$\mathrm{Mot}^{\mathrm{loc}}$ is not compactly generated
We prove that the $\infty$-category of localizing motives in the sense of Blumberg--Gepner--Tabuada is not compactly generated, extending a result of Efimov.
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arXiv subjects
Maxime Ramzi
Publications and source records attributed to Maxime Ramzi.
The Galois theory of $G$-spectra and the Burnside ring
We prove that the Galois groupoid of the category of $G$-spectra for a finite group $G$ is algebraic, i.e. equivalent to the \'etale fundamental groupoid of the Burnside ring of $G$. We implement an algorithm that computes the latter from the table of marks of $G$, and provide numerous examples.
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Free rigid commutative algebras
We describe free rigid commutative algebras in $2$-presentably symmetric monoidal $(\infty,2)$-categories as oplax colimits over the $1$-dimensional framed cobordism category. The special case of the $(\infty,2)$-category $\mathrm{Pr}^\mathrm{L}$ itself provides a description of the free symmetric monoidal $(\infty,1)$-category with duals on a given $(\infty,1)$-category, while the case of $\mathrm{Mod}_{\mathcal{V}}(\mathrm{Pr}^\mathrm{L})$ provides a description of a similar object in the $\mathcal{V}$-enriched context, for $\mathcal{V}$ a presentably symmetric monoidal $(\infty,1)$-category. As a byproduct, we obtain new proofs of some results about rigidification of locally rigid categories, as well as a proof that any rigid category over $\mathrm{Sp}$ embeds into a compactly-rigidly generated one.
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Chromatic Noshift
The chromatic redshift philosophy, introduced by Ausoni and Rognes, suggests that algebraic $K$-theory raises chromatic height by $1$. We show that the analogue of this philosophy fails in the case of rigid symmetric monoidal stable $\infty$-categories. More precisely, we construct examples of rigid $T(n)$-local categories $C$ where a refinement $\mathrm{Dim}$ of the dimension morphism induces an equivalence $$K(C)\to \mathrm{End}(\mathbf{1}_C)^{BS^1}$$ and for which $K(C)$ therefore vanishes $T(n+1)$-locally. In fact, we prove that this equivalence always holds for $\aleph_1$-Nullstellensatzian rigid $T(n)$-local categories in the sense of Burklund, Schlank and Yuan. We study more in depth the rational version of these results to find a rigid rational additive $1$-category witnessing the failure of redshift at height $0$. Finally, we use our methods to prove and generalize a conjecture of Levy about categorification of ordinary rings.
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Geometric Points in Tensor Triangular Geometry
In this paper, we study geometric points in tensor triangular geometry. In doing so, we construct a counter-example to Balmer's Nerves of Steel conjecture using free constructions in higher Zariski geometry. We then go on to introduce and discuss constructible spectra in the context of tensor triangular geometry. For tensor triangulated categories satisfying a mild enhancement condition, we use these spectra to construct geometric incarnations of (homological or triangular) primes via maps to "pointlike" tensor triangulated categories.
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Equivariant localizing motives and multiplicative norms on algebraic K-theory
We construct multiplicative norms on equivariant nonconnective algebraic $K$-theory for finite groups $G$. We also construct a genuine equivariant version of THH equipped with a Dennis trace map from K-theory compatible with the multiplicative norms. To do so, we follow the general strategy of Blumberg-Gepner-Tabuada in the nonequivariant case by generalizing their category of localizing motives to the genuine equivariant context, building upon the theory of perfect $G$-stable categories of the first-named author. Crucially, we proceed using the recent perspective on noncommutative motives by the second-named author with Sosnilo and Winges which allows us to deal with non-exact functors on this category of motives. Together with an isotropy separation argument for equivariant cubes, we prove our main theorem that norms of stable categories preserve equivariant motivic equivalences. As an immediate consequence, we obtain a unique equivariant multiplicative refinement of nonconnective algebraic $K$-theory. From these constructions and results, we draw several applications, namely: (1) that the endofunctor of (equivariant) tensor powers on ordinary perfect stable categories preserve motivic equivalences; (2) that the multiplicative norms also preserve the additive motivic equivalences, thus yielding a motivic refinement of a result of Elmanto-Haugseng and Cnossen-Haugseng-Lenz-Linskens that connective algebraic K-theory admits multiplicative norms; (3) we construct a genuine equivariant version of topological Hochschild homology equipped with a Dennis trace map that is compatible with multiplicative norms; and (4) we prove that every genuine $G$-spectrum is the K-theory of a perfect $G$-stable category.
