1.1 Previous results

Using a Koszul duality argument, Dwyer, Greenlees, and Iyengar [Reference Dwyer, Greenlees and IyengarDGI13] prove the following.

• – quasi-isomorphism classes of DGAs with homology $\Lambda_{\mathbb{F}_p}[x_{k}]$ ; and

• – quasi-isomorphism classes of DGAs with homology $\mathbb{F}_p[x_{-k-1}]$ .

The same also holds for topological equivalence classes in place of quasi-isomorphism classes.

Earlier, in [Reference Dugger and ShipleyDS07, Example 3.15], Dugger and Shipley have classified DGAs with homology $\Lambda_{\mathbb{F}_p}[x_k]$ for $k\gt0$ . By viewing them as square-zero extensions of $\mathbb{F}_p$ by $\mathbb{F}_p$ , they deduce that there is a unique such DGA if $k\gt0$ is odd and that there are two quasi-isomorphism classes if $k\gt0$ is even, and these two are topologically equivalent if $k\geq 2p-2$ . The case $k\lt0$ is more complicated, as such DGAs may not be given by square-zero extensions. Nevertheless, the case $k=-1$ is the main result of the work of Dwyer, Greenlees, and Iyengar [Reference Dwyer, Greenlees and IyengarDGI13]. Again using a Koszul duality argument, they prove the following.

• – equivalence classes of DGAs with homology $\Lambda_{\mathbb{F}_p}[x_{-1}]$ ; and

• – isomorphism classes of complete discrete valuation rings (CDVRs) with residue field $\mathbb{F}_p$ .

Here, equivalence refers to quasi-isomorphism or topological equivalence.

It follows that there are countably infinitely many DGAs with homology $\Lambda_{\mathbb{F}_p}[x_{-1}]$ . For the classification of DGAs with homology $\Lambda_{\mathbb{F}_p}[x_{k}]$ , the remaining case is therefore the case $k\lt-1$ . Equivalently, what remains is the classification of DGAs with polynomial homology $\mathbb{F}_p[x_{k}]$ for $k\gt0$ and the authors of [Reference Dwyer, Greenlees and IyengarDGI13] leave this problem open.

This classification question has been the subject of earlier work of the first author [Reference BayndrBay21], in which the main result states that there is a unique non-formal DGA with homology $\mathbb{F}_p[x_{2p-2}]$ and a non-formal $(2p-2)$ -Postnikov section, providing the first example of a non-formal DGA with polynomial homology $\mathbb{F}_p[x_k]$ with $k\gt0$ in the literature. Around the same time, Patchkoria has also constructed a non-formal DGA given by the DGA quotient $\mathbb{Z} /\!\!/ p$ , whose homology is $\mathbb{F}_p[x_2]$ . Incidentally, $\mathbb{Z}/\!\!/ p$ also appears in [Reference Land and TammeLT26, Example 4.31] as the $\odot$ -ring (first introduced in [Reference Land and TammeLT19]) associated to the Milnor square describing ${\mathbb{Z} \times_{\mathbb{F}_p} \mathbb{Z}}$ . In [Reference Davis, Frank and PatchkoriaDFP25], the authors compute the negative cyclic homology of $\mathbb{Z} /\!\!/ p$ and in his dissertation, Frank has also studied the classification of DGAs with polynomial homology and has proved that $\mathbb{Z} /\!\!/ p$ is not even topologically formal for $p\gt2$ , whereas the second author and Tamme [Reference Land and TammeLT26, Remark 4.33] have proved that $\mathbb{Z} /\!\!/ 2$ is in fact topologically formal.

1.2 Classification results

Let us now summarize our main results in regard to the classification problem alluded to above. First, we discuss under what circumstances we can show that a DGA is formal. We again stress that maps of DGAs always refer to derived maps, that is, maps in the $\infty$ -category of DGAs as described above, and that the term quasi-isomorphism is used for an equivalence in the $\infty$ -category of DGAs. In what follows, let $m\gt1$ be an integer and p be a prime.

1. (1) Assume that the homology of A is $\mathbb{Z}/m[x_{2n}]$ . If there is a map $\mathbb{Z}/m \to A$ of DGAs, then A is formal.

2. (2) Assume that the homology of A is $\mathbb{F}_p[x_{2n}]$ . If there is a map $\mathbb{F}_p \to A$ of ring spectra, then A is topologically formal.

