes the free O-module on the enclosed generators. From a multiplicative point of view, the DGA is as indicated a graded form of the ring of 2 × 2 matrices over O.
Proof of Lemma 3.1(1). Let Λ denote the graded algebra H * A and x ∈ Λ −1 an additive generator. Write Z/p for the (ordinary) graded Λ-module Λ/ x ∼ = H * X. For general reasons (see Subsection 3.2) there is a left-half plane Eilenberg-Moore spectral sequence (Note that Ext here is computed in the category of graded modules over a graded ring.) This is a conditionally convergent spectral sequence of bigraded algebras with differentials d r : E r i,j −→ E r i−r,j+r−1 , and it abuts to the graded algebra H * E. There is a free resolution of Z/p over Λ which leads by calculation to the conclusion that the bigraded algebra Ext i Λ (Σ j Z/p, Z/p) is a polynomial algebra on the extension class in Ext 1 Λ (ΣZ/p, Z/p). It follows that E 2 i,j vanishes in the above spectral sequence for i = −j, the spectral sequence collapses, and E ∞ = E 2 is concentrated in total degree 0. Hence, H i E vanishes for i = 0, and H 0 E is a ring O with a decreasing sequence of ideals Let π ∈ m 1 ⊂ O be an element that projects to a generator of m 1 /m 2 ∼ = Z/p, and map Z[s] to O by sending s to π. It is easy to argue by induction on k that the composite Z Proof of Lemma 3.1(2). By Subsection 3.1 and Lemma 3.1(1), End E (X) is a DGA of type β Fp . It is thus enough to show that the map H −1 A → H −1 End E (X) is an isomorphism, or even that this map is nonzero. We will use the notation of Section 2. Note that the mapping cone of the A-module map : A → JA is again A, and that the mapping cone of any nontrivial map A → X is again equivalent to X, this last by a homology calculation and the uniqueness result of Section 2. Consider the following diagram, in which the rows are exact triangles and the middle vertical map is provided by Section 2.
Here a denotes (right) multiplication by a generator of H −1 A. (In the language of [5], the lower row here shows that X is proxy-small over A, and Lemma 3.1(2) now follows fairly directly from [5,Proof of 4.10]. For convenience, we continue with a direct argument.) After a suitable identification of the mapping cone of with X, a homology calculation gives that the right vertical map is homotopic to . The right lower map is entitled to be labeled π, as in Subsection 3.1, because the homology of its mapping cone (namely H * ΣA = ΣH * A) evidently represents a nonzero element of Note that the element π of O is determined only up to multiplication by a unit, and this is reflected above in the fact that there are various ways to identify the mapping cone of with X.
Applying Hom A (-, X) to (3.2) gives The map * is surjective on homology, since by the argument of Section 2 any A-map A → X extends over to a map X → X. Let M be the O-module H 0 Hom A (JA, X). Applying H 0 to the solid arrows in diagram (3.3) gives a diagram of exact sequences The diagram implies that as an O-module, M is O/(π 2 ). In particular, the extension on the top line is nontrivial over O, which, by backing up the exact triangle as in (3.3), implies that a * : Σ −1 X → X represents a nonzero element of H −1 End E (X). But, by construction, a * is given by left multiplication with a generator of H −1 A.

Eilenberg-Moore spectral sequence
If A is a DGA and X, Y are modules over A, then there is an Eilenberg-Moore spectral sequence In a homotopy context, this can be constructed in precisely the same way as an Adams spectral sequence. In an algebraic context, it is constructed by inductively building an exact sequence of A-modules such that (i) each F (i) is a sum of shifts of copies of A and (ii) applying H * to (3.4) produces a free resolution of H * X over H * A.
See, for instance, [1, 9.11]. The totalization tF of the double complex F is then a projective (or cofibrant) model for X; filtering tF by {tF ( n)} n and applying Hom A (-, Y ) gives a filtration of Hom A (X, Y ) that yields the spectral sequence.

