Categorifying reduced rings

TL;DR

The paper constructs new monoidal stable $mbda$-categories with specified $K_0$ groups for various rings, addressing a problem posed by Khovanov.

Contribution

It provides functorial constructions of monoidal stable $mbda$-categories with prescribed $K_0$ groups for reduced rings, solving Khovanov's problem.

Findings

Constructed a rigid symmetric monoidal stable $mbda$-category with $K_0$ equal to a given domain of characteristic zero.

Built a rigid braided monoidal stable $mbda$-category with $K_0$ matching any reduced commutative ring.

Addressed and solved a problem posed by Khovanov regarding categorification of rings.

Abstract

Given a domain of characteristic zero $R$, we functorially construct a rigid symmetric monoidal stable $\infty$-category whose $K_0$ is $R$, solving a problem of Khovanov. We also functorially construct for any reduced commutative ring $R$ a rigid braided monoidal stable $\infty$-category whose $K_0$ is $R$.

Equations

$$K_0(D^b(C))\otimes K_0({\mathrm{StMod}}_{C_p}^{\omega })\to K_0(C^{\prime })$$

$$K_0(D^b(C))\stackrel{K_0(D^b(C)\otimes f)}{\to }{K}_0(D^b(C){\otimes }_{{\mathbb{F}}_p}{\mathrm{Rep}}_{{\mathbb{F}}_p}^{\omega }(C_p)$$

$$K_0({\mathrm{Mod}}^{\omega }({\mathbb{F}}_p[C_p])\to K_0({\mathrm{Rep}}_{{\mathbb{F}}_p}^{\omega }(C_p))\stackrel{g}{\to }{K}_0({\mathrm{Mod}}^{\omega }({\mathbb{F}}_p))$$

$$D_{(-)}:{\mathrm{Alg}}_{{\mathbb{E}}_n}(\mathrm{Mod}(K_0(C){)}^{♡}{)}_{K_0(C^{\prime })/}\to {\mathrm{Alg}}_{{\mathbb{E}}_n}(\mathrm{Mod}(C){)}_{C^{\prime }/}$$

$${\eta }_{(R)}:R\to K_0(D_R)$$

Full text

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Throughout this paper we use category to mean $\infty$-category in the sense of Joyal and Lurie.

One prevalent insight in mathematics is that many classical invariants admit categorifications. Namely, instead of assigning a number or polynomial to an object $X$, one assigns an object of some stable category222If the reader prefers, they may read the phrase ‘stable category’ as ‘dg category’. In our language, a dg category over $k$ is a $k$-linear stable category [Coh13]. All stable categories of interest in this paper are $k$-linear for some commutative ring $k$. $C$ which encodes richer information about $X$. The original invariant can then be recovered via applying an Euler characteristic, i.e a homomorphism out of $K_{0}(C)$333$K_{0}(C)$, i.e the Grothendieck group of $C$, is the abelian group generated by $[c]$ for each $c\in C$ with the relation $[c]=[c^{\prime}]+[c^{\prime\prime}]$ whenever there is a cofibre sequence $c^{\prime}\to c\to c^{\prime\prime}$., to the $K_{0}$ class of the categorified invariant.

A prototypical example of this is in algebraic topology, where the Euler characteristic of a space is categorified by its homology. Another example is in algebraic geometry, where the Hilbert polynomial is categorified by coherent cohomology. Finally, in low dimensional topology, classical knot invariants such as the Jones polynomial and Alexander polynomial are categorified respectively by Khovanov homology [Kho00] and Knot Floer homology [OS04].

A natural question to ask is whether the type of invariant can obstruct categorifications from existing. For example, the Witten–Reshetikhin–Turaev invariants constructed in [Wit89, RT91] are invariants taking values in the complex numbers, and it is not known whether these always admit a categorification.

One can build stable categories whose $K_{0}$ is a rational vector space, as is done in [BGHNS19]. The reason is that the multiplication by $n$ map on $K_{0}$ can be implemented by a functor: for example the functor sending $c\mapsto\oplus_{1}^{n}c$. By taking a filtered colimit along such a functor, since $K_{0}$ preserves filtered colimits, one obtains a category whose $K_{0}$ is that of $C$, but with $n$ inverted.

The method above doesn’t naturally produce monoidal categories, which is something usually desired for the target categories of categorifications of invariants, to allow for Künneth-type formulas for the invariants one is categorifying. This is also a fundamental feature of categorifications of manifold invariants that come from field theories. Along these lines, Khovanov posed the following problems:

[KT19] solved 2 in the case $n=2$, but the general case was previously open.

We solve both problems below in 7, even producing categories that are rigid symmetric monoidal. The fundamental input is the ability to produce a category whose $K_{0}$ is a characteristic zero field, which we construct as an ultraproduct of the stable module categories of $\mathbb{F}_{p}[C_{p}]$. We further refine this in A below.

