Localisations of rings of weak global dimension at most one

Introduction

Given a commutative ring $R$, we are interested in understanding the lattice of localising subcategories of the unbounded derived category over $R$: $\operatorname{Loc}(D(R))$. There are only a few classes of examples known in which this lattice has been fully classified. For instance, take $R$ to be a noetherian ring, or a semi-artinian absolutely flat ring. For these, the classification manifests itself as an isomorphism of lattices: $\operatorname{Loc}(D(R)) \cong\mathcal{P}(\operatorname{Spec}(R))$, where $\mathcal{P}(-)$ denotes the power-set functor. Moreover, such an isomorphism is obtained using a standard support theory. Derived categories admitting this kind of classification are called stratified; this terminology goes back to [BIK11]. In several other cases, only partial results are known [31, 40, 5].

Our work extends the list of examples where stratification can be obtained. In the sense of the abstract stratification theory within tensor-triangular geometry using Rickard idempotents [37, 4, 6], we give a complete characterisation of when the derived category is stratified for classes of rings of weak global dimension at most one. We refer to a key class of these rings as follows.

In the case that $R$ is not local (so not a valuation domain itself), we require more than $R$ being strongly discrete in our results. To this end, we call a flat ring epimorphism $f\colon R \to S \ne0$ sharp (see Lemma 5.9) if there exists a finitely generated ideal $J \subseteq S$ such that $S/J$ is a local ring. With this, we can formulate our first main result.

Furthermore, we provide an example of a ring $R$ of weak global dimension one (even of a valuation domain) that is not strongly discrete, and yet we have an abstract isomorphism of lattices $\operatorname{Loc}(\mathrm{D}(R)) \cong\mathcal{P}(\operatorname{Spec}(R))$. For this ring $R$, $\operatorname{Spec}(R)$ is not a weakly noetherian space in the sense of [6], the Gabriel dimension of $\operatorname{Mod}_{R}$ is undefined, the Telescope Conjecture for $\mathrm{D}(R)$ fails, yet the residue fields generate $\mathrm{D}(R)$ as a localising category. We also show that we cannot save the day by the finer smashing support theory in the sense of [13], as the smashing support does not stratify $\mathrm{D}(R)$ either. This is all Example 4.11.

Finally, our second main result resolves the classification problem of $\operatorname{Loc}(\mathrm{D}(R))$ via stratification by the Zariski spectrum for a class of coherent rings $R$ of weak global dimension at most one¹, e.g., Prüfer domains. We also put stratification problem into context with Gabriel dimension of the modules categories, partially extending results of Balchin–Gómez–Stevenson [5].

All considered rings are commutative and unital unless stated otherwise. If $R$ is a ring and $\mathfrak{p} \in\operatorname{Spec}(R)$, we denote the residue field $R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}}$ at $\mathfrak{p}$ by $k(\mathfrak{p})$. If $R$ is a domain, $Q$ stands for the fraction field (= the residue field at the zero ideal).

Acknowledgements & AI disclosure

Acknowledgements & AI disclosure. The first-named author was supported by the Czech Science Foundation grant GAČR 26-22734S. The second-named author was supported by the Deutsche Forschungsgemeinschaft (Project-ID 491392403 – TRR 358).

AI usage: Used solely for finding suitable references to the existing literature, spelling improvements and correction of typos.

Preliminaries

Commutative domains and ring epimorphisms

As alluded to, our main results are about rings of weak global dimension at most one. Here we will recall the necessary concepts about non-noetherian commutative rings and in particular domains, following the terminology of [19].

Standard but from the point of view here uninteresting, examples are discrete valuation rings (or discrete rank 1 valuation domains in the terminology of [19]) such as the ring of $p$-adic integers $\mathbb{Z}_p$ for a prime $p$ or the power series ring $k[[x]]$ for a field $k$.

In fact, the value group is a totally ordered abelian group, where we define the ordering by putting $aR^{\times} \le bR^{\times}$ for $a,b \in Q^{\times}$ if $ba^{-1} \in R$, or equivalently when $bR \subseteq aR$ as $R$-submodules of $Q$. Moreover, all totally ordered abelian groups occur as value groups of valuation domains by a result of Krull [29] (see also [19], Theorem II.3.8).

We will prove best results for strongly discrete valuation domains. We recall the corresponding definitions from [19], §II.8.

The relation between the two just defined notions is made precise by the following.

(iii) $R$ is discrete and the prime ideals of $R$ satisfy the ascending chain condition.

Valuation domains are, of course, local. For non-local commutative rings, we have the following result very much akin to the characterization of Dedekind domains in the noetherian case.

In the case of domains, the following terminology is well established.

Except for domains themselves, we will also need to understand certain class of ring homomorphisms between them.

Moreover, two ring epimorphisms $f_i:R\to S_i$, $i=1,2$ are called equivalent if there exists a (necessarily unique) ring isomorphism $g:S_1\to S_2$ such that $gf_1=f_2$

A ring homomorphism $f:R\to S$ is a ring epimorphism if and only if the multiplication in $S$ induces an isomorphism $S\otimes_R S\to S$; see [38 Proposition 1.1]. Each flat epimorphism is clearly a homological epimorphism but not vice versa—an explicit example was given in [26]. Moreover, if $R$ is commutative and $f:R\to S$ is a ring epimorphism, then $S$ is commutative by [38 Corollary 1.2] and the induced map of Zariski spectra $\operatorname{Spec}(f):\operatorname{Spec}(S)\to\operatorname{Spec}(R)$ is injective by [30 Proposition IV.1.5].

A typical example of a flat epimorphism of commutative rings is a localisation $R\to\Sigma^{-1}R$ at a multiplicative subset $\Sigma\subset R$, and flat epimorphisms originating from a valuation domain are all of this form by [BS17, Proposition 5.4]. If $f$ is a flat epimorphism, then $\operatorname{Spec}(f)$ is even a homeomorphism onto its image and $f$ is determined by the image of $\operatorname{Spec}(f)$ up to equivalence [30 Corollaire IV.2.2]. The latter is, however, not necessarily the case for homological epimorphisms. Indeed, the example from [26] is a valuation domain $R$ whose spectrum has two points: the zero ideal and the maximal ideal $\mathfrak m$. Then the canonical map $f:R\to\mathbb{Q}\times k(\mathfrak m)$ is a homological epimorphism, the map $\operatorname{Spec}(f)$ is a continuous bijection, but it is not a homeomorphism since $\operatorname{Spec}(\mathbb{Q}\times k(\mathfrak m))$ is discrete while $\operatorname{Spec}(R)$ is not. See also [BS17, Example 5.24].

