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Parikh-collinear morphisms have the property that all the Parikh vectors of the images of letters are collinear, i.e., the associated adjacency matrix has rank 1. In the conference DLT–WORDS 2023 we showed that fixed points of Parikh-collinear morphisms are automatic. We also showed that the abelian complexity function of a binary fixed point of such a morphism is automatic under some assumptions. In this note, we fully generalize the latter result. Namely, we show that the abelian complexity function of a fixed point of an arbitrary, possibly erasing, Parikh-collinear morphism is automatic. Furthermore, a deterministic finite automaton with output generating this abelian complexity function is provided by an effective procedure. To that end, we discuss the constant of recognizability of a morphism and the related cutting set.

The main result of this paper is the existence of Galois representations associated with the mod p (or mod pm) cohomology of the locally symmetric spaces for GLn over a totally real or CM field, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic 0 cohomology classes, one realizes the cohomology of the locally symmetric spaces for GLn as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of Shimura varieties. Thus, we prove new results on the p-adic geometry of Shimura varieties (of Hodge type). Namely, the Shimura varieties become perfectoid when passing to the inverse limit over all levels at p, and a new period map towards the flag variety exists on them, called the Hodge-Tate period map. It is roughly analogous to the embedding of the hermitian symmetric domain (which is roughly the inverse limit over all levels of the complex points of the Shimura variety) into its compact dual. The Hodge-Tate period map has several favorable properties, the most important being that it commutes with the Hecke operators away from p (for the trivial action of these Hecke operators on the flag variety), and that automorphic vector bundles come via pullback from the flag variety.

We show that the Kervaire invariant one elements $\\theta _j \\epsilon \\pi _{2^{j+1}-2}S^0$ exist only for j ≤ 6. By Browder's Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding problem in algebraic topology.

We prove that faithful traces on separable and nuclear C*-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C*-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms-Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal C*-algebras.

Various types of movability for abstract classical pro-morphisms or coherent mappings, and for abstract classical or strong shape morphisms was given by the same authors in some previous paper [10–12]. In the present paper we introduce and study the notions of (uniform) movability, and (uniform) co-movability for a new type of pro-morphisms and shape morphisms belonging to the so called enriched pro-category - and to the corresponding shape category , which were introduced by Uglešić [ 27 ].

This paper extends many conclusions based on phantom envelopes and Ext-phantom covers of modules, and we find that many important properties still hold after replacing phantom and Ext-phantom with n-phantom and n-Ext-phantom respectively. In addition, we also obtain some extra results. Specifically, we give a characterization of the weak dimensions of rings in terms of n-phantom envelopes and n-Ext-phantom covers of modules with the unique mapping property respectively. We show that wD(R) ≤ 2n whenever every right R-module has an n-phantom envelope with the unique mapping property or every left R-module has an n-Ext-phantom cover with the unique mapping property over left coherent rings.

We endow the set of lattices in ${\\mathrm{\\mathbb{Q}}}_{\\mathrm{p}}^{\\mathrm{n}}$ with a reasonable algebro-geometric structure. As a result, we prove the representability of affine Grassmannians and establish the geometric Satake equivalence in mixed characteristic. We also give an application of our theory to the study of Rapoport-Zink spaces.

We show that log canonical thresholds satisfy the ACC.

We prove Soergel's conjecture on the characters of indecomposable Soergel bimodules. We deduce that Kazhdan-Lusztig polynomials have positive coefficients for arbitrary Coxeter systems. Using results of Soergel one may deduce an algebraic proof of the Kazhdan-Lusztig conjecture.

Unbounded entailment relations, introduced by Paul Lorenzen (1951), are a slight variant of a notion which plays a fundamental role in logic (Scott 1974) and in algebra (Lombardi and Quitté 2015). We call systems of ideals their single-conclusion counterpart. If they preserve the order of a commutative ordered monoid 𝐺 and are equivariant with respect to its law, we call them equivariant systems of ideals for G: they describe all morphisms from 𝐺 to meet-semilattice-ordered monoids generated by (the image of) 𝐺. Taking a 1953 article by Lorenzen as a starting point, we also describe all morphisms from a commutative ordered group 𝐺 to lattice-ordered groups generated by 𝐺 through unbounded entailment relations that preserve its order, are equivariant, and satisfy a regularity property invented by Lorenzen; we call them regular entailment relations. In particular, the free lattice-ordered group generated by 𝐺 is described through the finest regular entailment relation for 𝐺, and we provide an explicit description for it; it is order-reecting if and only if the morphism is injective, so that the Lorenzen-Clifford-Dieudonné theorem fits into our framework. Lorenzen's research in algebra starts as an inquiry into the system of Dedekind ideals for the divisibility group of an integral domain 𝑅, and specifically into Wolfgang Krull's \"Fundamentalsatz\" that 𝑅 may be represented as an intersection of valuation rings if and only if 𝑅 is integrally closed: his constructive substitute for this representation is the regularisation of the system of Dedekind ideals, i.e. the lattice-ordered group generated by it when one proceeds as if its elements are comparable.