Let $\mathop{\rm Ring}\nolimits_{\mathop{\rm red}\nolimits}^{0}$ be the category of commutative rings which are reduced such that every minimal prime ideal is characteristic zero. Let $\mathbb{Z}\mathrm{Cat}^{\infty}_{\mathop{\rm rig}\nolimits}$ denote the category of rigid symmetric monoidal $\mathbb{Z}$-linear stable categories.

To illustrate the purpose of A, $C^{\infty}_{\overline{\mathbb{Q}}}$ is a rigid symmetric monoidal $\mathbb{Z}$-linear category with a continuous action of the absolute Galois group of $\mathbb{Q}$, such that $K_{0}(C^{\infty}_{\overline{\mathbb{Q}}})\cong\overline{\mathbb{Q}}$ with its standard action.

We next remove the characteristic zero assumption of A in the setting of braided monoidal categories. The key input here is the category of tilting modules in the mixed case for Lusztig’s quantum group $\mathfrak{sl}_{2}$, which we semisimplify and tensor with the stable module category of $\mathbb{F}_{p}[C_{p}]$ in order to categorify finite fields. Let $\mathop{\rm Ring}\nolimits_{\mathop{\rm red}\nolimits}$ be the category of reduced commutative rings, and $\mathbb{Z}\mathrm{Cat}^{2}_{\mathop{\rm rig}\nolimits}$ the category of rigid braided monoidal $\mathbb{Z}$-linear stable categories.

There is a natural condition on a category for which our constructions are essentially sharp.

Our constructions of braided monoidal categories as described above gives the following result:

The following result is an obstruction to producing trace-zero categorifications of rings that are more symmetric than those in C:

In particular, for trace-zero symmetric monoidal categories, the dimension homomorphism $K_{0}(C)\to\mathop{\rm End}\nolimits(\mathbbm{1}_{C})$ must land inside $\mathbb{F}_{p}$ (see [EK21, Lemma 3.13]), so $K_{0}(C)$ cannot contain any field extension of $\mathbb{F}_{p}$.

Despite the above obstruction, there are a number of possible directions for improvement to our theorems. Notably the categories of A are not idempotent-complete, so one could ask whether it is possible to build idempotent-complete categories. Another possible improvement to A would be an extension to all commutative rings, or at least the removal of the assumption about the minimal primes being characteristic zero. The method of proof of A works without the characteristic zero assumption given a positive solution to the following conjecture:

By 4, any positive solution to 5 must not be trace-zero.

Acknowledgements

I am very grateful to Nitu Kitchloo for introducing me to 2, and also to Mikhail Khovanov for posing the problem. I am also very grateful to Pavel Etingof for pointing out to me that $\mathfrak{sl}_{2}$-tilting modules in the mixed case could be used to give ribbon categorifications of finite fields. I would like to thank Pavel Etingof, David Gepner, Mikhail Khovanov, Nitu Kitchloo, and Vijay Srinivasan for discussions related to this problem. Finally, I would like to thank Pavel Etingof, Mikhail Khovanov, and Nitu Kitchloo for helpful feedback on earlier drafts.

1 The examples

In this section we exhibit symmetric monoidal categories whose $K_{0}$ is an arbitrary domain of characteristic zero. Combining this with categories built from $\mathfrak{sl}_{2}$-tilting modules in the mixed case, we produce ribbon categories whose $K_{0}$ is an arbitrary reduced ring.

We first introduce some categories and operations we use. Consider the category $\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}}$555The category $\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}}$ can also be described as the category of perfect modules over the $\mathbb{E}_{\infty}$-ring $\mathbb{F}_{p}^{tC_{p}}$, the ring giving rise to the Tate cohomology of $\mathbb{F}_{p}$ with a trivial $C_{p}$-action., the compact objects of the stable module category of the group ring $\mathbb{F}_{p}[C_{p}]$ (see for example [Mat15, Section 2] for an overview of this category). This is an idempotent-complete rigid symmetric monoidal $\mathbb{Z}$-linear category, and $K_{0}(\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}})\cong\mathbb{F}_{p}$.

The construction in 6 below uses an ultraproduct of the categories $\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}}$ over the set $\mathbb{P}$ of primes. We briefly recall how ultraproducts work, referring the reader to [BSS20, Section 3] or [EK21, Section 3.6] for more details. Given a family $C_{\alpha},\alpha\in A$ of objects in a category $D$ ($D$ could be the category of categories), we can choose a non-principal ultrafilter $U$ on the set $A$, which is the data of a maximal proper filter in the poset of subsets of $A$ containing all cofinite subsets. Then an ultraproduct666The ultraproduct depends on the choice of ultrafilter, but we disregard this in our notation., which we denote $\Pi^{\prime}_{\alpha\in A}C_{\alpha}$ is the object of $D$ defined as the filtered colimit along the subsets of $S$ in $U$ of the product $\Pi_{s\in S}C_{s}$. An ultrapower of an object $C$ is an ultraproduct where all of the $C_{\alpha}$ are the same object, $C$.