Hochster duality

Given a topological space $X$, we denote by $\Omega(X)$ the collection of all open subsets of $X$ and by $\Omega(X)^{\omega} \subseteq\Omega(X)$ the set of all quasi-compact open sets of $X$.

It is a classical result by Hochster [24] that a topological space $X$ is homeomorphic to one of the form $\operatorname{Spec}(R)$ for a commutative ring $R$ if and only if it is a spectral space in the following sense:

If $X$ is a spectral space, we can define a Hochster dual space $X^{\vee}$ with the same set of points, but whose quasi-compact open sets are precisely complements of the quasi-compact open subsets of $X$. Then $X^{\vee}$ is again a spectral space and $(X^{\vee})^{\vee}=X$. This is still due to Hochster [24] and details can be found in [17 Ch. 1]. The open sets of $X^{\vee}$ are often called the Thomason subsets of $X$, paying tribute to Thomason’s seminal paper [Tho97] classifying the thick subcategories of the perfect derived category of $R$ in terms of Thomason subsets of the Zariski spectrum.

By unraveling the definition, Thomason subsets of $\operatorname{Spec}(R)$ are precisely those of the form $\bigcup_{\lambda\in\Lambda}V(I_{\lambda})$, where $\{I_{\lambda}\subseteq R\mid\lambda\in\Lambda\}$ is a collection of finitely generated ideals of $R$. Thomason subsets are always closed under specialization in $\operatorname{Spec}(R)$, but not vice versa in general.

In connection with classification of localising subcategories of $D(R)$, we will need the following definition.

In other words, $X$ is weakly noetherian if and only if each point $x\in X^{\vee}$ is locally closed. The latter is a separation axiom between $T_0$ and $T_1$; this connection has been observed in [13].

Tensor-triangular geometry

Given any essentially small triangulated category $\mathcal{K}$ with a symmetric monoidal structure $(\mathcal{K},\otimes,\mathbb{1})$ such that $\otimes$ is exact in each variable, Balmer defined in [1] its spectrum $\operatorname{Spc}(\mathcal{K})$, nowadays called the Balmer spectrum. As he showed in [2], this allows one to uniformly treat a large class of examples from algebraic geometry, homotopy theory, representation theory, or motivic theory using geometric intuition.

More in detail, the points of $\operatorname{Spc}(\mathcal{K})$ are prime thick tensor ideals of $\mathcal{K}$. Here $\mathcal{J}\subseteq\mathcal{K}$ is a thick tensor ideal if $\mathcal{J}$ is a thick subcategory of $\mathcal{K}$ such that $x\otimes y\in\mathcal{J}$ whenever $x\in\mathcal{K}$ and $y\in\mathcal{J}$. A thick tensor ideal $\mathcal{P}\subset\mathcal{K}$ is called prime if $x\otimes y\in\mathcal{P}$ implies $x\in\mathcal{P}$ or $y\in\mathcal{P}$ for each pair $x,y\in\mathcal{K}$. The Balmer spectrum carries the Zariski topology that makes it a spectral space and whose quasi-compact open sets are precisely those of the form

where $x \in\mathcal{K}$. The corresponding Thomason sets are of the form

where $\mathcal{S}$ is a set of objects of $\mathcal{K}$.

Our prime example of $\mathcal{K}$ is as follows: If $R$ is a ring, the unbounded derived category $\mathcal{D}(R)$ is compactly generated and carries a monoidal structure $(\mathcal{D}(R), \otimes_R^{\mathbf{L}}, R)$ such that $\otimes_R^{\mathbf{L}}$ is exact in both variables. This is a compactly generated triangulated category and the full subcategory of compact objects $\mathcal{K} = \mathcal{D}(R)^\omega$ (which are known to be perfect complexes) is an essentially small category as above with the restricted structure. The key result in that context is as follows.

Proof. This goes back to [1 Corollary 5.6] for $R$ noetherian, and in general it was shown in [10 Theorem 9.5]; see also [2]. □

Next, we recall fragments of the results from [4, 6] which allow us to classify all localising subcategories or localising tensor ideals in some cases. To this end, let $(\mathcal{T}, \otimes, \mathbb{1})$ be a compactly generated triangulated category such that $\otimes$ is exact and preserves coproducts in each variable, and denote the full subcategory of compact objects by $\mathcal{T}^\omega$. We call a subcategory $\mathcal{L} \subseteq\mathcal{T}$ localising if it is closed under arbitrary direct sums, and we call $\mathcal{L}$ a localising tensor ideal if $\mathcal{L}$ is a localising subcategory such that $x \otimes y \in\mathcal{L}$ whenever $x \in\mathcal{T}$ and $y \in\mathcal{L}$. We will denote the smallest localising (resp., tensor) subcategory containing a class of objects $\mathcal{S} \subseteq\mathcal{T}$ by $\operatorname{loc}(\mathcal{S})$ (resp., $\operatorname{loc}^{\otimes}(\mathcal{S})$). Note that if $\operatorname{loc}(\mathbb{1}) = \mathcal{T}$, then every localising subcategory of $\mathcal{T}$ is a localising tensor ideal by [25 Lemma 1.4.6]. In particular, given a ring $R$, this holds for $\mathcal{T} = \mathcal{D}(R)$.

By the Brown representability theorem [32 Theorem 3.1], the monoidal structure on $\mathcal{T}$ is closed under the above assumptions. That is, for each $y,z \in\mathcal{T}$ there is a hom-object $\operatorname{RHom}(y,z) \in\mathcal{T}$ such that for each $x,y,z \in\mathcal{T}$ there is a natural isomorphism

This allows us to define dualisable objects of $\mathcal{T}$ as those $x \in\mathcal{T}$ such that the canonical homomorphism $\operatorname{RHom}(x,\mathbb{1}) \otimes y \to\operatorname{RHom}(x,y)$ is an isomorphism for each $y \in\mathcal{T}$. We then call $\mathcal{T}$ rigidly-compactly generated if the classes of compact and dualisable objects in $\mathcal{T}$ coincide. In that case, $\mathcal{T}^\omega$ is automatically an essentially small triangulated category, the monoidal structure restricts to $\mathcal{T}^\omega$, and it is well-known that $\mathcal{T} = \mathcal{D}(R)$ is rigidly-compactly generated. We refer to [4, 6] for more details.