Moreover, if $\tau_{\leq 2p-4}A$ is topologically formal, then there exists a map $\mathbb{F}_p \to A$ of ring spectra. In particular, A is topologically formal if and only if $\tau_{\leq 2p-4} A$ is. As a result, A is topologically formal if $n \geq p-1$ .

The final statement of Theorem A in fact generalizes to the odd-degree-generator case.

This fully resolves the topological classification of DGAs with (truncated) polynomial homology over $\mathbb{F}_p$ in a sufficiently large-degree generator, and in particular with exterior homology over $\mathbb{F}_p$ in a sufficiently small-degree generator. More precisely, equivalent to Theorem B is the statement that every DGA with homology $\Lambda_{\mathbb{F}_p}[x_n]$ is topologically formal whenever $n\lt -(2p-2)$ (Corollary 4.3).

Theorems A and B above say nothing about the existence (and uniqueness) of non-formal DGAs with polynomial homology. Our next result remedies this. To state it, we need to digress briefly. For $m\gt1$ and $n\gt0$ , in the main body of the text we construct canonical DGAs $S_{2n}^m$ in an inductive manner (over n) whose homology is $\mathbb{Z}/m[x_{2n}]$ . These DGAs are in fact also essential in the proof of Theorems A and B. Moreover, for these DGAs, we show that it is possible to adjoin suitable roots of the polynomial generator; we explain this in some more detail in § 1.3 below. In particular, for each $l\geq 1$ , we construct DGAs $S_{2nl}^m[\sqrt[l]{x_{2nl}}]$ whose homology is isomorphic to $\mathbb{Z}/m[x_{2n}]$ (for details, see Construction 2.33).

1. (1) The collection $\{S_{2nl}^m [\sqrt[l]{x_{2nl}}]\}_{l\geq 1}$ consists of pairwise non-quasi-isomorphic DGAs. In particular, up to quasi-isomorphism, there are infinitely many pairwise distinct DGAs with homology $\mathbb{Z}/m[x_{2n}]$ .

2. (2) For $l \geq ({p-1})/{n}$ , the DGA $S_{2nl}^p [\sqrt[l]{x_{2nl}}]$ is topologically equivalent to $\mathbb{F}_p[x_{2n}]$ , i.e., is topologically formal.

Consequently, for $n\gt0$ , we also obtain infinitely many pairwise distinct DGAs with homology $\Lambda_{\mathbb{F}_p}[x_{-2n-1}]$ , where all but finitely many are topologically equivalent to $\Lambda_{\mathbb{F}_p}[x_{-2n-1}]$ . To the best of our knowledge, these are the first examples of infinitely many pairwise non-quasi-isomorphic DGAs that are all topologically equivalent, i.e., infinitely many pairwise distinct $\mathbb{Z}$ -algebra structures on a single ring spectrum. We will later also leverage this result to construct exotic DG enhancements of certain triangulated categories (see § 1.4 in this introduction).

Remark 1.4. Let us remark on the case where we are given a DGA A with homology $\mathbb{Z}/p^s[x_{2n}]$ for $n\gt0$ and $s\geq 3$ or $s\geq 2 $ for p odd. By the (surprising) recent results of Burklund [Reference BurklundBur22], $\mathbb{S}/p^s$ is a ring spectrum (where $\mathbb{S}$ denotes the sphere spectrum). If there is a map of ring spectra $\mathbb{Z}/p^s \to A$ , then the adjoint of the canonical composite

\begin{align*}\mathbb{S}/p^s \to \mathbb{Z}/p^s \to A\end{align*}

is a DGA map $\mathbb{Z}/p^s\to A$ . From Theorem A, we deduce that A is topologically formal if and only if it is formal. More generally, we prove (Proposition 3.14).

(1) The collection $\{S_{2nl}^{p^s} [\sqrt[l]{x_{2nl}}]\}_{l\geq 1}$ consists of pairwise non-topologically equivalent DGAs with homology $\mathbb{Z}/p^s[x_{2n}]$ .

In particular, there are infinitely many topological equivalence classes of DGAs with homology $\mathbb{Z}/p^s[x_{2n}]$ . Since $\mathrm{MU}/m$ is an MU-algebra [Reference AngeltveitAng08] for each $m\gt1$ , the same arguments prove the following (Proposition 3.14).

(2) The collection $\{S_{2nl}^m [\sqrt[l]{x_{2nl}}]\}_{l\geq 1}$ consists of DGAs that are pairwise non-MU-algebra-equivalent.