Exterior algebras over Z
In this section, we prove the first claim of Theorem 1.3. To be specific, we show that any DGA A of type β Z is equivalent to the formal DGA F given by the following chain complex concentrated in degrees 0, −1.
The multiplication on F is the only possible one consistent with the requirement that 1 ∈ Z = F 0 act as a unit.
We will only sketch the line of reasoning, since it is similar to that in sections Sections 2 and 3, although the conclusion is very different. Along the lines of Section 2 there exists a module X of type (Z, 0) over A, and this is unique up to noncanonical equivalence. Let E = End A (X). The argument in the proof of Lemma 3.1 (1) shows that H i E vanishes for i = 0, and that H 0 E is a ring R with a decreasing sequence of ideals In this section, we complete the proof of Theorem 1.3. The uniqueness statement for A was proved in Section 4, so we only have to handle the classification of modules of type (Z/p, 0). Let R = Z[s] and let R act on Z via Z ∼ = R/(s). Clearly, Ext i R (Z, Z) is Z for i = 0, 1 and zero otherwise, so that E = End R (Z) is a DGA of type β Z . By uniqueness, we may as well take A = E; the following proposition explains why this is useful.
Say that a chain complex N of R-modules is s-torsion if for each i ∈ Z and each x ∈ H i N there is an integer k(x) > 0 such that s k(x) x = 0. Remark 5.2. Here E acts on Hom R (Z, N) through its action on Z. The inverse to this bijection assigns to a right E-module X the derived tensor product X ⊗ E Z.
It is clear from Section 4 that E ∼ F is equivalent as a DGA to its opposite algebra, so we can pass over the distinction between right and left E-modules. The question of studying

Other exterior algebras
Suppose that R is a commutative ring, and say that a DGA A is of type β R (n) if H * A is an exterior algebra over R on a class of degree n. In this section, we briefly consider the problem of classifying such DGAs if n = −1. If n = 0, then A is formal (Subsection 1.1) and hence determined up to equivalence by the ring H 0 A, so we may as well also assume n = 0.
Say that a DGA is of type P R (n) if H * A is isomorphic to a polynomial algebra over R on a class of degree n. Proposition 6.1. If n / ∈ {0, −1}, then there is a natural bijection between equivalence classes of DGAs of type β R (n) and equivalence classes of DGAs of type P R (−n − 1).
Proof (Sketch). If A is of type β R (n), then the inductive technique of Section 2 produces a left module X A over A of type (R, 0), that is, a module X A such that H 0 (X A ) is a free (ordinary) module of rank 1 over R = H 0 A and H i (X A ) = 0 for i = 0. If B is of type P R (n), then it is even easier to produce a right module X B over B of type (R, 0): just take the mapping cone of f : Σ n B → B, where f represents left multiplication by a generator of H n B. In both cases, the modules are unique up to possibly noncanonical equivalence. Along the lines of Section 3 (cf. Subsection 1.2), calculating with appropriate collapsing Eilenberg-Moore spectral sequences now gives the desired bijection.
o o Remark 6.2. It is clear from Theorem 1.1 that Proposition 6.1 fails for n = −1 and R = F p , essentially because if A is of type β Fp (−1), then the nonvanishing entries of the Eilenberg-Moore spectral sequence for H * End A (X A ) accumulate in degree 0. This accumulation creates extension possibilities that allow for a profusion of complete discrete valuation rings in the abutment. Similarly, Proposition 6.1 fails for n = 0 and R = F p , because if B is of type P Fp (−1), then the Eilenberg-Moore spectral sequence for H * End B (X B ) accumulates in degree 0. In this case, though, the accumulation has consequences which are less drastic, because up to isomorphism there are only two possibilities for a ring R with an ideal I such that I 2 = 0 and such that the associated graded ring {R/I, I} is an exterior algebra on one generator over F p . The conclusion is that up to equivalence there are only two DGAs of type P Fp (−1), one corresponding to the true exterior algebra F p [t]/t 2 , and the other to the fake exterior algebra Z/p 2 . 6.1. DGAs of type β Z (n), all n Up to equivalence, there is only one of these for each n. The case n = 0 is trivial (Subsection 1.1), while n = −1 is Theorem 1.3. By Proposition 6.1, for other n these correspond to DGAs of type P Z (n), but there is only one of these for each n, because they are all formal (Subsection 1.1).
6.2. DGAs of type β Fp (n), n 0 As usual, the case n = 0 is trivial: up to equivalence there is only one example. We sketch an argument that if n > 0 is odd, then there is only one example, while if n > 0 is even, then there are two. In [3], Dugger and Shipley describe a Postnikov approach to constructing a connective (Remark 1.5) DGA A; the technique involves starting with the ring H 0 A (considered as a DGA with trivial higher homology) and attaching one homology group at a time, working from low dimensions to high. If A is of type β Fp (n), then there is only a single homology group