Given a rigidly-compactly generated category $\mathcal{T}$ and $\mathcal{P} \in\operatorname{Spc}(\mathcal{T}^\omega)$ weakly visible, Balmer and Favi define in [4 Definition 7.9] a certain tensor idempotent object $g(\mathcal{P}) \in\mathcal{T}$ (i.e. one has that $g(\mathcal{P}) \otimes g(\mathcal{P}) \simeq g(\mathcal{P})$). Paying tribute to a similar idea of Rickard [37] this time, we will call such $g(\mathcal{P})$ the Rickard idempotent associated with $\mathcal{P}$. We will not reproduce the general definition here, but show instead how to construct such an object directly for $\mathcal{T} = \mathcal{D}(R)$. Note also that in that case $\mathcal{P}$ corresponds to a unique prime $\mathfrak{p} \in\operatorname{Spec}(R)$ by Theorem 2.11, and $\mathfrak{p}$ being weakly visible in $\operatorname{Spec}(R)$ precisely means that there is a finitely generated ideal $I \subseteq R$ such that $V(IR_{\mathfrak{p}})=\{\mathfrak{p}R_{\mathfrak{p}}\}$ in $\operatorname{Spec}(R_{\mathfrak{p}})$.

Proof. Upon identifying $\operatorname{Spc}(D(R)^\omega)$ with $\operatorname{Spec}(R)$, the two Thomason sets of $\operatorname{Spec}(R)$ from [4 Lemma 7.8], witnessing that $\mathfrak{p}$ is weakly visible, are $U=V(I)$ and $W=\operatorname{Spec}(R)\setminus\operatorname{Spec}(R_{\mathfrak{p}})$ (in the sense that $\{\mathfrak{p}\}$ equals the intersection of $U$ and the complement of $W$). Then $g(P)$ is introduced in [4 Definition 7.9] as $e(U)\otimes_R^{\mathbb{L}}f(W)$, where $e(U)$ is the so-called left idempotent corresponding to $U$ and $f(W)$ is the so-called right idempotent corresponding to $W$. It follows from [16 Proposition 6.10] that

Finally, one observes that the localisation functor corresponding to $W$ is none other than the usual localisation at $\mathfrak{p}$, and so $f(W)\simeq R_{\mathfrak{p}}$ in $D(R)$. It follows that $g(P)=e(U)\otimes_R^{\mathbb{L}}f(W)$ is the complex from the statement.

If now $\mathcal{T}$ is rigidly-compactly generated and $\operatorname{Spc}(\mathcal{T}^\omega)$ is weakly noetherian, we can define a support of each object $x\in\mathcal{T}$.

Finally, we recall a key definition and result for us from [6].

In other words, for stratified categories, one can classify all localising subcategories in terms of the Balmer-Favi support. If, moreover, $\mathcal{T} = \mathrm{D}(R)$, then localising tensor ideals coincide with localising subcategories by the discussion above. It is a classical result that $\mathrm{D}(R)$ is stratified if $R$ is noetherian [31 Theorem 2.8], but it is also well-known that stratification for $\mathrm{D}(R)$ may badly fail if $R$ is not noetherian, [33]. In the rest of the paper, we will characterise when $\mathrm{D}(R)$ is stratified for valuation domains and, more generally, for a large class of rings of weak global dimension bounded by one.

Stratification for valuation domains

Our aim in the section is to prove that for a valuation domain $R$, the category $\mathrm{D}(R)$ is stratified if and only if $R$ is strongly discrete. We begin by proving the local-to-global principle for strongly discrete valuation domains.

Proof. Every noetherian topological space is weakly noetherian by [6 Remark 2.2]. If $\operatorname{Spec}(R)$ is weakly noetherian, then the primes of $R$ satisfy the ascending chain condition by [3 Proposition 4.13].

Finally, suppose that the primes of $R$ satisfy the ascending chain condition. Proving that $\operatorname{Spec}(R)$ is noetherian is the same as proving that each non-empty Zariski open subset $\varnothing\ne U \subseteq\operatorname{Spec}(R)$ is quasi-compact. Now, if $U \subseteq\operatorname{Spec}(R)$ is Zariski open, it is a down-set with respect to the inclusion of primes, and by the ascending chain condition there is a maximal prime ideal $\mathfrak{p}_U \in U$ with respect to inclusion. In particular, we must have $U = \{ \mathfrak{q} \in\operatorname{Spec}(R) \mid\mathfrak{q} \subseteq\mathfrak{p}_U \}$. Thus, if $U = \bigcup_{\lambda\in\Lambda} W_\lambda$ is an open cover of $U$, we must have $\mathfrak{p}_U \in W_\lambda$ for some $\lambda\in\Lambda$ and, consequently, $U = W_\lambda$. □

Proof. If $R$ is strongly discrete, then $\operatorname{Spec}(R)$ is a noetherian space by Theorem 2.4 and Lemma 3.1. Hence also $\operatorname{Spc}(\mathrm{D}(R)^\omega)$ is noetherian by Theorem 2.11, and $\mathrm{D}(R)$ satisfies the local-to-global principle by [39 Theorem 6.9]. □

After proving the local-to-global principle, our next goal is to prove minimality of the localising subcategories generated by the Rickard idempotents $g(P)$ (recall Proposition 2.12). To this end, let us fix some notation. The set of primes $(\operatorname{Spec}(R), \subseteq)$ is totally ordered by inclusion, is non-empty and satisfies the ascending chain condition by Theorem 2.4. If we denote by $(I, \leq)$ a totally ordered set opposite to $(\operatorname{Spec}(R), \subseteq)$, we can use the anti-isomorphism $I \simeq\operatorname{Spec}(R)^\circ$ to index primes; then $\mathfrak{p}_j \subset\mathfrak{p}_i$ whenever $i < j$ in $I$. If we denote $0 = \min I$ and $\infty= \max I$, then $\mathfrak{p}_0$ is the maximal ideal and $\mathfrak{p}_\infty= (0)$. Similarly, using the inclusion-reversing bijection $\operatorname{Spec}(R) \simeq\operatorname{Spc}(\mathrm{D}(R)^\omega)$ from Theorem 2.11, we index elements of $\operatorname{Spc}(\mathrm{D}(R)^\omega)$ by $I$ in such a way that $\mathcal{P}_0 = \{0\}$, $\mathcal{P}_\infty$ consists of all perfect complexes with torsion cohomology, and $\mathcal{P}_i \subset\mathcal{P}_j$ whenever $i < j$ in $I$.

