Telescope Conjecture Research Articles

Homotopy Theory

The workshop brought together experts in homotopy theory from many areas, including chromatic homotopy theory, algebraic K-theory, derived algebra and equivariant homotopy theory. We had a lecture series on the recent disproof of the telescope conjecture. In addition to a total of 24 research talks we had two gong-shows where participants presented their research in 10-minute talks.

The Telescope Conjecture for von Neumann regular rings

In this note, we show that any epimorphism originating at a von Neumann regular ring (not necessary commutative) is a universal localization. As an application, we prove that the Telescope Conjecture holds for the unbounded derived categories of von Neumann regular rings (not necessary commutative).

On stratification for spaces with noetherian mod p cohomology

Let $X$ be a topological space with Noetherian mod $p$ cohomology and let $C^*(X;\mathbb{F}_p)$ be the commutative ring spectrum of $\mathbb{F}_p$-valued cochains on $X$. The goal of this paper is to exhibit conditions under which the category of module spectra on $C^*(X;\mathbb{F}_p)$ is stratified in the sense of Benson, Iyengar, Krause, providing a classification of all its localizing subcategories. We establish stratification in this sense for classifying spaces of a large class of topological groups including Kac--Moody groups as well as whenever $X$ admits an $H$-space structure. More generally, using Lannes' theory we prove that stratification for $X$ is equivalent to a condition that generalizes Chouinard's theorem for finite groups. In particular, this relates the generalized telescope conjecture in this setting to a question in unstable homotopy theory.

The telescope conjecture at height 2 and the tmf resolution

Mahowald proved the height 1 telescope conjecture at the prime 2 as an application of his seminal work on bo-resolutions. In this paper, we study the height 2 telescope conjecture at the prime 2 through the lens of tmf-resolutions. To this end, we compute the structure of the tmf-resolution for a specific type 2 complex Z. We find that, analogous to the height 1 case, the E 1 -page of the tmf-resolution possesses a decomposition into a v 2 -periodic summand, and an Eilenberg–MacLane summand which consists of bounded v 2 -torsion. However, unlike the height 1 case, the E 2 -page of the tmf-resolution exhibits unbounded v 2 -torsion. We compare this to the work of Mahowald–Ravenel–Shick, and discuss how the validity of the telescope conjecture is connected to the fate of this unbounded v 2 -torsion: either the unbounded v 2 -torsion kills itself off in the spectral sequence, and the telescope conjecture is true, or it persists to form v 2 -parabolas and the telescope conjecture is false. We also study how to use the tmf-resolution to effectively give low-dimensional computations of the homotopy groups of Z. These computations allow us to prove a conjecture of the second author and Egger: the E ( 2 ) -local Adams–Novikov spectral sequence for Z collapses.

Definable coaisles over rings of weak global dimension at most one

In the setting of the unbounded derived category D(R) of a ring R of weak global dimension at most one we consider t-structures with a definable coaisle. The t-structures among these which are stable (that is, the t-structures which consist of a pair of triangulated subcategories) are precisely the ones associated to a smashing localization of the derived category. In this way, our present results generalize those of [8] to the non-stable case. As in the stable case [8], we confine for the most part to the commutative setting, and give a full classification of definable coaisles in the local case, that is, over valuation domains. It turns out that, unlike in the stable case of smashing subcategories, the definable coaisles do not always arise from homological ring epimorphisms. We also consider a non-stable version of the Telescope Conjecture for t-structures and give a ring-theoretic characterization of the commutative rings of weak global dimension at most one for which it is satisfied.

Weak global dimension and CM-freeness of rings

Let $R$ be a ring with finite weak global dimension. We prove that $R$ is (strongly) CM-free. As applications,we obtain some triangulated equivalences between some compactly-generated homotopy categories and relative derived categories.We classify the definable categories of such equivalent triangulated categories when the weak global dimension of $R$ is at most 1, and show that such equivalent triangulated categoriessatisfy Telescope conjecture when $R$ is a von Neumann regular ring. We also improve the condition such that a triangulated equivalence about the extended Grothendieck duality holdsfrom the finiteness of the left and right global dimension of $R$ to only the finiteness of the right global dimension of $R$.

Telescope conjecture for homotopically smashing t-structures over commutative noetherian rings

We show that any homotopically smashing t-structure in the derived category of a commutative noetherian ring is compactly generated. This generalizes the validity of the telescope conjecture for commutative noetherian rings due to Neeman. As another consequence, we obtain a cofinite type result for pure-injective cosilting objects. We also give a formulation of telescope conjecture for homotopically smashing t-structures in underlying triangulated categories of certain Grothendieck derivators.

Algebraic chromatic homotopy theory for BP∗BP-comodules

In this paper, we study the global structure of an algebraic avatar of the derived category of ind-coherent sheaves on the moduli stack of formal groups. In analogy with the stable homotopy category, we prove a version of the nilpotence theorem as well as the chromatic convergence theorem, and construct a generalized chromatic spectral sequence. Furthermore, we discuss analogs of the telescope conjecture and chromatic splitting conjecture in this setting, using the local duality techniques established earlier in joint work with Valenzuela.

