Equivalence of categories
View article: Alperin's weight conjecture, Galois automorphisms, alternating sums, and functorial equivalences
Alperin's weight conjecture, Galois automorphisms, alternating sums, and functorial equivalences Open
We show that functorial equivalences can offer new insight into the blockwise Galois Alperin weight conjecture (BGAWC). Inspired by Knörr and Robinson's work, we first formulate the BGAWC in terms of alternating sums indexed by chains of $…
View article: Groupoid graded rings and their categories of graded modules
Groupoid graded rings and their categories of graded modules Open
Let $G$ be a groupoid acting on a set $X$ and let $R$ be a $G$-graded ring with graded local units. We study the main properties of the category $gr-(R,G,X)$ of $X$-graded $R$-modules and adjoint functors between categories of this kind. W…
View article: Groupoid graded rings and their categories of graded modules
Groupoid graded rings and their categories of graded modules Open
Let $G$ be a groupoid acting on a set $X$ and let $R$ be a $G$-graded ring with graded local units. We study the main properties of the category $gr-(R,G,X)$ of $X$-graded $R$-modules and adjoint functors between categories of this kind. W…
View article: Alperin's weight conjecture, Galois automorphisms, alternating sums, and functorial equivalences
Alperin's weight conjecture, Galois automorphisms, alternating sums, and functorial equivalences Open
We show that functorial equivalences can offer new insight into the blockwise Galois Alperin weight conjecture (BGAWC). Inspired by Knörr and Robinson's work, we first formulate the BGAWC in terms of alternating sums indexed by chains of $…
View article: Axiomatic Categorical Curvature Theory: Curvature, Invariants, and Structural Rigidity
Axiomatic Categorical Curvature Theory: Curvature, Invariants, and Structural Rigidity Open
This paper extends Axiomatic Universal Second Variation (AUSV) into its categorical manifestation, Axiomatic Categorical Curvature Theory (ACCT), demonstrating that categorical structures uniquely express invariants from the intrinsic refi…
View article: Axiomatic Categorical Curvature Theory: Curvature, Invariants, and Structural Rigidity
Axiomatic Categorical Curvature Theory: Curvature, Invariants, and Structural Rigidity Open
This paper extends Axiomatic Universal Second Variation (AUSV) into its categorical manifestation, Axiomatic Categorical Curvature Theory (ACCT), demonstrating that categorical structures uniquely express invariants from the intrinsic refi…
View article: Unveiling the Abelian Spectrum of Triangulated Categories
Unveiling the Abelian Spectrum of Triangulated Categories Open
In this paper, we delve into the intricate structure of triangulated categories, focusing on the construction and properties of their Abelian spectrum. We introduce a novel approach to understanding the local behavior of these categories b…
View article: A Homological Theory of Krull Dimension: Unifying Finiteness Conditions
A Homological Theory of Krull Dimension: Unifying Finiteness Conditions Open
This paper develops a comprehensive homological theory for Krull dimension, extending its applicability beyond classical commutative Noetherian settings and providing a unified framework for various finiteness conditions across different a…
View article: A Kunz-type theorem for formal unramification
A Kunz-type theorem for formal unramification Open
We prove that for a morphism of schemes of positive characteristic whose relative Frobenius is a morphism of locally noetherian schemes, being formally unramified (resp. formally étale) is equivalent to its Frobenius morphism being a close…
View article: The Étale Geometrization of Motives
The Étale Geometrization of Motives Open
This paper explores the profound concept of étale geometrization within the theory of motives. Motives, originally introduced by Alexander Grothendieck, aim to provide a universal cohomology theory for algebraic varieties and classify them…
View article: Generalized Stone Dualities for Non-Classical Logical Systems
Generalized Stone Dualities for Non-Classical Logical Systems Open
This paper explores the generalization of Stone duality from classical Boolean algebras to the algebraic semantics of various non-classical logical systems. Stone duality establishes a fundamental equivalence between the category of Boolea…
View article: The Étale Geometrization of Motives
The Étale Geometrization of Motives Open
This paper explores the profound concept of étale geometrization within the theory of motives. Motives, originally introduced by Alexander Grothendieck, aim to provide a universal cohomology theory for algebraic varieties and classify them…
View article: Étale Motivic Homotopy Theory: Foundations, Advances, and Applications
Étale Motivic Homotopy Theory: Foundations, Advances, and Applications Open
Étale motivic homotopy theory represents a profound synthesis of algebraic geometry, algebraic topology, and number theory, extending the celebrated work of Morel and Voevodsky by incorporating the powerful tools of 'etale cohomology. This…
View article: Categorical Homotopy Theory for Universal Algebra
Categorical Homotopy Theory for Universal Algebra Open
This paper explores the integration of categorical homotopy theory with universal algebra, providing a robust framework for studying algebraic structures up to homotopy. We delve into the foundational aspects of universal algebra, characte…
View article: The Geometry of Triangulated Categories Through Abelian Lenses
The Geometry of Triangulated Categories Through Abelian Lenses Open
Triangulated categories are fundamental structures in modern mathematics, providing a unifying framework for studying diverse areas such as algebraic geometry, representation theory, and topology. Despite their power, their intrinsic non-a…