Proof. If $i = \infty$, then $\mathfrak p_{\infty} = 0$ and we can put $n = 1$ and $x_1 = 0$ in Proposition 2.12, giving $g(P_{\infty}) = Q$.

If $i < \infty$, choose any element $s \in\mathfrak p_i \setminus\mathfrak p_{i+1}$ so that we have $V(x_1R_{\mathfrak p_i}) = \{\mathfrak p_i\}$ in $\operatorname{Spec}(R_{\mathfrak p_i})$. Then the intersection $\bigcap_{n\geq1}(s^n)$ is a prime ideal by [19 Lemma II.1.3]. Notice that for every $n$ we must have $\mathfrak p_{i+1} \subseteq(s^n) \subseteq(s)$. Otherwise, the total ordering of the ideals implies $s^n \in\mathfrak p_{i+1}$ and thus $s \in\mathfrak p_{i+1}$; a contradiction. Hence, $\mathfrak p_{i+1} = \bigcap_{n\geq1}(s^n)$, as there are no prime ideals in between $\mathfrak p_{i+1} \subset\mathfrak p_i$. Now consider $R_{\mathfrak p_{i+1}} = (R_{\mathfrak p_i})_{\mathfrak p_{i+1}}$, where we invert

In particular, we may replace $S$ by $T = \bigcup_{n\geq2}(sR_{\mathfrak p_i} \setminus s^nR_{\mathfrak p_i})$, as $S^{-1}R_{\mathfrak p_i} = T^{-1}R_{\mathfrak p_i}$. Recall that $T^{-1}R_{\mathfrak p_i} \cong\operatorname*{colim}_{t\in T}\frac{1}{t}R_{\mathfrak p_i}$, where the system runs over the directed system given by morphisms $\frac{1}{t}R_{\mathfrak p_i} \to\frac{1}{t'}R_{\mathfrak p_i}$ which exist if and only if there exists some $u \in T$ such that $t' = tu$. By the description of $T$ we see that the powers of $s$ are final in this system, i.e., $R_{\mathfrak p_{i+1}} \simeq S^{-1}R_{\mathfrak p_i} = R_{\mathfrak p_i}\left[\frac{1}{s}\right]$. Finally, the conclusion follows from Proposition 2.12 applied to $n = 1$, $x_1 = s$ since then $g(P_i) \simeq(R_{\mathfrak p_i} \to R_{\mathfrak p_{i+1}})$. □

Proof. The case $i = \infty$, i.e., $\operatorname{loc}(g(P_i)) = \operatorname{loc}(Q)$ follows from Lemma 3.3. For a fixed $i < \infty$ we also obtain $\operatorname{loc}(g(P_i)) = \operatorname{loc}(R_{\mathfrak p_{i+1}}/R_{\mathfrak p_i})$, as $g(P_i) \simeq R_{\mathfrak p_{i+1}}/R_{\mathfrak p_i}[-1]$.

By Theorem 2.4(ii) we have $\mathfrak p_iR_{\mathfrak p_i} = sR_{\mathfrak p_i}$ for some non-zero $s \in R_{\mathfrak p_i}$. Without loss of generality, we may assume $s = s/1$ for some $s \in\mathfrak p_i \setminus\mathfrak p_{i+1}$. Then, by the same argument as in the proof of Lemma 3.3, $\mathfrak p_{i+1} = \bigcap_{n\geq1}(s^n)$ and $R_{\mathfrak p_{i+1}} = R_{\mathfrak p_i}\left[\frac{1}{s}\right]$. Now the union of the filtration

whose terms are isomorphic to $R_{\mathfrak p_i}$ is precisely $R_{\mathfrak p_i}\left[\frac{1}{s}\right] = R_{\mathfrak p_{i+1}}$. Passing the filtration to quotient modules, we get

This gives us a filtration of the module $R_{\mathfrak p_{i+1}}/R_{\mathfrak p_i}$ we are interested in. Now we wish to relate this to $k(\mathfrak p_i)$. To this end, notice that for any $n \geq1$ the composite

is surjective and has kernel $R_{\mathfrak p_i}$. In particular, $s^{-n}R_{\mathfrak p_i}/R_{\mathfrak p_i} \cong R_{\mathfrak p_i}/s^nR_{\mathfrak p_i}$ as $R_{\mathfrak p_i}$ modules, which implies they are also isomorphic as $R$-modules. A similar argument reveals that multiplication by $s^{1-n}$ induces an isomorphism $s^{n-1}R_{\mathfrak p_i}/s^nR_{\mathfrak p_i} \cong R_{\mathfrak p_i}/sR_{\mathfrak p_i} = k(\mathfrak p_i)$ of $R$-modules. Jointly, these modules participate in a short exact sequence

which witnesses $R_{\mathfrak p_i}/s^nR_{\mathfrak p_i} \cong s^{-n}R_{\mathfrak p_i}/R_{\mathfrak p_i}$ as an $n$-fold extension of $k(\mathfrak p_i) \cong\ker(\varphi)$. Similarly, multiplication by $s$ induces an isomorphism $R_{\mathfrak p_{i+1}}/R_{\mathfrak p_i} \cong R_{\mathfrak p_{i+1}}/sR_{\mathfrak p_i}$ and these modules form an exact sequence

If follows that $k(\mathfrak{p}_i)$ is the kernel of a surjective endomorphism of $R_{\mathfrak{p}_{i+1}}/R_{\mathfrak{p}_i}$. Finally, since localising subcategories are closed under extensions, cokernels of monomorphisms of $R$-modules, and countable direct limits, we use (1) and the exact sequences to conclude

But since $\operatorname{loc}(g(P_i)) = \operatorname{loc}(R_{\mathfrak{p}_{i+1}}/R_{\mathfrak{p}_i})$, the desired result follows. □

Proof. The minimality at $P_i$ holds because $\operatorname{loc}(k(\mathfrak{p}_i))$ is minimal and equals $\operatorname{loc}(g(P_i))$ by Proposition 3.4. Then by Lemma 3.2 the stratification follows. □

Remark 3.8. Theorem 3.7 gives a new proof that the Telescope Conjecture holds for $\mathrm{D}(R)$, where $R$ is a strongly discrete valuation domain, independent of [BŠ17]. Indeed, Proposition 3.6 shows that $\mathrm{D}(R)$ is stratified and, since $\operatorname{Spec}(R) \simeq\operatorname{Spc}(\mathrm{D}(R)^\omega)$ is a noetherian topological space, the Telescope Conjecture also follows from [6 Theorem 9.11]. In forthcoming work [8], Barthel–Heard–Sanders–Zou provide an independent proof of Theorem 3.7.