Lemma 6**.**

$\Pi^{\prime}_{p\in\mathbb{P}}\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}}$* is an idempotent-complete rigid symmetric monoidal $\mathbb{Z}$-linear category with $K_{0}$ a field of characteristic zero. It is possible to choose an ultrafilter so that $K_{0}$ contains $\overline{\mathbb{Q}}$.*

Proof.

The functor $K_{0}$ preserves arbitrary products and filtered colimits, so it preserves ultraproducts. Since $K_{0}(\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}})$ is $\mathbb{F}_{p}$, the result follows, as an ultraproduct of fields of characteristic $p$ for different $p$ is one of characteristic [math]. It follows from [Ax67, Theorem 5] that one can choose an ultrafilter so that $K_{0}$ contains $\overline{\mathbb{Q}}$. ∎

Theorem 7**.**

For any reduced commutative ring $R$ such that every minimal prime ideal of $R$ is characteristic [math], there is a rigid symmetric monoidal $\mathbb{Z}$-linear category $C$ such that $K_{0}(C)\cong R$, and the dual functor is the identity on $K_{0}$.

Proof.

We claim that the set of commutative rings $R$ satisfying the theorem is closed under products, ultraproducts, and subrings. The claim for products and ultraproducts follows since $K_{0}$ commutes with products and ultraproducts. For subrings, if $K_{0}(C)\cong R$, $R^{\prime}\subset R$ is a subring, and the dual is the identity on $K_{0}(C)$, then the full subcategory of $C$ consisting of objects whose $K_{0}$ class is in $R^{\prime}$ is a category showing that the result is true for $R^{\prime}$.

Any reduced ring embeds into the product of its localizations at all of its minimal primes, which by assumption is a field of characteristic zero, so we have thus reduced the theorem to the case $R$ is a characteristic zero field. An arbitrary field of characteristic zero embeds into an ultrapower of $\overline{\mathbb{Q}}$, so it suffices to produce a rigid symmetric monoidal $\mathbb{Z}$-linear category with the dual acting by the identity such that $K_{0}$ contains $\overline{\mathbb{Q}}$. Now we apply 6, and conclude by observing that the dual functor induces the identity on $K_{0}(\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}})$.

∎

We next explain an improvement of 7 in the setting of ribbon braided monoidal categories. The key example of a ribbon braided monoidal category that we use is constructed in the following proposition:

Proposition 8**.**

For every pair of distinct primes $p,\ell>2$, there is an idempotent-complete semisimple $\overline{\mathbb{F}}_{p}$-linear ribbon category $C$ with $K_{0}(C)\cong\mathbb{Z}[\zeta_{\ell}+\zeta_{\ell}^{-1}]$.

Proof.

Let $q$ be a primitive $\ell^{th}$ root of unity in $\overline{\mathbb{F}}_{p}$. We consider the category $\mathop{\rm Tilt}\nolimits^{\overline{\mathbb{F}}_{p},q}$ of tilting modules of Lusztig’s divided power quantum group associated to $\mathfrak{sl}_{2}$ over the field $\overline{\mathbb{F}}_{p}$. We refer the reader to [STWZ21] for a good reference for this category: it is an idempotent-complete $\overline{\mathbb{F}}_{p}$-linear ribbon category with indecomposable objects $T(v)$ for $v\geq 0$, where $T(0)$ is the unit.

Let $C^{\prime}$ be the semisimplification of this category: see [EO21] for an overview of this construction. $C^{\prime}$ is then an idempotent-complete $\overline{\mathbb{F}}_{p}$-linear semisimple ribbon category, and it remains to show that $K_{0}(C^{\prime})\cong\mathbb{Z}[\zeta_{\ell}+\zeta_{\ell}^{-1}]$. The simple objects in the semisimplification $C^{\prime}$ correspond to indecomposable objects in $\mathop{\rm Tilt}\nolimits^{\overline{\mathbb{F}}_{p},q}$ with nonzero categorical dimension, which is exactly $T(i)$ for $0\leq i\leq\ell-2$ by [STWZ21, Proposition 3.23]. The tensor product by [STWZ21, Lemma 4.1] agrees with the truncated Clebsch–Gordon rule for the tensor product in the Verlinde category $\mathop{\rm Ver}\nolimits_{p}$ [EK21, Section 4.2]. Let $C$ be the full subcategory generated by $T(i)$ for $i$ even: this is also an idempotent-complete $k$-linear semisimple ribbon category since the even objects are closed under tensor product, duals, and contain the unit. Then $K_{0}(C)$ then agrees with $K_{0}(\mathop{\rm Ver}\nolimits_{p}^{+})$, which is indeed $\mathbb{Z}[\zeta_{\ell}+\zeta_{\ell}^{-1}]$ by [BEO23, Theorem 4.5]. ∎

Remark 1.1**.**

The key properties of the categories of 8 that we use are that their mod $p$ reductions can contain arbitrarily large field extensions of $\mathbb{F}_{p}$. If we were interested in constructing categories that are just monoidal as opposed to braided monoidal, there are other candidates, such as the categories constructed in [LQ18], whose $K_{0}$ is an arbitrary cyclotomic extension of the integers.

