A family of infinite degree tt-rings

Juan Omar Gómez Centro de Investigación en Matemáticas, A.C., Unidad Mérida
Parque Científico y Tecnológico de Yucatán
Carretera Sierra Papacal–Chuburná Puerto Km 5.5
Sierra Papacal, Mérida, YUC 97302
Mexico. [email protected]

Abstract.

We construct a family of infinite degree tt-rings, giving a negative answer to an open question by P. Balmer.

Key words and phrases:

Degree, tt-ring, tensor triangulated category.

2020 Mathematics Subject Classification:

18G80

The author acknowledges support by CONACYT under the program “Becas Nacionales de Posgrado”.

Introduction

The abstraction of tensor-triangular geometry makes it possible to connect ideas and unify techniques in the study of tensor triangulated categories arising in different areas of mathematics, including algebraic geometry, commutative algebra, modular representation theory and stable homotopy theory. We refer to [Bal10] for an account of standard tensor triangulated categories.

A tt-ring is a separable commutative algebra object in a tensor triangulated category. These objects play an important role in tensor-triangular geometry. For instance, the Eilenberg-Moore category of modules over a tt-ring remains a tensor triangulated category, and extension of scalars is a tt-functor. The notion of degree introduced by Balmer in [Bal14] has been successfully exploited in many applications: notably, in establishing a connection between the Going-Up Theorem and Quillen’s Stratification Theorem (see [Bal16]) and generalizing the étale topology (see [NP23]), both in the setting of tensor-triangular geometry.

Remarkably, all tt-rings in standard tensor triangulated categories have finite degree [Bal14, Section 4]. It is an open question in [Bal14, Page 2] whether the degree of a tt-ring must always be finite. We provide a family of infinite degree tt-rings, giving a negative answer to this question. In fact, this family extends to a family of infinite degree rigid-compact tt-rings in the framework of rigidly-compactly generated tensor triangulated categories.

1. Infinite degree tt-rings

For $i\in\mathbb{N}$ , let ${\mathcal{K}}_{i}$ be a non-trivial essentially small tensor triangulated category. Define

{\mathcal{K}}:=\prod_{i\in\mathbb{N}}{\mathcal{K}}_{i}.

It is clear that ${\mathcal{K}}$ is essentially small; the product of small skeletons in each component defines a small skeleton of ${\mathcal{K}}$ . We give ${\mathcal{K}}$ a triangulated structure and a symetric monoidal structure, both component-wise. In particular, ${\mathcal{K}}$ is a non-trivial essentially small tensor triangulated category.

Theorem 1.1.

Let ${\mathcal{K}}_{n}$ and ${\mathcal{K}}$ as above and let $\mathbb{1}_{n}$ denote the monoidal unit of ${\mathcal{K}}_{n}$ . Then the tt-ring

A:=(\mathbb{1}_{n}^{\times n})_{n\in\mathbb{N}}\in{\mathcal{K}}

has infinite degree with the component-wise tt-ring structure.

Proof.

It is clear that $A$ is a tt-ring with component-wise multiplication, and a component-wise bilinear section. On the other hand, by the definition of ${\mathcal{K}}$ , the projection functor

\mathrm{pr}_{n}\colon{\mathcal{K}}\to{\mathcal{K}}_{n}

is a tensor triangulated functor for each $n\geq 1$ . In particular, $\mathrm{pr}_{n}(A)=\mathbb{1}_{n}^{\times n}$ which has finite degree $n$ (see [Bal14, Theorem 3.9]). Then $A$ has infinite degree, otherwise it contradicts [Bal14, Theorem 3.7]. ∎

Remark 1.2.

By [Bal14, Theorem 3.8], it follows that there exists a prime ${\mathcal{P}}$ in ${\mathcal{K}}$ such that the tt-ring $q_{\mathcal{P}}(A)$ has infinite degree in ${\mathcal{K}}_{\mathcal{P}}$ . Therefore placing the adjective local on an essentially small tensor triangulated category is not enough to guarantee that tt-rings have finite degree.

At first glance, our example of a tt-ring of infinite degree seems to live in an artificial tensor triangulated category. However, it is possible to find this type of example in practice, for instance in the study of stable module categories for infinite groups.

Let $\mathrm{CAlg}(\mathrm{Pr}^{L}_{\textrm{st}})$ denote the $\infty$ -category of stable homotopy theories, that is, presentable, symmetric monoidal, stable $\infty$ -categories with cocontinuous tensor product in each variable¹¹1Also known as stable homotopy theories.. Let $2\textrm{-}\mathrm{Ring}$ denote the $\infty$ -category of of essentially small, symmetric monoidal, stable $\infty$ -categories with exact tensor product in each variable. We refer to [Mat16, Definition 2.14]) for further details about these $\infty$ -categories.