The Gabriel dimension

Instead of arguing by elementary means as in the previous section, we shall provide an alternative proof of Proposition 3.6 using the Gabriel dimension. We will briefly recall the concept here and refer to [34 §5.5], [20] for more background.

If $\mathcal{C}$ is a locally coherent Grothendieck category, it is natural to ask how the filtration of the Gabriel dimension translates to filtrations of Serre subcategories in the full subcategory $\mathcal{C}^{\mathrm{fp}}$ formed by the finitely presented objects. This is discussed in [15], where Burke introduces the so-called f.p.-dimension.

Remark 4.3. There is a classical one-to-one correspondence between localising subcategories of finite type of $\mathcal{C}$ and Serre subcategories of $\mathcal{C}^{\mathrm{fp}}$; see [23 Theorem 2.8] or [27 Corollary 2.10]. In particular, each Serre subcategory $\mathcal{F}_{\alpha} \subseteq\mathcal{C}^{\mathrm{fp}}$ extends by direct limits to a localising subcategory $\mathcal{H}_{\alpha} \subseteq\mathcal{C}$ (keeping the notation of [15]). So $\mathcal{C}$ has f.p.-dimension $\alpha$ if and only if $\alpha$ is the smallest ordinal such that $\mathcal{C} = \mathcal{H}_{\alpha}$.

A closely related concept is also that of $m$-dimension of a lattice and of a locally coherent Grothendieck category.

By transfinite induction, we can now define congruences $\sim_{\alpha}$ and lattice quotients $L_{\alpha} = L/\sim^{\alpha}$ as follows:

• $L_0 := L$ and $\sim_0$ is the trivial congruence.

• If $\alpha+ 1$ is a successor ordinal, we put $L_{\alpha+1} = L_{\alpha}/\sim_{L_{\alpha}}$ and $\sim_{\alpha+1}$ is the congruence corresponding to the composite quotient map

• If $\beta$ is a limit ordinal, we set $\sim_{\beta} = \bigcup_{\alpha<\beta} \sim_{\alpha}$.

The $m$-dimension of $L$ is defined to be the least ordinal number $\beta$ such that $L_{\beta+1}$ is the trivial (one-point) lattice, or undefined if no such ordinal exists. If $L = \{\ast\}$, we set it to $-1$.

Remark 4.5. If $L \ne\{\ast\}$, the definition implicitly uses the fact that the smallest ordinal $\gamma$ such that $L_{\gamma} = \{\ast\}$ is always a successor ordinal; see [35 §7.1, p. 282].

Now we can summarise a key result of [15] as follows.

(iii) $\mathcal{C}$ has $m$-dimension, i.e., for every $X \in\mathcal{C}^{\mathrm{fp}}$ the modular lattice of finitely presented subobjects, denoted by $\mathcal{L}_{\mathrm{fp}}(X)$, has $m$-dimension.

Let $R$ be a coherent ring, e.g., a valuation domain, and recall that $\mathcal{C} = \operatorname{Mod}_R$ is a locally coherent Grothendieck category. Then we have:

Proof. The connection between $m$-dimension and dense chains is standard and can be found in [35 Proposition 7.2.3] or [28 Lemma 14.1.11]. Now assume $\mathcal{L}_{\mathrm{fp}}(R)$ has $m$-dimension. The case when this dimension is $-1$ is clear. Otherwise, $R$ must have non-zero finite length in $\operatorname{Mod}_R/\mathcal{H}_\alpha$ for some ordinal $\alpha$ by [15 Theorem 4.5]. In particular, by definition of the filtration $R \in\mathcal{H}_{\alpha+1}$, and thus $\operatorname{Mod}_R$ has f.p.-dimension, hence, Gabriel dimension. The converse is Theorem 4.6. □

Getting back to our alternative approach to Proposition 3.6, we are going to use a key result of [5].

We are now in the position to show that for valuation domains $R$, Gabriel dimension of $\operatorname{Mod}_R$ exists precisely when $R$ is strongly discrete. Hence, we provide a converse to Theorem 4.8 in case $R$ is a valuation domain.

Proof. Using Corollary 4.7, this reduces to the question of the existence of dense chains of the lattice of (non-zero) finitely generated ideals. Such ideals are principal, and so are classified by non-zero elements of $R$ up to multiplication by a unit of $R$, and so also by elements of the positive cone $\Gamma(R)^+ = \{v \in\Gamma(R) \mid v \ge0\}$ of the value group $\Gamma(R)$ of $R$ (cp. [14 Proposition 2.8]). Thus, it suffices to show: $R$ is strongly discrete if and only if $(\mathbb{Q}, \le)$ does not embed into $(\Gamma(R), \le)$. By [Pun92, Corollary 4.2] the latter is equivalent to the non-existence of superdecomposable pure-injective $R$-modules. Then [19 Theorem XIII.5.11] yields the result. □

Proof. Again, (i) and (ii) are equivalent by [BŠ17, §7], and (i) and (iv) are equivalent by Corollary 4.9. If (i) holds, then $\operatorname{Spec}(R)^\vee$ is a $T_D$-space by Theorem 2.4 and Lemma 3.1, so it follows from Theorem 4.8 that (i) implies (iii). Finally, we can use the same argument as in Theorem 3.7 for the fact that (iii) implies (ii). If (iii) holds, then $\operatorname{Spec}(R)$ is weakly noetherian, so noetherian by Lemma 3.1, and then (ii) follows from [6 Theorem 9.11]. □

In particular, by Theorem 4.10 there is an isomorphism $\operatorname{Loc}(\operatorname{D}(R)) \cong\mathcal{P}(\operatorname{Spec}(R))$ for strongly discrete valuation domains $R$. However, the following example shows that the mere existence of such an isomorphism for a given valuation domain $R$ does not necessarily imply that $R$ is strongly discrete.

Example 4.11. By [14 Proposition 5.1, Remark 5.2] there exists a valuation domain $R$ with the following properties:

• (i) The value group is the lexicographic product of copies of $\mathbb{Z}$, indexed by $\mathbb{N}$.