The next goal is to realize the operation $K_{0}\mapsto K_{0}/(p)$ at the level of categories. To do this, we use the tensor product of $\mathbb{F}_{p}$-linear stable categories. If $C,D$ are $\mathbb{F}_{p}$-linear stable categories, then $C\otimes_{\mathbb{F}_{p}}D$ is also such a category that is generated as a stable category by objects $c\otimes d,c\in C,d\in D$, and $\mathop{\rm map}\nolimits(c\otimes d,c^{\prime}\otimes d^{\prime})\cong\mathop{\rm map}\nolimits(c,c^{\prime})\otimes_{\mathbb{F}_{p}}\mathop{\rm map}\nolimits(d,d^{\prime})$. In particular, $K_{0}(C\otimes_{\mathbb{F}_{p}}D)$ has a surjective map from $K_{0}(C)\otimes K_{0}(D)$.

Proposition 9**.**

Let $C$ be an $\mathbb{F}_{p}$-linear abelian category. Then if $D^{b}(C)$ is the bounded derived category of $C$, then $K_{0}(D^{b}(C)\otimes_{\mathbb{F}_{p}}\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}})\cong K_{0}(C)/(p)$.

Proof.

Let $C^{\prime}=D^{b}(C)\otimes_{\mathbb{F}_{p}}\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}}$. There is a surjective map from

[TABLE]

By the Gillet-Waldhausen theorem, $K_{0}(D^{b}(C))\cong K_{0}(C)$, and since $K_{0}(\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}})\cong\mathbb{F}_{p}$, the above map is really a surjective map $K_{0}(C)/(p)\to K_{0}(C^{\prime})$. It suffices to show that this map is injective.

Let $\mathop{\rm Rep}\nolimits^{\omega}_{\mathbb{F}_{p}}(C_{p})$ denote the bounded derived category of finite dimensional representations of the cyclic group $C_{p}$ over the field $\mathbb{F}_{p}$, so that $\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}}$ is the quotient of $\mathop{\rm Rep}\nolimits^{\omega}_{\mathbb{F}_{p}}(C_{p})$ by the full stable subcategory generated by the projective representation, which is equivalent to $\mathop{\rm Mod}\nolimits^{\omega}(\mathbb{F}_{p}[C_{p}])$.

Tensoring with $D^{b}(C)$, and applying the (connective) $K$-theory functor, we get an sequence

{K(D^{b}(C)\otimes_{\mathbb{F}_{p}}\mathop{\rm Mod}\nolimits^{\omega}(\mathbb{F}_{p}[C_{p}]))}$${K(D^{b}(C)\otimes_{\mathbb{F}_{p}}\mathop{\rm Rep}\nolimits^{\omega}_{\mathbb{F}_{p}}(C_{p}))}$${K(C^{\prime})}

This is a cofibre sequence since connective $K$-theory sends localization sequences of categories to cofibre sequences if the quotient is surjective on $K_{0}$. It suffices then to show that the natural map

[TABLE]

is an equivalence, and that the image of $K_{0}(D^{b}(C)\otimes_{\mathbb{F}_{p}}\mathop{\rm Mod}\nolimits^{\omega}(\mathbb{F}_{p}[C_{p}]))$ under the second map is divisible by $p$, so that the kernel of the map $K_{0}(C)/(p)\to K_{0}(C^{\prime})$ is contained in the ideal $(p)$ and hence is zero.

The first claim follows from 10 below. For the second, we note that the functor $f$ has a retraction $g:\mathop{\rm Rep}\nolimits^{\omega}_{\mathbb{F}_{p}}(C_{p})\to\mathop{\rm Mod}\nolimits^{\omega}(\mathbb{F}_{p})$ given by taking the underlying $\mathbb{F}_{p}$-module. $K_{0}(D^{b}(C)\otimes g)$ is an equivalence since it is an inverse of $K_{0}(D^{b}(C)\otimes f)$, which we have already seen is an isomorphism. Thus it suffices to show that the composite

[TABLE]

has image in the ideal $(p)$. This is because this functor has a filtration given by the filtration of $\mathbb{F}_{p}[C_{p}]$ by the powers of the maximal ideal, whose associated graded is a direct sum of $p$ copies of the functor given by basechange along the map of rings $\mathbb{F}_{p}[C_{p}]\to\mathbb{F}_{p}$. ∎

Lemma 10**.**

Let $C$ be an $\mathbb{F}_{p}$-linear category with bounded $t$-structure. Then the natural map $K(C)\xrightarrow{K(C\otimes f)}K(C\otimes_{\mathbb{F}_{p}}\mathop{\rm Rep}\nolimits_{\mathbb{F}_{p}}^{\omega}(C_{p}))$ is an equivalence, and $K_{-1}$ of both categories vanish.

