Related papers: Nilpotent approximation (completion) of $\mathbb{E…

The notion of conformal algebras was introduced by Victor G. Kac using the axiomatic description of the operator product expansion of chiral fields in conformal field theory. The structure theory, representations and cohomology of Lie and…

We study the interaction between various analytification functors, and a class of morphisms of rings, called homotopy epimorphisms. An analytification functor assigns to a simplicial commutative algebra over a ring $R$, along with a choice…

This paper provides an extensive study of the homotopy theory of types of algebras with units, like unital associative algebras or unital commutative algebras for instance. To this purpose, we endow the Koszul dual category of curved…

We define the space of nearly holomorphic automorphic forms on a connected reductive group $G$ over $\mathbb{Q}$ such that the homogeneous space $G(\mathbb{R})^1/ K_\infty^\circ$ is a Hermitian symmetric space. By Pitale, Saha and Schmidt's…

Given a compact Lie group $G$ acting on a space $X$, the classical Atiyah-Segal completion theorem identifies topological $K$-theory of the homotopy quotient $X/G$ with an explicit completion of $G$-equivariant topological $K$-theory of…

Let $M$ be a topological monoid with homotopy group completion $\Omega BM$. Under a strong homotopy commutativity hypothesis on $M$, we show that $\pi_k (\Omega BM)$ is the quotient of the monoid of free homotopy classes $[S^k, M]$ by its…

Let $\Gamma$ be a discrete group. To every ideal in $\ell^{\infty}(\G)$ we associate a C$^*$-algebra completion of the group ring that encapsulates the unitary representations with matrix coefficients belonging to the ideal. The general…

For an atomic orbital base category in the sense of Barwick-Dotto-Glasman-Nardin-Shah, we introduce the category of parametrised perfect-stable categories and use it to construct the parametrised version of noncommutative motives in which…

In this note, we study the $p$-complete topological cyclic homology of the affine line relative to a ring $A$ which is smooth over a perfectoid ring $R$. Denoting by $NTC(A; \mathbb{Z}_p)$ the spectrum which measures the failure of…

For a category $\mathcal E$ with finite limits and well-behaved countable coproducts, we construct a model structure, called the effective model structure, on the category of simplicial objects in $\mathcal E$, generalising the Kan--Quillen…

In a previous paper, we constructed a category of (phi, Gamma)-modules associated to any adic space over Q_p with the property that the etale (phi, Gamma)-modules correspond to etale Q_p-local systems; these involve sheaves of period rings…

Let $G$ be a finite group of Lie type. In studying the cross-characteristic representation theory of $G$, the (specialized) Hecke algebra $H=\End_G(\ind_B^G1_B)$ has played a important role. In particular, when $G=GL_n(\mathbb F_q)$ is a…

Recollements of abelian categories are used as a basis of a homological and recursive approach to quasi-hereditary algebras. This yields a homological proof of Dlab and Ringel's characterisation of idempotent ideals occuring in heredity…

Let $V$ be a complete discrete valuation ring, and let $G$ be either a word-hyperbolic group or a reductive $p$-adic group. We prove that the canonical morphism $V[G] \to V[G]^\dagger$ from the group algebra to its dagger completion is an…

This article is a survey of algebra in the $\infty$-categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing symmetric monoidal stable…

We consider a large class of monomial maps respecting an action of the infinite symmetric group, and prove that the toric ideals arising as their kernels are finitely generated up to symmetry. Our class includes many important examples…

In this paper we elaborate a general homotopy-theoretic framework in which to study problems of descent and completion and of their duals, codescent and cocompletion. Our approach to homotopic (co)descent and to derived (co)completion can…

In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of…

In this paper, we present an infinity-categorical version of the theory of monoidal categories. We show that the infinity category of spectra admits an essentially unique monoidal structure (such that the tensor product preserves colimits…

The symmetric spectra introduced by Hovey, Shipley and Smith are a convenient model for the stable homotopy category with a nice associative and commutative smash product on the point set level and a compatible Quillen closed model…