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Localizing motives of Azumaya algebras
We prove that the natural map from the derived Brauer group of a qcqs scheme $X$ to the Picard group of $X$-linear motives is injective, extending results of Tabuada and Tabuada-Van den Bergh.
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$K(1)$-local $K$-theory of Azumaya algebras
We compute certain strict Picard spectra of $K(1)$-local $K$-theory spectra of schemes in terms of Brauer groups, using the map that takes an Azumaya algebra to its $K(1)$-local $K$-theory and proving a K\"unneth formula in that setting. For example, we prove that for semi-local rings of characteristic $\neq p$, $\mathbf{Br}(R)[p^\infty]\simeq \mathbb{G}_{\mathrm{pic}}(L_{K(1)}K(R)\otimes\mathbb{S}_{W(\overline{\mathbb F}_p)})[p^\infty]$, where $\mathbb{G}_{\mathrm{pic}}$ is Carmeli's strict Picard spectrum. We prove the same result for $R[\frac{1}{p}]$, when $R$ is $p$-henselian.
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$p$-perfection and group completion of $\mathbb{E}_\infty$-monoids
We study $\mathbb{E}_\infty$-monoids on which a prime $p$ acts invertibly, which we call $p$-perfect, in the non-group-complete situation. In particular, we prove that in many examples, they almost embed in their group-completion. We further study the $p$-perfection functor, and describe it in terms of Quillen's $+$-construction, similarly to group-completion. This gives an alternative description of the $p$-inverted higher algebraic $K$-theory of a ring.
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Every motive is the motive of a stable $\infty$-category
We define a class of motivic equivalences of small stable $\infty$-categories $W_{\mathrm{mot}}$ and show that the Dwyer--Kan localization functor $\mathrm{Cat}^{\mathrm{perf}}_\infty \to \mathrm{Cat}^{\mathrm{perf}}_\infty[W_{\mathrm{mot}}^{-1}]$ is the universal localizing invariant in the sense of Blumberg--Gepner--Tabuada. In particular, we show that every object in its target $\mathcal{M}_{\mathrm{loc}}$ can be represented as $\mathcal{U}_{\mathrm{loc}}(\mathcal{C})$ for some small stable $\infty$-category $\mathcal{C}$. As another consequence, and using work of Efimov, we improve the universal property of $\mathcal{M}_{\mathrm{loc}}$ and show that any $\aleph_1$-finitary localizing invariant factors uniquely through it.
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On endomorphisms of topological Hochschild homology
We compute endomorphisms of topological Hochschild homology ($\mathrm{THH}$) as a functor on stable $\infty$-categories, as well as variants thereof: we also compute endomorphisms of the $k$-linear Hochschild homology functor $\mathrm{HH}_k$ over some base $k$; and endomorphisms of $\mathrm{THH}$ as a functor on stably symmetric monoidal $\infty$-categories.
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Fully faithful functors and pushouts of $\infty$-categories
We study stability properties of fully faithful functors, and compute mapping anima in pushouts of $\infty$-categories along fully faithful functors. We provide applications of these calculations to pushouts along Dwyer functors and Reedy categories.