Example 1.3.

Let $G$ be the fundamental group of the following graph of finite groups,

where $G_{n}$ is a non-trivial finite group, for $n\geq 1$ . In other words, the group $G$ corresponds to the free product of the groups $G_{n}$ . In particular, $G$ is a group of type $\Phi$ (see [Tal07]). By [Góm23, Theorem 3.3], the stable module $\infty$ -category $\operatorname{StMod}\nolimits(kG)$ (see [Góm23, Definition 2.13]) decomposes in terms of the above graph of groups, that is, we have an equivalence

\mathrm{StMod}(kG)\simeq\prod_{n\in\mathbb{N}}\mathrm{StMod}(kG_{n})

in $\mathrm{CAlg}(\mathrm{Pr}^{L}_{\textrm{st}})$ . Note that dualizable objects in $\operatorname{StMod}\nolimits(kG)$ are detected component-wise via this equivalence. In other words, we have a similar decomposition in $2\textrm{-}\mathrm{Ring}$ for the dualizable part of $\operatorname{StMod}\nolimits(kG)$ , i.e., the symmetric monoidal, stable $\infty$ -category on the dualizable objects of $\operatorname{StMod}\nolimits(kG)$ . Moreover, this factorization induces a product decomposition at the level of homotopy categories. Hence the homotopy category of the dualizable part of $\operatorname{StMod}\nolimits(kG)$ satisfies the hypothesis of Theorem 1.1.

In practice, essentially small tensor triangulated categories arise as the dualizable part of a bigger tensor triangulated category which, for instance, admits small coproducts, just as in Example 1.3. Then we can consider tt-rings in a tensor triangulated category which sits inside a bigger one. In particular, the framework of rigidly-compactly generated tensor triangulated categories has been extensively studied (see for instance [BHS21]). In fact, all tt-rings that have been proved to have finite degree in [Bal14, Section 4] sit in the dualizable part a rigidly-compactly generated tensor triangulated category, so we might think these are the conditions we should impose on a tensor triangulated category to guarantee that any tt-ring has finite degree. We will see in Example 1.6 that this is not the case.

Recall that an object $x$ in a triangulated category ${\mathcal{K}}$ with small coproducts is compact if the functor $\mathrm{Hom}(x,-)$ commutes with small coproducts. In particular, the subcategory ${\mathcal{K}}^{c}$ of compact objects remains triangulated.

Definition 1.4.

A tensor triangulated category ${\mathcal{K}}$ is rigidly-compactly generated if ${\mathcal{K}}^{c}$ is essentially small, the smallest triangulated subcategory containing ${\mathcal{K}}^{c}$ which is closed under small coproducts is ${\mathcal{K}}$ , and the class of compact objects coincides with the class of dualizable objects. In this case, ${\mathcal{K}}^{c}$ remains tensor triangulated.

Remark 1.5.

For a general tensor triangulated category ${\mathcal{K}}$ with small coproducts, compact objects are not necessarily dualizable, and vice versa, dualizable objects are not necessarily compact. However, if the monoidal unit of ${\mathcal{K}}$ is compact, then any dualizable object in ${\mathcal{K}}$ is compact. This follows from the fact that a dualizable object $x$ and its dual $y$ determine adjoint functors $x\otimes-\dashv y\otimes-$ .

Example 1.6.

For $i\in\mathbb{N}$ , let ${\mathcal{K}}_{i}$ be a non-trivial rigid $2\textrm{-ring}$ . Define ${\mathcal{K}}:=\prod_{i\in\mathbb{N}}{\mathcal{K}}_{i}$ in $2\textrm{-}\mathrm{Ring}$ . Note that ${\mathcal{K}}$ is a rigid $2\textrm{-ring}$ . Let $\mathcal{L}$ denote the $\mathrm{Ind}$ -completion of ${\mathcal{K}}$ which lies in $\mathrm{CAlg}(\mathrm{Pr}^{L}_{\textrm{st}})$ (see [NP23, Section 2]). In particular, the compact objects of $\mathcal{L}$ are precisely the elements of ${\mathcal{K}}$ . Since the inclusion functor

{\mathcal{K}}\hookrightarrow\mathcal{L}

is strongly monoidal, we deduce that any compact element in $\mathcal{L}$ is dualizable. Therefore the homotopy category of $\mathcal{L}$ is a rigidly-compactly generated tensor triangulated category. In particular, we can construct a tt-ring in the dualizable part of $\mathcal{L}$ , just as in Theorem 1.1, which has infinite degree.

Acknowledgments. I deeply thank my supervisor José Cantarero for his support and for many interesting conversations on this work. I thank Paul Balmer and Luca Pol for helpful comments on this project. This work is part of the author’s PhD thesis.

References

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