View article: The Geometry of Triangulated Categories Through Abelian Lenses
The Geometry of Triangulated Categories Through Abelian Lenses Open
Triangulated categories are fundamental structures in modern mathematics, providing a unifying framework for studying diverse areas such as algebraic geometry, representation theory, and topology. Despite their power, their intrinsic non-a…
View article: Unveiling the Abelian Spectrum of Triangulated Categories
Unveiling the Abelian Spectrum of Triangulated Categories Open
In this paper, we delve into the intricate structure of triangulated categories, focusing on the construction and properties of their Abelian spectrum. We introduce a novel approach to understanding the local behavior of these categories b…
View article: A Kunz-type theorem for formal unramification
A Kunz-type theorem for formal unramification Open
We prove that for a morphism of schemes of positive characteristic whose relative Frobenius is a morphism of locally noetherian schemes, being formally unramified (resp. formally étale) is equivalent to its Frobenius morphism being a close…
View article: Étale Motivic Homotopy Theory: Foundations, Advances, and Applications
Étale Motivic Homotopy Theory: Foundations, Advances, and Applications Open
Étale motivic homotopy theory represents a profound synthesis of algebraic geometry, algebraic topology, and number theory, extending the celebrated work of Morel and Voevodsky by incorporating the powerful tools of 'etale cohomology. This…
View article: Generalized Stone Dualities for Non-Classical Logical Systems
Generalized Stone Dualities for Non-Classical Logical Systems Open
This paper explores the generalization of Stone duality from classical Boolean algebras to the algebraic semantics of various non-classical logical systems. Stone duality establishes a fundamental equivalence between the category of Boolea…
View article: A Homological Theory of Krull Dimension: Unifying Finiteness Conditions
A Homological Theory of Krull Dimension: Unifying Finiteness Conditions Open
This paper develops a comprehensive homological theory for Krull dimension, extending its applicability beyond classical commutative Noetherian settings and providing a unified framework for various finiteness conditions across different a…
View article: Categorical Homotopy Theory for Universal Algebra
Categorical Homotopy Theory for Universal Algebra Open
This paper explores the integration of categorical homotopy theory with universal algebra, providing a robust framework for studying algebraic structures up to homotopy. We delve into the foundational aspects of universal algebra, characte…
View article: The Exact Structure of Triangulated Categories: Unveiling Abelian Foundations
The Exact Structure of Triangulated Categories: Unveiling Abelian Foundations Open
Triangulated categories are foundational structures in algebraic geometry, representation theory, and homological algebra. While their axiomatic definition provides a powerful framework for studying objects up to quasi-isomorphism, their u…
View article: The Exact Structure of Triangulated Categories: Unveiling Abelian Foundations
The Exact Structure of Triangulated Categories: Unveiling Abelian Foundations Open
Triangulated categories are foundational structures in algebraic geometry, representation theory, and homological algebra. While their axiomatic definition provides a powerful framework for studying objects up to quasi-isomorphism, their u…
View article: Krull-Artinian Duality in Homological Dimension Theory
Krull-Artinian Duality in Homological Dimension Theory Open
This paper explores the existence and nature of a Krull-Artinian duality within the framework of homological dimension theory. We investigate how the algebraic invariants of Krull dimension for rings and modules, which measure the complexi…
View article: Krull-Artinian Duality in Homological Dimension Theory
Krull-Artinian Duality in Homological Dimension Theory Open
This paper explores the existence and nature of a Krull-Artinian duality within the framework of homological dimension theory. We investigate how the algebraic invariants of Krull dimension for rings and modules, which measure the complexi…
View article: Unveiling the Homotopical Architecture of Non-Associative Algebras
Unveiling the Homotopical Architecture of Non-Associative Algebras Open
This paper delves into the intricate homotopical architecture of non-associative algebras, a class of algebraic structures where the fundamental associativity axiom is relaxed or altogether absent. While traditional algebraic topology and …
View article: Unveiling the Homotopical Architecture of Non-Associative Algebras
Unveiling the Homotopical Architecture of Non-Associative Algebras Open
This paper delves into the intricate homotopical architecture of non-associative algebras, a class of algebraic structures where the fundamental associativity axiom is relaxed or altogether absent. While traditional algebraic topology and …
View article: Exact Structures on Triangulated Categories
Exact Structures on Triangulated Categories Open
This paper explores the intricate relationship between exact structures and triangulated categories, two fundamental concepts in modern homological algebra and category theory. Triangulated categories provide a robust framework for studyin…
View article: The Derived Core of a Triangulated Category: Reconstructing Abelian Structures
The Derived Core of a Triangulated Category: Reconstructing Abelian Structures Open
Triangulated categories have emerged as a fundamental framework in modern algebra, algebraic geometry, and representation theory, providing a flexible setting to study derived functors and homological invariants. However, their inherent la…