Proof.

It suffices to show that $K(C\otimes f)$ is an equivalence and that $K_{-1}$ of both categories vanish. To do this, we will show that $C\otimes f$ satisfies the conditions of [BL23, Theorem 1.3]. It is clear that the image of $f$ generates the target, so it suffices to show that the functor is fully faithful on the heart. If $a,b\in C^{\heartsuit}$, then $\pi_{*}\mathop{\rm map}\nolimits(C\otimes f(a),C\otimes f(b))\cong\pi_{*}\mathop{\rm map}\nolimits(a,b)\otimes_{\mathbb{F}_{p}}H^{-*}(C_{p};\mathbb{F}_{p})$. Since $\mathop{\rm map}\nolimits(a,b)$ is coconnective and $H^{-*}(C_{p};\mathbb{F}_{p})$ is concentrated in nonpositive degrees with $H^{0}\cong\mathbb{F}_{p}$, it follows that $\pi_{0}\mathop{\rm map}\nolimits(a,b)\cong\pi_{0}\mathop{\rm map}\nolimits(C\otimes f(a),C\otimes f(b))$, i.e that the functor is fully faithful. ∎

Trace-zero categories are closed under the operations we use:

Lemma 11**.**

If $C$ is a spherical monoidal abelian category, then $C$ is trace-zero. Any stable category with bounded $t$-structure is also trace-zero. Trace-zero stable categories are also closed under idempotent completion, products, and filtered colimits.

Proof.

The fact that $C$ is trace-zero is well-known: given a nilpotent endomorphism $f:c\to c$, one can filter $c$ by the kernel of the powers of $f$ to find that the map $f$ is zero on the associated graded. Thus the trace of $f$ is zero since it is on the associated graded.

To see that a category with bounded $t$-structure is trace-zero, we use the fact that every map is canonically filtered by the Postnikov tower, with associated graded maps in shifts of the heart. In particular, an endomorphism is nilpotent iff it is on all homotopy groups, as the condition of the trace being zero can also be checked on homotopy groups. The heart of the $t$-structure is an abelian category, so the previous case applies.

We omit the proof that trace-zero categories are closed under idempotent completion, products, and filtered colimits, as it is straightforward. ∎

We are now ready to produce ribbon categorifications of all reduced rings.

Theorem 12**.**

For any reduced ring $R$, there exists a trace-zero ribbon braided monoidal $\mathbb{Z}$-linear stable category $C$ with $K_{0}(C)\cong R$ such that the dual is the identity on $K_{0}$.

Proof.

As in 7, The set of $R$ satisfying the theorem are closed under subrings and ultraproducts. The collection of reduced rings is generated under subrings and ultraproducts by the collection of finite fields $\mathbb{F}_{p^{n}}$.

By 9, taking $C$ to be a tensor product of categories coming from 8, $D^{b}(C)\otimes\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}}$ is a ribbon braided monoidal $\mathbb{Z}$-linear stable category with $K_{0}$ a tensor product of the mod $p$ reductions of $\mathbb{Z}[\zeta_{\ell}+\zeta_{\ell}^{-1}]$ for various $\ell>2$. To see it is trace-zero, we first observe that since $C$ is semisimple, the underlying stable category of $D^{b}(C)\otimes\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}}$ is a product of copies of $\overline{\mathbb{F}}_{p}\otimes_{\mathbb{F}_{p}}\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}}$ corresponding to the simple objects of $C$. Now the category is trace-zero for the same reason that $\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}}$ is, which we now explain.

Given an endomorphism $f$ of an object $x\in D^{b}(C)\otimes\mathop{\rm StMod}\nolimits^{\omega}_{C_{p}}$, we can choose a lift $\tilde{f}$ of $f$ to an endomorphism of an object $\tilde{x}$ in the heart of $D^{b}(C)\otimes\mathop{\rm Rep}\nolimits_{\mathbb{F}_{p}}(C_{p})$ containing no projective summand. Traces of endomorphisms factor through the semisimplification, which kills the projective representation. By assumption $\tilde{f}^{n}$ factors through some projective representation for sufficiently large $n$, so $\tilde{f}$ is nilpotent in the semisimplification, and thus has zero trace.

Since $\mathbb{F}_{p^{n}}$ is the tensor product of finite fields of the form $\mathbb{F}_{p^{q^{n}}}$, it suffices to show that for each $p,q,n$ with $p,q$ prime, there is an $\ell>2$ such that $\mathbb{F}_{p^{q^{n}}}$ is contained in the mod $p$ reduction of $\mathbb{Z}[\zeta_{\ell}]$.