In addition, we show that for $l,l'\lt ({p-1})/{n}$ , the DGAs $S_{2nl}^p[\sqrt[l]{x_{2nl}}]$ and $S_{2nl'}^p[\sqrt[l']{x_{2nl'}}]$ are topologically equivalent if and only if $l=l'$ and neither are topologically formal (Proposition 3.11), showing that part (2) of Theorem C above is sharp.

The following is still open.

Also, we do not know whether every DGA with homology $\mathbb{Z}/p^s[x_{2k}]$ is equivalent to one of the form $S_{2nl}^{p^s}[\sqrt[l]{x_{2nl}}]$ . In particular, Theorem C is not a complete classification.

Question 1.6. Is every DGA with homology $\mathbb{Z}/p^s[x_{2n}]$ ( $n\gt0$ ) quasi-isomorphic to a DGA of the form $S_{2nl}^{p^s}[\sqrt[l]{x_{2nl}}]$ for some l?

However, there is one situation in which we can say something in this direction.

This result generalizes the previously mentioned earlier work of the first author [Reference BayndrBay21], which has treated the case $n=p-1$ .

1.3 Proof ingredients

We now spell out the basic constructions and ideas that go into the proofs of the above results. After this, we end the introduction with a number of applications.

As indicated earlier, our results build on the construction of certain DGAs $S_{2n}^m$ whose homology is $\mathbb{Z}/m[x_{2n}]$ . We briefly explain this construction. For the rest of this paper, we denote the homology of a DGA A by $\pi_*(A)$ , as it is also the homotopy of the underlying ring spectrum.

We then set $S_2^m := \mathbb{Z}/\!\!/ m$ , which has homology $\mathbb{Z}/m[x_2]$ as indicated above, and inductively define

\begin{align*}S_{2n}^{m} := S_{2n-2}^m/\!\!/ x_{2n-2}.\end{align*}

Part of this definition is, of course, to show that indeed $\pi_*(S_{2n}^m) =\mathbb{Z}/m[x_{2n}]$ . We then obtain a sequence of DGAs

\begin{align*} S_2^m \to S_4^m \to \cdots \to S_{2n}^m \to \cdots \end{align*}

whose colimit is $\mathbb{Z}/m$ . By construction, we therefore obtain an odd cell decomposition of each $S_{2n}^m$ and consequently also of $\operatorname{colim}_nS_{2n}^m \simeq \mathbb{Z}/m$ (the reader may want to contrast this with the Hopkins–Mahowald theorem, which states that $\mathbb{F}_p$ is obtained from $\mathbb{S}$ by attaching a single $\mathbb{E}_2$ -cell in dimension 1). This gives obstructions to the existence of maps $S_{2n}^m \to A$ for another DGA A. If $\pi_*(A)$ is concentrated in even degrees, this also implies that if there is a map $S_{2n}^m\to A$ (or $\mathbb{Z}/m\to A$ ), then it is unique up to homotopy. We then show that if A has homology $\mathbb{Z}/m[x_{2n}]/x_{2n}^k$ , there is a map $S_{2l}^m \to A$ carrying the generator $x_{2l}$ to a non-trivial element in $\pi_{2l}(A)$ if and only if l is the smallest integer such that $\tau_{\leq 2l}(A)$ is not formal (Theorem 2.26). Theorem D is an immediate consequence of this result.

For the proof of Theorem A(1), given a DGA map $\mathbb{Z}/m\to A$ , we would like to extend it to an equivalence $\mathbb{Z}/m\otimes_{\mathbb{Z}}\mathbb{Z}[x_{2n}]\xrightarrow{\simeq} A$ . If these were classical associative rings, such an extension would exist if there was a map $\mathbb{Z}[x_{2n}]\to A$ whose image commuted with the image of $\mathbb{Z}/m$ in A (which is, of course, automatic). This idea also applies to DGAs (and ring spectra) through the theory of centralizers à la Lurie; in the case at hand, the centralizers identify with (topological) Hochschild cohomology. We then compute enough about the centralizer of the given map $\mathbb{Z}/m\to A$ to run the above argument and obtain Theorem A(1).

