Homology Theory on Algebraic Varieties

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

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Blackbox computation of A ∞-algebras

Georgian Mathematical Journal ◽

10.1515/gmj.2010.005 ◽

2010 ◽

Vol 17 (2) ◽

pp. 391-404

Author(s):

Mikael Vejdemo-Johansson

Keyword(s):

Homology Theory ◽

Algebra Structure ◽

Finite Amount ◽

Computational Work ◽

Dg Algebra ◽

Dg Algebras

Abstract Kadeishvili's proof of theminimality theorem [T. Kadeishvili, On the homology theory of fiber spaces, Russ. Math. Surv. 35:3 (1980), 231–238] induces an algorithm for the inductive computation of an A ∞-algebra structure on the homology of a dg-algebra. In this paper, we prove that for one class of dg-algebras, the resulting computation will generate a complete A ∞-algebra structure after a finite amount of computational work.

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Path homology theory of edge-colored graphs

Open Mathematics ◽

10.1515/math-2021-0049 ◽

2021 ◽

Vol 19 (1) ◽

pp. 706-723

Author(s):

Yuri V. Muranov ◽

Anna Szczepkowska

Keyword(s):

Spectral Sequence ◽

Edge Coloring ◽

Homology Theory ◽

Homotopy Category ◽

Homology Groups ◽

Colored Graphs ◽

Natural Filtration ◽

Definition Of ◽

Exact Sequences

Abstract In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

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