To see this, we first observe that whenever there is a prime $\ell$ dividing $|\mathbb{F}_{p^{q^{n}}}|^{\times}$ that doesn’t divide $|\mathbb{F}_{p^{q^{n-1}}}|^{\times}$, then $\mathbb{Z}[\zeta_{\ell}]/p$ contains $\mathbb{F}_{p^{q^{n}}}$.

We thus need to show that for infinitely many values of $n$, $|\mathbb{F}_{p^{q^{n}}}|^{\times}=p^{q^{n}}-1$ contains primes not dividing $p^{q^{n-1}}-1$. This will be true if $\gcd(\frac{p^{q^{n}}-1}{p^{q^{n-1}}-1},p^{q^{n-1}}-1)=\gcd(q,p^{q^{n-1}}-1)$ is $1$, i.e if $p$ is not congruent to $1$ mod $q$.

If $q|p-1$, then for large $n$, the $q$-adic valuation of $p^{q^{n-1}}-1$ is larger than $1$. Thus since $\gcd(\frac{p^{q^{n}}-1}{p^{q^{n-1}}-1},p^{q^{n-1}}-1)=q$, it follows that for large $n$, the $q$-adic valuation of $\frac{p^{q^{n}}-1}{p^{q^{n-1}}-1}$ is $1$. Thus $\frac{p^{q^{n}}-1}{p^{q^{n-1}}-1}$ must have a prime factor that is not a prime factor of $p^{q^{n-1}}-1$, allowing us to conclude. ∎

The categories constructed in 12 and 7 are stable categories that do not arise from additive categories. Moreover, they are not idempotent-complete, and the dual functor is the identity. Therefore we ask:

Question 13**.**

Given $R$ a commutative ring, when does there exist an additive rigid symmetric monoidal category $C$ with $K_{0}(C)\cong R$?

Question 14**.**

Given $R$ a commutative ring, when does there exist an idempotent-complete rigid symmetric monoidal stable category $C$ with $K_{0}(C)\cong R$?

Question 15**.**

Given $R$ a commutative ring with an involution, when does there exist a rigid symmetric monoidal stable category $C$ such that $K_{0}(C)$ acted on by the dual functor is equivalent to $R$?

2 Universal and functorial examples

Next, we show it is possible to produce categories whose $K_{0}$ admit maps from a ring $R$ equipped with universal witnesses of relations in $K_{0}$. We then combine this with the results of the previous theorem to prove A and B. Given a symmetric monoidal stable category $C$, let $\mathop{\rm Alg}\nolimits_{\mathbb{E}_{n}}(\mathop{\rm Mod}\nolimits(C))$ denote the category of $\mathbb{E}_{n}$-monoidal $C$-linear categories. For a discrete commutative ring $R$, we use $\mathop{\rm Alg}\nolimits_{\mathbb{E}_{n}}(\mathop{\rm Mod}\nolimits(R)^{\heartsuit})$ to denote the category of (discrete) associative $R$-algebras for $n=1$ and commutative $R$-algebras for $n>1$.

Proposition 16**.**

Let $1\leq n\leq\infty$, $C$ be a symmetric monoidal stable category, and $C^{\prime}$ a $\mathbb{E}_{n}$-monoidal $C$-linear category. There is a functor

[TABLE]

and a natural transformation

[TABLE]

*such that $\eta_{R}$ makes $D_{R}$ a versal777or weakly initial object in $\mathop{\rm Alg}\nolimits_{\mathbb{E}_{n}}(\mathop{\rm Mod}\nolimits(C))_{C^{\prime}/}$ equipped with a map $R\to K_{0}(D_{R})$ in $\mathop{\rm Alg}\nolimits_{\mathbb{E}_{n}}(\mathop{\rm Mod}\nolimits(K_{0}(C))^{\heartsuit})_{K_{0}(C^{\prime})/}$. The functor $D_{(-)}$ functorially depends on $C,C^{\prime}$, and a choice of representative of each $K_{0}$-class of $K_{0}(C^{\prime})$, and preserves filtered colimits with respect to all parameters. We can moreover require that $D_{(-)}$ take values and have a versal property instead in the subcategory of rigid categories. *

Proof.

We will prove the proposition simultaneously with and without the rigid assumption, using (rigid) to indicate which operations must be done as rigid categories with the rigid assumption.

For each class in $x\in K_{0}(C^{\prime})$, let us fix a choice of representative of the $K_{0}$-class of $x$, which we call $O_{x}$. Let us fix $R\in\mathop{\rm Alg}\nolimits_{\mathbb{E}_{n}}(\mathop{\rm Mod}\nolimits(K_{0}(C))^{\heartsuit})_{K_{0}(C^{\prime})/}$. We choose a presentation of $R$ as an associative $K_{0}(C)$-algebra under $C^{\prime}$ as follows: Let each $r\in R$ itself be the set of generators. To write the set of relations consider all formal expression of the form $\sum_{i=1}^{n}\prod_{j=1}^{m_{i}}x_{ij}$, where $x_{ij}$ is an element of the disjoint union $R\coprod K_{0}(C^{\prime})$. For each such data, we get a relation if the relation $\sum_{i=1}^{n}\prod_{j=1}^{m_{i}}x_{ij}=0\in R$ holds where for elements of $K_{0}(C^{\prime})$, we consider their image in $R$.