Moreover, we compute enough about the Hochschild cohomology of $S_{2n}^m$ (i.e., the centralizer of the identity of $S_{2n}^m$ ) as to construct a $\mathbb{Z}[X_{2n}]$ -algebra structure on $S_{2n}^m$ where $X_{2n}$ acts via $x_{2n}$ and then define, for $l\geq 1$ ,

\begin{align*}S_{2nl}^m[\sqrt[l]{x_{2nl}}] := S_{2nl}^m \otimes_{\mathbb{Z}[X_{2nl}]} \mathbb{Z}[X_{2n}]\end{align*}

following [Reference Ausoni, Bayndr and MoulinosABM26], where $\mathbb{Z}[X_{2nl}] \to \mathbb{Z}[X_{2n}]$ carries $X_{2nl}$ to $X_{2n}^l$ ; this is then a DGA with homology $\mathbb{Z}/m[x_{2n}]$ . By considering the maps that they receive from $S_{2nl}^m$ , we deduce $S_{2nl}^m[\sqrt[l]{x_{2nl}}] \not \simeq S_{2nl'}^m[\sqrt[l']{x_{2nl'}}]$ for $l \neq l'$ , which gives Theorem C(1).

We then show that $p=0$ in the Hochschild cohomology of $S_{2p-2}^p$ and deduce from the Hopkins–Mahowald theorem mentioned earlier that there is a map of ring spectra $\mathbb{F}_p \to S_{2p-2}^p$ . By our earlier results, this implies the topological formality of $S_{2p-2}^p$ . For a DGA A with homology $\mathbb{F}_p[x_{2n}]$ ( $n\gt0$ ) and formal $(2p-4)$ -Postnikov truncation, the odd cell decomposition of $S_{2p-2}^p$ gives a map $S_{2p-2}^p\to A$ , providing the desired map $\mathbb{F}_p\to A$ of ring spectra for Theorem A (2). Theorem B (in the case of odd-degree generators) is obtained similarly by analyzing the centralizer of the map $S_{2p-2}^p \to A$ itself.

We finish this introduction with a number of applications of the aforementioned classification results.

1.4 Exotic DG enhancements

Triangulated categories are ubiquitous in many areas, such as representation theory, homotopy theory, and algebraic geometry. However, often it is advantageous (or necessary) to enhance a triangulated category with a higher categorical structure; classically, these arise in the form of DG enhancements and after the work of Lurie in the form of stable $\infty$ -categories. In this language, a DG enhancement amounts to equipping a stable $\infty$ -categorical lift with a $\mathbb{Z}$ -linear structure. It is then natural to ask whether a given triangulated category admits an enhancement, and if so, how many. For instance, in [Reference Muro, Schwede and StricklandMSS07, Reference Rizzardo and Van den BerghRV20], triangulated categories without enhancements are constructed, and in [Reference SchlichtingSch02, Reference Dugger and ShipleyDS09, Reference Rizzardo and Van den BerghRV19], examples of triangulated categories with non-unique DG enhancements are constructed.

Here, we obtain (to our knowledge) the first example of a triangulated category with infinitely many distinct DG enhancements. Let $F_{2n}^p(l)$ denote the localization $S_{2nl}^p[\sqrt[l]{x_{2nl}}][x_{2nl}^{-1}]$ , a DGA with homology $\mathbb{F}_p[x_{2n}^{\pm1}]$ . As an application of Theorem C, we obtain the following.

Corollary E. The collection

\begin{align*}\{\mathrm{Mod}(F_{2n}^p(l))\}_{l\geq({p-1})/{n}}\end{align*}

consists of pairwise distinct $\mathbb{Z}$ -linear structures on the stable $\infty$ -category $\mathrm{Mod}(\mathbb{F}_p[x_{2n}^{\pm1}])$ . In particular, the triangulated category $\operatorname{\text{Ho}}(\mathrm{Mod}(\mathbb{F}_p[x_{2n}^{\pm1}]))$ admits infinitely many pairwise distinct DG enhancements.

1.5 Classification of prime DG fields

The fields $\mathbb{F}_p$ and $\mathbb{Q}$ are prime fields. Any map of fields with codomain $\mathbb{F}_p$ or $\mathbb{Q}$ is an isomorphism. We reflect this idea at the level of DGAs. We say that a DGA is a DG division ring (DGDR) if every non-zero homogeneous element in its homology is a unit (we warn, however, that our definitions are different than the recent ones given in [Reference ZimmermannZim26]). For instance, $F_{2n}^p := F_{2n}^p(1) = S_{2n}^p[x_{2n}^{-1}]$ is a DGDR. Furthermore, we say that a DGDR D is prime if every map of DGDRs with codomain D is an equivalence.

Corollary F. The collection of all prime DGDRs is given by

\begin{align*} \{ \mathbb{Q},\mathbb{F}_p, F_{2n}^p \mid p \ \textrm{prime and } n\gt0 \}.\end{align*}

Every DGDR receives a map from at least one of the prime DGDRs. Furthermore, every DGDR with even homology receives a map from exactly one of the prime DGDRs and this map is unique up to homotopy.

