---
date: 2022-04-19 12:51
modification date: Tuesday 19th April 2022 12:51:06
title: "tensor triangulated category"
aliases: [tensor triangulated category, tensor ideal, tensor thick ideal, thick ideal, Balmer spectrum, tensor triangular geometry]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #todo/untagged
- Refs:
	- UROP overview: <https://math.mit.edu/research/undergraduate/urop-plus/documents/2020/Benson.pdf#page=1> #resources/notes #resources/summaries
- Links:
	- [compactly generated](Unsorted/compact%20object%20of%20a%20category.md)
	- [SHC](SHC)

---

# tensor triangulated category

A tensor-triangulated category is a triple $(\mathcal{K}, \otimes, 1)$ consisting of a triangulated category $\mathcal{K}$, a symmetric monoidal product $\otimes: \mathcal{K} \times \mathcal{K} \rightarrow \mathcal{K}$ which is exact in each variable. 

# Thickness

Given a $\cat T\in \triang\Cat$, a **thick subcategory** $S$ is a [full](full) subcategory of $T$ which is closed under finite direct sums and summands. 

![](attachments/Pasted%20image%2020220419145945.png)

# Spectrum

- A subcategory $I\leq \cat{T}$ is an **ideal** iff whenever $i\in I$ and $k\in \cat{T}$ is compact, $k\tensor i\in I$.
- A proper thick ideal $X\leq \cat{T}$ is **prime** iff $x\tensor y\in I \implies x\in I$ or $y\in I$.
- The **spectrum** of a thick ideal is $\Spc(I) = \ts{I\leq \cat{T} \st I \text{ is prime}}$.

![](attachments/Pasted%20image%2020220213164001.png)
![](attachments/Pasted%20image%2020220508204846.png)

# Localization

- A subcategory $S \subset T$ is **localizing** if it is thick and closed under small coproducts. 
- A subcategory **colocalizing** if it is thick and closed under small products.

![](attachments/Pasted%20image%2020220419131437.png)

# SHC

See [chromatic homotopy](Unsorted/chromatic%20homotopy%20theory.md)
![](attachments/Pasted%20image%2020220419150128.png)

![](attachments/Pasted%20image%2020220508204937.png)