Levels and Thick Subcategories in Triangulated Categories

Definitions and mathematical statement

Let $\mathcal T$ be a triangulated category, with suspension $\Sigma$, let $\mathcal C$ be a class of objects, and let $M\in\mathcal T$. The thick subcategory $\operatorname{thick}_{\mathcal T}(\mathcal C)$ is the smallest full triangulated subcategory containing $\mathcal C$ and closed under direct summands.

One standard level convention is the following. Put $\langle\mathcal C\rangle_0=\{0\}$ and let $\langle\mathcal C\rangle_1$ consist of direct summands of finite direct sums of objects $\Sigma^j C$, with $C\in\mathcal C$ and $j\in{\mathbb{Z}}$. For $n\geq1$, let $\langle\mathcal C\rangle_{n+1}$ be the closure under finite sums, shifts and direct summands of objects $X$ fitting into a distinguished triangle \[ U\longrightarrow X\longrightarrow V\longrightarrow\Sigma U, \qquad U\in\langle\mathcal C\rangle_n,\quad V\in\langle\mathcal C\rangle_1. \] Define \[ \operatorname{level}_{\mathcal C}^{\mathcal T}(M) =\inf\{n\geq0:M\in\langle\mathcal C\rangle_n\},\qquad \inf\varnothing=\infty. \] Thus the level measures a finite number of extension layers, allowing shifts, finite sums and direct summands; arbitrary coproducts are not included. With this convention, \[ \operatorname{level}_{\mathcal C}^{\mathcal T}(M)<\infty \quad\Longleftrightarrow\quad M\in\operatorname{thick}_{\mathcal T}(\mathcal C). \] The exact numerical indexing of level is conventional and is not supplied by the record.

The source problem asks for ways to determine this level for given $\mathcal T$, $\mathcal C$ and $M$. Its more modest question asks for ways to decide finiteness, or equivalently membership in the generated thick subcategory. No category class, effective encoding, representation of morphisms or distinguished triangles, or computability hypotheses are given. Consequently "decide" is retained as a request for criteria or methods, not strengthened into an assertion that a single terminating algorithm exists for every abstract triangulated category.

The supplied literature assessment checked 2026-08-21 labels the problem open and says that the report supplies methods for special projective-class cases without a general decision theorem. This is a dated attributed assessment, not a result established in this draft.

Short English statement

Find methods to compute the number of extension layers needed to build $M$ from shifts, finite sums and summands of $\mathcal C$. At minimum, decide whether finitely many layers suffice, equivalently whether $M\in\operatorname{thick}(\mathcal C)$. No general effective presentation is supplied.

Sources

• Local UnsolvedMath registry, permanent ID 30000931, source code catalogue entry; supplied complete record, including its corrected-verified source-check note and dated assessment of projective-class methods.
• Representation Theory of Finite Dimensional Algebras (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/09, pp. 401--472: https://doi.org/10.4171/owr/2008/09.