• (ii) The prime ideals $\mathfrak{p}_i$ are indexed by $\mathbb{N} \cup\{\infty\}$, form an ascending chain by the natural order, $\mathfrak{p}_0 = (0)$, and $\mathfrak{p}_\infty= \bigcup_{i \in\mathbb{N}} \mathfrak{p}_i$ is maximal.

• (iii) The only idempotent prime ideals are zero and $\mathfrak{p}_\infty$.

As $\mathfrak{p}_\infty$ is idempotent and non-zero, $\operatorname{D}(R)$ cannot satisfy the Telescope Conjecture by [BS17, §7], hence Gabriel dimension for $\operatorname{Mod}_R$ is undefined by Theorem 4.10. Moreover, $\operatorname{Spec}(R) \cong\operatorname{Spc}(\operatorname{D}(R)^\omega)$ is not weakly noetherian by Lemma 3.1 (in fact $\mathfrak{p}_\infty\in\operatorname{Spec}(R)$ is not weakly visible).

However, we claim that $L := \operatorname{loc}(k(\mathfrak{p}_i) \mid i \in\mathbb{N} \cup\{\infty\})$ equals $\operatorname{D}(R)$, and therefore $\operatorname{Loc}(\operatorname{D}(R)) \cong\mathcal{P}(\operatorname{Spec}(R))$. Thus, $R$ provides a counterexample to the converse of [5 Theorem 2.1] and shows that having Gabriel dimension is not necessary for residue fields to generate.

To prove $L = \operatorname{D}(R)$, let $i \in\mathbb{N}$ and define $L_i := \operatorname{loc}(k(\mathfrak{p}) \mid\mathfrak{p} \subseteq\mathfrak{p}_i) \subseteq\operatorname{D}(R)$. As $R_{\mathfrak{p}_i}$ has discrete value group and is of finite Krull dimension, it is strongly discrete [19 Proposition II.8.4]. Hence, using Lemma 3.2 and Proposition 3.4, (or alternatively Theorem 4.10 together with [5 Theorem 2.1]), we find that $\operatorname{D}(R_{\mathfrak{p}_i})$ is generated by residue fields. As the restriction functor $\operatorname{D}(R_{\mathfrak{p}_i}) \to\operatorname{D}(R)$ is fully faithful and coproduct-preserving, we obtain $L_i = \operatorname{loc}(R_{\mathfrak{p}_i})$. Since $L_i \subset L$, we have $R_{\mathfrak{p}_i} \in L$ for any $i \in\mathbb{N}$. Now, crucially, $\mathfrak{p}_iR_{\mathfrak{p}_i} = \mathfrak{p}_i$ as subsets of $R_{\mathfrak{p}_i}$ since $\mathfrak{p}_i$ is canonically an $R_{\mathfrak{p}_i}$-module by [19 Lemma 4.3(iv)], and $\mathfrak{p}_iR_{\mathfrak{p}_i} = s_iR_{\mathfrak{p}_i}$ for some $s_i \in R$ since $R_{\mathfrak{p}_i}$ is strongly discrete. Hence, $\mathfrak{p}_i \cong R_{\mathfrak{p}_i}$ as $R$-modules for $i \ne0$ and so $\mathfrak{p}_i \in L$. As $\mathfrak{p}_\infty$ is the countable filtered colimit of $\mathfrak{p}_i$, we have $\mathfrak{p}_\infty\in L$. As $k(\mathfrak{p}_\infty) = R/\mathfrak{p}_\infty\in L$, this forces $R \in L$ and $L = \operatorname{D}(R)$. This proves the claim.

We have now proven that the small Foxby support stratifies in this case, which replaces the Balmer-Favi support in Theorem 2.15 and realizes the isomorphism $\operatorname{Loc}(\operatorname{D}(R)) \cong\mathcal{P}(\operatorname{Spec}(R))$. Further, we will provide a negative answer saying that the smashing support as defined in [13 §3.2] is strictly finer than the Balmer-Favi support in our example, does not do the job for stratification.

In order to analyse the smashing support, we first need to compute the lattice $\operatorname{Sm}(\operatorname{D}(R))$ of smashing localising subcategories of $\operatorname{D}(R)$, which is known to be a frame (i.e. a complete lattice where finite meets distribute over arbitrary joins) by the main result of [11]. The following construction is due to [BS17, §5], but for ease we will refer to [14 Construction 3.3, Theorem 3.4]. For our ring $R$, let $j \geq-1$. The elements of the frame, which are representatives of equivalence classes of homological epimorphisms originating from $R$, can be described as follows:

A homological epimorphism $f:R\to S$ corresponds to the smashing localising subcategory $S=\ker(S\otimes_R^{\mathbb{L}}-)\subseteq\mathrm{D}(R)$. Adopting notation of [14 §4.1], we obtain Figure 1. Note that the bottom element $0=\ker(R\otimes_R^{\mathbb{L}}-)$ of the smashing frame is a meet-prime as the intersection of two non-zero smashing subcategories is non-zero.

The compactly generated localisations of $\mathrm{D}(R)$ correspond to the flat epimorphisms and are encircled in the figure. Let $C_j\in\operatorname{Sm}(\mathrm{D}(R))$ correspond to $f_j$ for $-1\leq j\leq\infty$, which is the upper row in Figure 1, and analogously $S_j$ to $g_j$ for $-1\leq j<\infty$; the lower row. By the kernel description and $\operatorname{Loc}(\mathrm{D}(R))\cong\mathcal{P}(\operatorname{Spec}(R))$ we get

Having described the smashing frame, we can construct the smashing spectrum $\operatorname{Spc}^{s}(\mathrm{D}(R))$ that we obtain by Stone duality; we refer to [13 §2.1 and §3] for details. The points of $\operatorname{Spc}^{s}(\mathrm{D}(R))$ are the meet-primes of $\operatorname{Sm}(\mathrm{D}(R))$, i.e. the coloured points in Figure 1. We also have the smashing support: if $x\in\mathrm{D}(R)$, then $\operatorname{supp}^{s}(x)=\{P\in\operatorname{Spc}^{s}(\mathrm{D}(R))\mid x\notin P\}$. An open basis of $\operatorname{Spc}^{s}(\mathrm{D}(R))$ is given by the sets

where $S$ runs over the elements of the smashing frame, i.e. the smashing localising subcategories of $\mathrm{D}(R)$. Note also that for each $S\in\operatorname{Sm}(\mathrm{D}(R))$, the following holds tautologically by [13 Remark 3.2.5]:

