# On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes

Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex of free $R$-modules of finite type. The category of perfect complexes, $D(R)_{\mathrm{parf}}$, is the full subcategory of $D(R)$ consisting of all complexes that are quasi-isomorphic to a strictly perfect complex on $R$. It is well-known, e.g. by [SGA6, Expose I, Proposition 4.10], that $D(R)_{\mathrm{parf}}$ is a thick subcategory of $D(R)$.

This question is related to the following theorem:

Theorem. Let $E, F\in D(R)_{\mathrm{parf}}$ be two perfect complexes on $R$. Suppose $\operatorname{Supph}(E)\subseteq\operatorname{Supph}(F)$ (cohomological support). Then $E$ is in the smallest thick subcategory of $D(R)_{\mathrm{parf}}$ containing $F$.

This theorem was first proved by Hopkins in [M. J. Hopkins: Global methods in homotopy theory (1985)]. Neeman reproduced the proof in Lemma 1.2 in [A. Neeman: The chromatic tower in $D(R)$ (1991)]. Thomason proved it in a more general setting for quasi-compact and quasi-separated schemes, in Lemma 3.14 in [R. W. Thomason: The classification of triangulated subcategories (1997)].

Now let $\mathcal{T}_{\mathrm{Parf}}$ be the full subcategory of $D(R)_{\mathrm{parf}}$ consisting of complexes all of whose cohomology groups have finite lengths. One can quickly see that $\mathcal{T}_{\mathrm{Parf}}$ is a thick subcategory of $D(R)_{\mathrm{parf}}$. By the above-mentioned theorem any nonzero object of $\mathcal{T}_{\mathrm{Parf}}$ is a generator of this category, meaning that if $G$ is a nonzero object of $\mathcal{T}_{\mathrm{Parf}}$, then the smallest thick subcategory of $D(R)_{\mathrm{parf}}$ containing $G$ is $\mathcal{T}_{\mathrm{Parf}}$ itself. Saying that $G$ is a generator is equivalent to saying that for every object $E$ of $\mathcal{T}_{\mathrm{Parf}}$ there is an object $E_1$ and a tower of triangles in $\mathcal{T}_{\mathrm{Parf}}$

Question. My question concerns the shifts $n_1,\ldots,n_k$ that occur in such a tower. What can be said about these shifts? Can they be bounded by invariants related to objects $G$ and $E$?

Remark. Obviously this question could be asked in a more general setting about a generator of a thick subcategory of any triangulated category. But by specializing to $\mathcal{T}_{\mathrm{Parf}}$ I am hoping that there may be a positive answer to the question in this case.