Now we let $D^{\prime}_{R}$ be the free (rigid) $\mathbb{E}_{n}$-monoidal $C$-linear category under $C^{\prime}$ equipped with objects $X_{r}$ for each generator $r\in K_{0}(C^{\prime})$. These objects give a natural map $\eta_{R}^{\prime}:K_{0}(C^{\prime})\{R\}\to K_{0}(D^{\prime}_{R})$ in $\mathop{\rm Alg}\nolimits_{\mathbb{E}_{n}}(\mathop{\rm Mod}\nolimits(K_{0}(C))^{\heartsuit})_{K_{0}(C^{\prime})/}$.

Next, for each relation $\sum_{i=1}^{n}\prod_{j=1}^{m_{i}}x_{ij}$, consider the object $O=\oplus_{i=1}^{n}(\otimes_{j=1}^{m_{i}}P_{ij})$ where $P_{ij}$ is $O_{x_{ij}}$ whenever $x_{ij}\in K_{0}(C^{\prime})$ and is $X_{x_{ij}}$ whenever $x_{ij}\in R$. Note that we consider the operations $\oplus$ and $\otimes$ as only giving monoidal structures (as opposed to symmetric monoidal and $\mathbb{E}_{n}$-monoidal) so that $O$ is an object defined up to a contractible space of choices from the data defining the relation.

For each relation, we freely adjoin to $D^{\prime}_{R}$ as a (rigid) $\mathbb{E}_{n}$-monoidal $C$-linear category objects $Y,Z$, maps

{Y}$${O\oplus Z}$${Y}$${Z}$$\scriptstyle{f}$$\scriptstyle{g}

and an isomorphism $\mathop{\rm cof}\nolimits(f)\cong\mathop{\rm cof}\nolimits(g)$. Let $D_{R}$ be the object in $\mathop{\rm Alg}\nolimits_{\mathbb{E}_{n}}(\mathop{\rm Mod}\nolimits(C))_{C^{\prime}/}$ constructed via these operations.

The fact that $\mathop{\rm cof}\nolimits(f)\cong\mathop{\rm cof}\nolimits(g)$ implies that $[O]+[Z]-[Y]=[Z]-[Y]$, i.e $[O]=0$. Thus the composite $K_{0}(C^{\prime})\{R\}\xrightarrow{\eta_{R}^{\prime}}K_{0}(D^{\prime}_{R})\to K_{0}(D_{R})$ factors through $R$, so we obtain the natural transformation $\eta_{R}$ as this factorization.

It remains to show the versal property of $D_{R}$. To do this, let $E$ be a (rigid) $\mathbb{E}_{n}$-monoidal $C$-linear category under $C^{\prime}$ with a factorization $K_{0}(C^{\prime})\to R\to K_{0}(E)$. We must then produce a map $D_{R}\to E$ in $\mathop{\rm Alg}\nolimits_{\mathbb{E}_{n}}(\mathop{\rm Mod}\nolimits(C))_{C^{\prime}/}$ compatible with this factorization.

First, observe that by choosing representatives of the $K_{0}$-classes of each element of $R$, we obtain a map $D^{\prime}_{R}\to E$ by sending the object $X_{r},r\in R$ to this object. It suffices to show then that this functor $F:D^{\prime}_{R}\to E$ admits a factorization through $D_{R}$. To do this, we use the following lemma:

Lemma 17** (Heller’s criterion [KW19, Lemma 2.12]).**

Let $C$ be a stable category. For any relation $[A]=[B]$ in $K_{0}(C)$, there exist cofibre sequences of the form

{Y}$${A\oplus Z}$${J}$${Y}$${B\oplus Z}$${J}$$\scriptstyle{f}$$\scriptstyle{g}

For a given relation $\sum_{i=1}^{n}\prod_{j=1}^{m_{i}}x_{ij}$, by applying 17 to $[F(O)]=[0]$ in $E$, we obtain the data as in the lemma. For each relation, by sending the $Y,Z,f,g$ used to form $D_{R}$ to these objects and maps, and by choosing an isomorphism between the $J$s appearing in the cofibre sequences, we obtain the desired factorization through $D_{R}$. ∎

Remark 18**.**

There is a version of 16 for additive categories: one need merely replace the application of 17 with the more basic fact that in an additive category, the relation $[A]=[B]$ iff there is some $Y$ such that $A\oplus Y\cong B\oplus Y$.

Now we combine 7, 12, and 16 to prove A and B, whose statements we recall for convenience.