We claim that $\operatorname{supp}^{s}$ cannot induce an isomorphism of lattices given by mapping a localising subcategory to its support,

To this end, notice that by our explicit computations, we have

Furthermore, $\bigcap_{j\geq0}S_j=C_\infty$, while $\bigcap_{j\geq0}\operatorname{supp}^{s}(S_j)=\{C_\infty\}$. Hence, if $\operatorname{supp}^{s}$ induced an order isomorphism, and thus stratified $\mathrm{D}(R)$ by $\operatorname{Spc}^{s}(\mathrm{D}(R))$, it would preserve these meets. This is absurd, however, as we would obtain

Stratification for rings of weak global dimension at most one

In this section we characterise for a certain class of commutative rings of weak global dimension at most one when $\operatorname{Spc}(\mathrm{D}(R)^{\omega})^{\vee}$ stratifies $\mathrm{D}(R)$. Recall from Lemma 2.5 that a commutative ring has weak global dimension at most one if and only if the localisation at any prime is a valuation domain. By [BS17, §7] we know that for such rings, $\mathrm{D}(R)$ satisfies the Telescope Conjecture precisely when $R_{\mathfrak{p}}$ is a strongly discrete valuation domain for all prime ideals $\mathfrak{p}$.

Recall from Theorem 2.4 that the ascending chain condition on primes holds for strongly discrete valuation domains. The good news is:

Proof. We shall prove that for any prime ideal $\mathfrak{p}$ there are Thomason subsets $U$, $V$ in $\operatorname{Spec}(R)$ satisfying $U \cap V^{\mathrm{c}} = \{\mathfrak{p}\}$. As $R$ is strongly discrete, there exists by Lemma 2.5 and Theorem 2.4 some $x \in\mathfrak{p}$ such that $xR_{\mathfrak{p}} = \mathfrak{p}R_{\mathfrak{p}}$, as $R_{\mathfrak{p}}$ is local. Note that $\mathfrak{p}$ is minimal over $(x)$. Let $V := V(x) \setminus\{\mathfrak{p}\}$ and $\mathfrak{q} \in V$. Then by construction, we cannot have $\mathfrak{q} \subsetneq\mathfrak{p}$. Thus, there exists some $y \in\mathfrak{q} \setminus\mathfrak{p}$, and we have $(x,y) \nsubseteq\mathfrak{p}$, $V(x,y) \subseteq V$. Therefore, we may choose the Thomason subsets

and $U = V(x)$, which satisfy $U \cap V^{\mathrm{c}} = \{\mathfrak{p}\}$. □

Now, the bad news is that there is no analogous statement of Theorem 3.7 for strongly discrete rings of weak global dimension at most one. This is because it already fails for absolutely flat rings, i.e. rings of weak global dimension zero. Explicitly for a field $K$, the infinite product $\prod_{\mathbb{N}} K$ serves as a counterexample; see [40 Example 4.13]. However, we can rely on the following.

Notation 5.5. Let $R$ be strongly discrete of weak global dimension at most one. For a non-empty generalisation-closed subset $C \subseteq\operatorname{Spec}(R)$ let $\operatorname{Max}(C)$ denote the maximal elements of $C$. Note that this is non-empty as $R$ is strongly discrete.

Proof. Assume $\mathfrak{p}$ is isolated in $C$. Then Remark 5.4 gives $\mathfrak{p} \in\operatorname{Max}(C)$. By definition, there exists an open $U \subseteq\operatorname{Spec}(R)^{\vee}$ such that $C \cap U = \{\mathfrak{p}\}$. Since $V(I)$, with $I$ finitely generated, form a basis, we may replace $U$ by some $V(I)$. Hence, $C \cap V(I) = \{\mathfrak{p}\}$, i.e., $\mathfrak{p}$ contains $I$ and no other $\mathfrak{q} \in\operatorname{Max}(C)$ contains $I$.

Conversely, assume that $I$ is a finitely generated ideal such that $I \subseteq\mathfrak{p}$, but $I \not\subseteq\mathfrak{q}$ for $\mathfrak{p} \ne\mathfrak{q} \in\operatorname{Max}(C)$. Let $t \in\mathfrak{p}$ such that $tR_{\mathfrak{p}}=\mathfrak{p}R_{\mathfrak{p}}$, which exists as $R$ is strongly discrete, and consider the finitely generated ideal $I+tR$. From the assumptions $C \setminus\{\mathfrak{p}\}=C\cap(\operatorname{Spec}(R)\setminus V(I+tR))$ is an intersection of two closed sets in $\operatorname{Spec}(R)^{\vee}$, so is itself closed in $\operatorname{Spec}(R)^{\vee}$. But then $\mathfrak{p}\notin(C\setminus\{\mathfrak{p}\})^{\vee}$, i.e., $\mathfrak{p}$ is isolated in $C$. □

The property of Lemma 5.6 has been studied in the context of Prüfer domains.

Let us provide a non-trivial example where the local-to-global principle holds.

Let us also discuss how sharpness can be detected by flat ring epimorphisms. By [BŠ17, Proposition 6.7] there is a bijection between non-empty closed subsets in $\operatorname{Spec}(R)^{\vee}$ and equivalence classes of flat ring epimorphisms $f:R\to S\ne0$. The correspondence sends a representative $f$ to $C_S=\{\mathfrak{q}\in\operatorname{Spec}(R)\mid\mathfrak{q}S\ne S\}$.

Proof. As $f$ is a flat epimorphism of rings, $\operatorname{Spec}(f)$ is by [30 Corollaire IV.2.2] a homeomorphism onto its image, which is $C_S$. As such, it is straightforward to show that the condition of Definition 5.7 translates to the existence of a finitely generated ideal $J\subseteq S$ such that $S/J$ is local. In particular, denoting by $\mathfrak{n}$ the unique maximal ideal of $V(J)\subseteq\operatorname{Spec}(S)$, $(S/J,\mathfrak{n}/J)$ is as required.

Suppose conversely that $J \subseteq S$ is a finitely generated ideal and $(S/J,\mathfrak{n}/J)$ is a local ring. Since $S_{\mathfrak{n}}$ is a strongly discrete valuation domain, there exists $t \in S$ such that $\mathfrak{n}S_{\mathfrak{n}}=tS_{\mathfrak{n}}$. In particular, $V(J+tS)=\{\mathfrak{n}\}$ in $\operatorname{Spec}(S)\cong C_S$.