Approximations and Mittag-Leffler conditions the tools

Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [20], [14], [19]. If R is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class of all flat Mittag-Leffler modules is not deconstructible [16], and it does not provide for approximations when R has cardinality ≤ ℵ0, [8]. We remove the cardinality restriction on R in the latter result. We also prove an extension of the Countable Telescope Conjecture [23]: a cotorsion pair (A, B) is of countable type whenever the class B is closed under direct limits. In order to prove these results, we develop new general tools combining relative Mittag-Leffler conditions with set-theoretic homological algebra. They make it possible to trace the above facts to their ultimate, countable, origins in the properties of Bass modules. These tools have already found a number of applications: e.g., they yield a positive answer to Enochs’ problem on module approximations for classes of modules associated with tilting [4], and enable investigation of new classes of flat modules occurring in algebraic geometry [26]. Finally, the ideas from Section 3 have led to the solution of a long-standing problem due to Auslander on the existence of right almost split maps [22].

The telescope conjecture for algebraic stacks

Using Balmer--Favi's generalized idempotents, we establish the telescope conjecture for many algebraic stacks. Along the way, we classify the thick tensor ideals of perfect complexes of stacks.

Smashing localizations of rings of weak global dimension at most one

We show for a ring R of weak global dimension at most one that there is a bijection between the smashing subcategories of its derived category and the equivalence classes of homological epimorphisms starting in R. If, moreover, R is commutative, we prove that the compactly generated localizing subcategories correspond precisely to flat epimorphisms. We also classify smashing localizations of the derived category of any valuation domain, and provide an easy criterion for the Telescope Conjecture (TC) for any commutative ring of weak global dimension at most one. As a consequence, we show that the TC holds for any commutative von Neumann regular ring R, and it holds precisely for those Prüfer domains which are strongly discrete.

Affine weakly regular tensor triangulated categories

We prove that the Balmer spectrum of a tensor triangulated category is homeomorphic to the Zariski spectrum of its graded central ring, provided the triangulated category is generated by its tensor unit and the graded central ring is noetherian and regular in a weak sense. There follows a classification of all thick subcategories, and the result extends to the compactly generated setting to yield a classification of all localizing subcategories as well as the analog of the telescope conjecture. This generalizes results of Shamir for commutative ring spectra.

Derived categories of representations of small categories over commutative noetherian rings

We study the derived categories of small categories over commutative noetherian rings. Our main result is a parametrization of the localizing subcategories in terms of the spectrum of the ring and the localizing subcategories over residue fields. In the special case of representations of Dynkin quivers over a commutative noetherian ring we give a complete description of the localizing subcategories of the derived category, a complete description of the thick subcategories of the perfect complexes and show the telescope conjecture holds. We also present some results concerning the telescope conjecture more generally.

Variations of the telescope conjecture and Bousfield lattices for localized categories of spectra

We investigate several versions of the telescope conjecture on localized categories of spectra, and implications between them. Generalizing the "finite localization" construction, we show that on such categories, localizing away from a set of strongly dualizable objects is smashing. We classify all smashing localizations on the harmonic category, HFp-local category and I-local category, where I is the Brown-Comenetz dual of the sphere spectrum; all are localizations away from strongly dualizable objects, although these categories have no nonzero compact objects. The Bousfield lattices of the harmonic, E(n)-local, K(n)-local, HFp-local and I-local categories are described, along with some lattice maps between them. One consequence is that in none of these categories is there a nonzero object that squares to zero. Another is that the HFp-local category has localizing subcategories that are not Bousfield classes.

A local-global principle for the telescope conjecture

We prove an étale local-global principle for the telescope conjecture and use it to show that the telescope conjecture holds for derived categories of Azumaya algebras on noetherian schemes as well as for many classifying stacks and gerbes. This specializes to give another proof of the fact that the telescope conjecture holds for noetherian schemes.

Subcategories of singularity categories via tensor actions

AbstractWe obtain, via the formalism of tensor actions, a complete classification of the localizing subcategories of the stable derived category of any affine scheme that has hypersurface singularities or is a complete intersection in a regular scheme; in particular, this classifies the thick subcategories of the singularity categories of such rings. The analogous result is also proved for certain locally complete intersection schemes. It is also shown that from each of these classifications one can deduce the (relative) telescope conjecture.

Stratifying modular representations of finite groups

We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equiv- alently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.

Generalized tensor idempotents and the telescope conjecture

We transpose Rickard's construction of idempotent representations to the general context of tensor triangulated categories. We connect our construction to the telescope conjecture and prove that the latter is of local nature. As an illustration, we extend the affine telescope theorem of Neeman to arbitrary noetherian schemes. We also develop supports for non-compact objects.

The telescope conjecture for hereditary rings via Ext-orthogonal pairs

For the module category of a hereditary ring, the Ext-orthogonal pairs of subcategories are studied. For each Ext-orthogonal pair that is generated by a single module, a 5-term exact sequence is constructed. The pairs of finite type are characterized and two consequences for the class of hereditary rings are established: homological epimorphisms and universal localizations coincide, and the telescope conjecture for the derived category holds true. However, we present examples showing that neither of these two statements is true in general for rings of global dimension 2.

Telescope conjecture, idempotent ideals, and the transfinite radical

We show that for an artin algebra Λ \Lambda , the telescope conjecture for module categories is equivalent to certain idempotent ideals of mod ⁡ Λ \operatorname {mod}\Lambda being generated by identity morphisms. As a consequence, we prove the conjecture for domestic standard selfinjective algebras and domestic special biserial algebras. We achieve this by showing that in any Krull-Schmidt category with local d.c.c. on ideals, any idempotent ideal is generated by identity maps and maps from the transfinite radical.