1.6 Applications to algebraic K-theory

As a first application, we obtain a computation of the algebraic K-theory of the mixed characteristic coordinate axes $\mathbb{Z}[x]/px = \mathbb{Z} \times_{\mathbb{F}_p} \mathbb{F}_p[x]$ .

Corollary G. Let p be a prime. We then have

\begin{align*} K_n(\mathbb{Z}[x]/px) = \begin{cases} K_n(\mathbb{Z}) \oplus \mathbb{W}_{n/2}(\mathbb{F}_p) & \text{ if }\textit{n}\text{ is even}, \\ K_n(\mathbb{Z}) & \text{ if }\textit{n}\text{ is odd}, \end{cases}\end{align*}

where $\mathbb{W}_r(\mathbb{F}_p)$ denotes the ring of big Witt vectors of length r.

Indeed, a special case of [Reference Land and TammeLT26, Lemma 4.30] gives a pullback diagram

with $\pi_*(\odot) \cong \mathbb{F}_p[t_2]$ . By construction, there is a map $\mathbb{F}_p \to \odot$ and by Theorem A, we deduce that $\odot$ is the formal DGA $\mathbb{F}_p[t_2]$ . The K-theory of $\mathbb{F}_p[t_2]$ is known due to [Reference Bayndr and MoulinosBM22] or independently due to [Reference Land and TammeLT26, Example 4.29] giving the above corollary.

As another application, we have the following. Let $l\geq 1$ and $f \in \mathbb{Z}[x]$ be a polynomial in $x^l$ with constant term p, e.g., $f=x^l-p$ .

We prove this by showing that the DGA $\odot$ obtained from [Reference Land and TammeLT26, Lemma 4.30] in the present situation is topologically formal, as an application of Theorem B using a grading trick involving the assumption $l\geq p-1$ . We note that, a priori, neither of the two maps $\mathbb{Z}[x]/f \to \odot \leftarrow \mathbb{Z}$ factors through the unit $\mathbb{F}_p \to \mathbb{F}_p[t_2]$ .

Finally, we have the following.

The case $n=1$ is [Reference Land and TammeLT26, Example 4.32] and is independently due to Krause–Nikolaus; we do not reprove it here. Before applying K-theory, the square for $n=1$ maps to the square for general n; therefore, the $\odot$ -ring for the case $n=1$ maps to the general one. In particular, for all n, we obtain a map $\mathbb{F}_p \to \odot$ of ring spectra and deduce formality of $\odot$ by Theorem A(2).

Notation and conventions

1. (1) We work in the $\infty$ -categorical setting, so all tensor products, maps, (co)limits, mapping spaces/spectra, etc. are derived.

2. (2) When we speak of DGAs, we shall always mean $\mathbb{Z}$ -DGAs.

3. (3) For a (commutative) ring R, there is a corresponding (commutative) ring spectrum characterized by the property that all of its homotopy groups are trivial except degree zero, where it is given by R. This ring spectrum is often denoted by HR but we drop H from our notation and also denote HR by R.

4. (4) With this notation, for commutative R, the $\infty$ -category $\operatorname{Alg}(R)$ Footnote ⁴ of R-algebra spectra is equivalent to the $\infty$ -category of R-DGAs; that is, of differential graded R-algebras with quasi-isomorphisms formally inverted. We shall not distinguish between R-DGAs and R-algebra spectra for that reason. For A an R-algebra spectrum, we also denote by A its underlying ring spectrum.

5. (5) We denote the sphere spectrum by $\mathbb{S}$ ; this is the monoidal unit of the $\infty$ -category of spectra and hence an $\mathbb{S}$ -algebra is the same thing as a ring spectrum.

6. (6) For a DGA A, the homology ring of A agrees with the homotopy ring $\pi_* A$ of the corresponding $\mathbb{Z}$ -algebra (or $\mathbb{S}$ -algebra). For this reason, we also denote the homology of A by $\pi_* A$ .

7. (7) We regard a graded ring as a DGA by equipping it with trivial differentials. DGAs that are equivalent to their homology (viewed as DGAs in the manner just mentioned) are called formal.

8. (8) For generators of homotopy rings (or homology rings), we will use subscripts to indicate the homotopical (or homological) degree.

9. (9) The $k=\infty$ case of the truncated polynomial algebra $R[y]/y^{k}$ denotes the usual polynomial algebra R[y].

10. (10) Whenever we write $\mathbb{Z}/m$ , we assume that $m\gt1$ .