From now on, we will consider only rigid-compactly generated tensor-triangulated categories. Let $(\mathcal{T}, \otimes, 1)$ be one of these categories, it is known that the thick tensor ideals of compact objects are completely classified by the Thomason subsets of $\text{Spc}(\mathcal{T}^c)$, the Balmer spectrum associated to the subcategory of compact objects $\mathcal{T}^c$. The correspondence is given by \begin{align} \{\text{Thomason subsets of} \ \text{Spc}(\mathcal{T}^c) \} &\rightarrow \{ \text{thick tensor-ideals of} \ \mathcal{T}^c\}\\ Y &\mapsto \{t\in \mathcal{T}^c : \text{supp}(t)\subseteq Y \}. \end{align} One could wonder if we can extend this classification to the localizing tensor-ideals of the whole $\mathcal{T}$. The idea behind the notion of $\textit{stratification}$ is that the localizing ideals of $\mathcal{T}$ are similarly parametrized by some object (usually a space).
One of the most recent versions of stratification is the one formulated by Barthel, Heard and Sanders in their paper "Stratification in tensor triangular geometry with applications to spectral Mackey functors". They use the notion of Balmer-Favi support, extending the previous notion of Balmer support, to define the following function \begin{align} \{\text{subsets of} \ \text{Spc}(\mathcal{T}^c) \} &\rightarrow \{ \text{localizing tensor-ideals of} \ \mathcal{T}\}\\ Z &\mapsto \{t\in \mathcal{T} : \text{Supp}(t)\subseteq Z \} \end{align} and define the category $\mathcal{T}$ to be stratified (via the Balmer-Favi support) if this assignment is a bijection.
The Balmer-Favi support is defined as follows. First, we need the Balmer spectrum to be weakly noetherian: this allows us to define for each Balmer prime $\mathfrak{p} \in \text{Spc}(\mathcal{T}^c)$ an idempotent $g(\mathfrak{p})\in \mathcal{T}$ and we set $\text{Supp}(t)=\{ \mathfrak{p} \in \text{Spc}(\mathcal{T}^c) : t\otimes g(\mathfrak{p}) \neq 0 \}$.
Barthel, Heard and Sanders prove that $\mathcal{T}$ being stratified is equivalent to two conditions: the local-to-global principle and the localizing tensor-ideal generated by $g(\mathfrak{p})$ being minimal for each $\mathfrak{p}\in \text{Spc}(\mathcal{T}^c)$.
The local-to-global principle is the following statement: for each $t \in \mathcal{T}$ it holds $t \in \text{Locid}(t\otimes g(\mathfrak{p}): \mathfrak{p} \in \text{Supp}(t))$. That is, $t$ belongs to the localizing ideal generated by the objects $t\otimes g(\mathfrak{p})$. This means that at a categorical level each object can be decomposed in simpler objects with support at a single point.
There are various known cases where this condition is verified. For example, if the Balmer spectrum $\text{Spc}(\mathcal{T}^c)$ is noetherian, then the local-to-global principle holds.
What I am interested in are examples where the local-to-global principle is proven not to hold. In his paper "Derived categories of absolutely flat rings", Stevenson shows that if $R$ is an absolutely flat ring which is not semi-artinian then the derived category $D(R)$ does not satisfy the local-to-global principle.
Do you know of similar results, proposing cases where this principle is shown not to hold?
I should point out that the one above is not the only notion of stratification in the literature. There are other versions which do not rely on the Balmer-Favi support. For example, Quillen, Carlson employ for their classification results in modular representation theory the notion of support varieties. The trio Benson, Iyengar and Krause developed another theory of support for compactly generated triangulated categories $\mathcal{T}$ equipped with an action of a graded noetherian commutative ring $R$.
In some of these cases, we can show the stratification is equivalent to appropriate adaptations of the local-to-global principle and minimality. Even if I am working with the notion of Barthel, Heard and Sanders, I am happy to hear about examples using one of the alternative notions of stratification.