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Universality of Barwick's unfurling construction
Given an $\infty$-category $\mathcal{C}$ with pullbacks, its $(\infty,2)$-category $\mathbf{Span}(\mathcal{C})$ of spans has the universal property of freely adding right adjoints to morphisms in $\mathcal{C}$ satisfying a Beck--Chevalley condition. We show that this universal property is implemented by an $(\infty,2)$-categorical refinement of Barwick's \emph{unfurling construction}: For any right adjointable functor $\mathcal{C} \to \mathrm{Cat}_{\infty}$, the unstraightening of its unique extension to $\mathbf{Span}(\mathcal{C})$ can be explicitly written down as another span $(\infty,2)$-category, and on underlying $(\infty,1)$-categories this recovers Barwick's construction. As an application, we show that the constructions of cartesian normed structures by Nardin--Shah and Cnossen--Haugseng--Lenz--Linskens coincide.
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Separable commutative algebras in equivariant homotopy theory
Given a finite group $G$ and a commutative ring $G$-spectrum $R$, we study the separable commutative algebras in the category of compact $R$-modules. We isolate three conditions on the geometric fixed points of $R$ which ensure that every separable commutative algebra is standard, i.e. arises from a finite $G$-set. In particular we show that all separable commutative algebras in the categories of compact objects in $G$-spectra and in derived $G$-Mackey functors are standard provided that $G$ is a $p$-group. In these categories we also show that for a general finite group $G$, not all separable commutative algebras are standard. We finally discuss how the classification of separable commutative algebras in compact $G$-spectra varies if we require the existence of multiplicative norms. We show that if $G$ is solvable, then any separable commutative algebra therein that is normed is automatically standard. However, if $G$ is not solvable, we provide examples of separable commutative algebras that are normed but not standard.
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A symmetric monoidal fracture square
Given a symmetric monoidal stable $\infty$-category $\mathcal{C}$ which is rigidly-compactly generated and a set of compact objects $\mathcal{K}$ of $\mathcal{C}$, one can form the subcategories of $\mathcal{K}$-complete and $\mathcal{K}$-local objects. The goal of this paper is to explain how to recover $\mathcal{C}$ from its $\mathcal{K}$-local and $\mathcal{K}$-complete subcategories while retaining the symmetric monoidal structure. Specializing to the case where $\mathcal{C}$ is the $\infty$-category of $G$-spectra for a finite group $G$, our result can be viewed as a symmetric monoidal variant of the isotropy separation decomposition, a version of which appeared previously in work of Krause.
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Locally rigid $\infty$-categories
We develop the theory of locally rigid and rigid symmetric monoidal $\infty$-categories over an arbitrary base $\mathcal{V}\in\mathrm{CAlg}(\mathbf{Pr}^\mathrm{L})$. Among other things, we prove that every locally rigid commutative $\mathcal{V}$-algebra arises as a ``completion'' of a rigid commutative $\mathcal{V}$-algebra. Along the way, we introduce and study ``$\mathcal{V}$-atomic morphisms'', which are analogues of compact morphisms over an arbitrary base $\mathcal{V}$.
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Dualizable presentable $\infty$-categories
We prove that for any presentably symmetric monoidal $\infty$-category $\mathcal{V}$, the $\infty$-category $\mathbf{Mod}_\mathcal{V}(\mathbf{Pr}^{\mathrm{L}})^{\mathrm{dbl}}$ of dualizable presentable $\mathcal{V}$-modules and internal left adjoints between them is itself presentable. Along the way, we survey formal properties of these dualizable $\mathcal V$-modules. We pay close attention to the case of the $\infty$-category of spectra, where we survey the foundational properties of ``compact morphisms''.
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Every spectrum is the K-theory of a stable $\infty$-category
We prove that any spectrum is equivalent to the nonconnective K-theory of a stable $\infty$-category. We use these results to construct a stable $\infty$-category $\mathcal{C}$ with a bounded t-structure such that $\operatorname{K}(\mathcal{C})$ is not equivalent to $\operatorname{K}(\mathcal{C}^\heartsuit)$, disproving a conjecture of Antieau, Gepner, and Heller.
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