Definition 5.10. Let $R$ be a ring of weak global dimension at most one. We shall call the flat epimorphisms satisfying the condition in Lemma 5.9 sharp.

Let us summarise our findings of this section in the following result.

Proof. Suppose $R$ is strongly discrete and the local-to-global principle holds. Then $\operatorname{Spc}(D(R)^{\omega})^{\vee}$ is a $T_D$-space by Lemma 5.2 and $D(R)$ is stratified at each local category $D(R_{\mathfrak{p}})$ by definition and Theorem 3.7. Thus, $D(R)$ is stratified by [6 Corollary 5.3]. The other implications are straightforward given Theorem 5.3 and Lemma 5.9.

Finally, using Theorem 5.11 we may extend the result of Theorem 4.10 to coherent rings of weak global dimension at most one, e.g., Prüfer domains.

Proof. Assume $R$ is strongly discrete. Using Theorem 5.11 and Theorem 5.3, $X=\operatorname{Spec}(R)^{\vee}$ is scattered. As usual, denote $X^0=X$, $X^{\alpha+1}=X^\alpha\setminus\operatorname{Isol}(X^\alpha)$ for successor ordinals $\alpha$, where $\operatorname{Isol}(X^\alpha)\subseteq X^\alpha$ denotes the set of isolated points, and $X^\lambda=\bigcap_{\alpha<\lambda}X^\alpha$ for limit ordinals $\lambda$. This procedure gives $X^\gamma=\varnothing$ for some ordinal $\gamma$, so for $\varnothing\ne U\in\Omega(X)^\omega$ there is a least ordinal $\beta$ with $U\cap X^\beta=\varnothing$. Note that $\beta$ cannot be a limit ordinal. Otherwise $(U\setminus(U\cap X^\alpha))_{\alpha<\beta}$ yields an open cover of $U$, thus there is a finite subcover, which contradicts $\varnothing=U\cap X^\beta=\bigcap_{\alpha<\beta}U\cap X^\alpha$. Hence, $\alpha(U):=\beta-1$ is the largest ordinal satisfying $U_X:=U\cap X^{\alpha(U)}\ne\varnothing$. In particular, $U_X$ is closed in $U$ and it must by finite quasi-compactness of $U$.

Let us now prove that (i) implies (iv). If Gabriel dimension of $\operatorname{Mod}_R$ is undefined, there exists a dense chain $\mathcal{C}$ of finitely generated ideals by Corollary 4.7. Pick $I\subsetneq J$ in $\mathcal{C}$. Since $J/I\ne0$ is finitely generated, we have $\varnothing\ne\operatorname{Supp}(J/I)=V(\operatorname{Ann}(J/I))$. The annihilator of $J/I=m_1R+\cdots+m_kR$ is finitely presented by coherence, as it is the kernel of $R\to(J/I)^k$, $r\mapsto(m_1r,\ldots,m_kr)$. In particular, $V(\operatorname{Ann}(J/I))=V(I:J)$ is Zariski closed with quasi-compact complement in $\operatorname{Spec}(R)$. Thus, $\varnothing\ne\operatorname{Supp}(J/I)\subseteq\Omega(X)^\omega$ and we may use the previous paragraph for any pair $I\subsetneq J$ in $\mathcal{C}$. To show some $R_{\mathfrak{p}}$ is not strongly discrete for some prime $\mathfrak{p}$, it suffices by Corollary 4.9 to find $S\subsetneq T$ such that $(J/I)_{\mathfrak{p}}\ne0$ for any $S\subseteq I\subsetneq J\subseteq T$, where $S,I,J,T\in\mathcal{C}$. For ideals $I\subsetneq J$ in $\mathcal{C}$ define $f(I,J):=(\alpha(\operatorname{Supp}(J/I)),\#\operatorname{Supp}(J/I)_X)$. Let the codomain of $f$ be its image and endow it with the lexicographical order. Out of this image, pick a pair, $S\subsetneq T$, in the preimage of the minimal value under this order. Let $S \subseteq I \subsetneq J \subseteq T$ be in $\mathcal{C}$, and set $U = \operatorname{Supp}(T/S)$. Then $\operatorname{Supp}(J/I) \subseteq U$ as $J/I$ is a subquotient of $T/S$. Hence, $\alpha(\operatorname{Supp}(J/I)) \leq\alpha(U)$, and by construction this forces equality. Thus, $\operatorname{Supp}(J/I)_{X} \subseteq U_{X}$. By construction we must have $\#U_{X} \leq\#\operatorname{Supp}(J/I)_{X}$, and finiteness gives $\operatorname{Supp}(J/I)_{X} = U_{X}$, i.e., $f(I,J) = f(S,T)$. By design, we have now for each $\mathfrak{p} \in U_{X}$ that $\mathfrak{p} \in\operatorname{Supp}(J/I)$, i.e., $(J/I)_{\mathfrak{p}} \ne0$. This finishes the proof of the implication.

Assume $\operatorname{Mod}_{R}$ has Gabriel dimension. Then $R$ must be strongly discrete by Theorem 4.10, so $(i)$ and $(iv)$ are equivalent.

If $R$ is strongly discrete then, with the assumption and Theorem 5.11, the stratification of $\operatorname{D}(R)$ follows, showing that $(i)$ implies $(iii)$.

If $\operatorname{D}(R)$ is stratified, $\operatorname{Spec}(R)$ must be weakly noetherian by definition, and so must be $\operatorname{Spec}(R_{\mathfrak{p}})$ for each $\mathfrak{p}$. By Lemma 3.1, this implies that $\operatorname{Spec}(R_{\mathfrak{p}})$ is noetherian for each $\mathfrak{p}$. In particular, $\operatorname{Spec}(R)$ is generically noetherian in the sense of [6 Definition 9.5], so [6 Theorem 9.11] applies, showing that the stratification implies the Telescope Conjecture. In other words, $(iii)$ implies $(ii)$.

The final implication from $(ii)$ to $(i)$ follows by [BŠ17, §7], since strongly discrete rings $R$ are characterised among those of weak global dimension at most one by $\operatorname{D}(R)$ satisfying the Telescope Conjecture.

References

Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague, Czech Republic

Email address: [email protected]

Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany

Email address: [email protected]