Theorem A**.**

There is a filtered colimit preserving functor $C^{\infty}_{(-)}:\mathop{\rm Ring}\nolimits_{\mathop{\rm red}\nolimits}^{0}\to\mathbb{Z}\mathrm{Cat}^{\infty}_{\mathop{\rm rig}\nolimits}$ and a natural isomorphism $K_{0}(C^{\infty}_{R})\cong R$.

Theorem B**.**

There is a filtered colimit preserving functor $C^{2}_{(-)}:\mathop{\rm Ring}\nolimits_{\mathop{\rm red}\nolimits}\to\mathbb{Z}\mathrm{Cat}^{2}_{\mathop{\rm rig}\nolimits}$ and a natural isomorphism $K_{0}(C^{2}_{R})\cong R$.

Proof of A and B.

We first prove A, and then indicate the necessary changes to prove B. Letting $C=C^{\prime}=\mathop{\rm Mod}\nolimits(\mathbb{Z})^{\omega},n=\infty$, choosing any representatives of $K_{0}(C^{\prime})$, and applying 16, we obtain a functor $D_{(-)}$, taking a discrete commutative ring $R$ to $D_{R}$, a $\mathbb{Z}$-linear rigid symmetric monoidal category with a natural map $R\to K_{0}(D_{R})$. The category $D_{R}$ comes functorially equipped with objects $X_{r},r\in R$ (see the proof of 16) whose $K_{0}$-class is the image of $r$.

We consider the ideal $I_{R}$ of $K_{0}(D_{R})$ generated by $r-r^{*}$, where $r^{*}$ is the class of the dual of $r$. We apply 16 again with $C=\mathop{\rm Mod}\nolimits(\mathbb{Z})^{\omega},C^{\prime}=D_{R},n=\infty$ and $X_{r}$ as our choice of representatives to obtain a category $D^{{}^{\prime}}_{R}$ versally equipped with a map $K_{0}(D_{R})/I_{R}\to K_{0}(D^{{}^{\prime}}_{R})$. Note that the image of the composite $R\to K_{0}(D_{R})/I_{R}\to K_{0}(D^{{}^{\prime}}_{R})$ by construction has the property that it is fixed by the dual functor, so the subcategory of $D^{{}^{\prime}}_{R}$ consisting of objects whose $K_{0}$-class is in this image is a rigid symmetric monoidal subcategory, which we define to be $C_{R}^{\infty}$.

It remains to show that the natural surjection $R\to K_{0}(C_{R}^{\infty})$ is an isomorphism, which is equivalent to the claim that $R\to K_{0}(D_{R}^{{}^{\prime}})$ is injective. We now assume that each minimal prime of $R$ is characteristic [math]. Let $C$ be a $\mathbb{Z}$-linear rigid symmetric monoidal category as in 7 with the property that $K_{0}(C)\cong R$ and the action of the dual functor on $K_{0}$ is trivial. The fact that $K_{0}(C)\cong R$ by versality gives the existence of a symmetric monoidal $\mathbb{Z}$-linear functor $D_{R}\to C$. Because the action of the dual functor on $C$ is trivial, versality of $D^{\prime}_{R}$ further allows us to extend this to a symmetric monoidal functor $D^{\prime}_{R}\to C$. It then follows that the composite $R\to K_{0}(D^{\prime}_{R})\to K_{0}(C)$ is an isomorphism, so $R\to K_{0}(D^{\prime}_{R})$ is injective as desired.

To prove B, we run the same proof, except changing $n$ from $\infty$ to $2$, and replacing the use of 7 with 12. ∎

Remark 19**.**

One can use 16 to make an ‘obstruction theory’ for constructing idempotent-complete categories with a specified $K_{0}$. We briefly indicate how this works for a symmetric monoidal categories.

Given $C_{0}$ an idempotent-complete symmetric monoidal rigid stable category and a map $K_{0}(C_{0})\to R$, we can by 16 construct a versal idempotent-complete rigid symmetric monoidal functor $C_{0}\to C_{1}$ with a factorization $K_{0}(C_{0})\to R\to K_{0}(C_{1})$. The first obstruction to constructing an idempotent-complete $C_{0}$-linear category with $K_{0}$ isomorphic to $R$ is whether the map $R\to K_{0}(C_{1})$ admits a retraction.

If a retraction exists, we can choose a retraction, and form $C_{2}$ as a versal idempotent-complete $C_{1}$-linear category with a factorization $K_{0}(C_{1})\to R\to K_{0}(C_{2})$. The second obstruction is whether the map $R\to K_{0}(C_{2})$ admits a retraction.

One can keep going, and if all obstructions vanish, one can produce for each $i$ a category $C_{i}$, and the filtered colimit of $C_{i}$ is an idempotent-complete $C_{0}$-linear category with the desired $K_{0}$. On the other hand, versality shows that if an idempotent-complete $C_{0}$-linear category exists with the desired $K_{0}$, then choices of retractions can be made so that all obstructions vanish.