# The frame of smashing tensor-ideals

## TL;DR

This paper establishes a correspondence between flat tensor-idempotents in module categories over compact objects and smashing ideals in tensor-triangulated categories, showing the lattice of smashing ideals forms a frame.

## Contribution

It proves that each flat tensor-idempotent uniquely corresponds to a smashing ideal, revealing the lattice structure of smashing ideals as a frame.

## Key findings

- Every flat tensor-idempotent arises from a unique smashing ideal.
- The lattice of smashing ideals is a frame.
- Provides a classification of smashing ideals via tensor-idempotents.

## Abstract

Given a tensor-triangulated category $T$, we prove that every flat tensor-idempotent in the module category over $T^c$ (the compacts) comes from a unique smashing ideal in $T$. We deduce that the lattice of smashing ideals forms a frame.