Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry

10.1142/12495 ◽  
2021 ◽  
Author(s):  
Jean Gallier ◽  
Jocelyn Quaintance
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Algebraic Topology ◽  
Sheaf Cohomology

scholarly journals THE UNITY AND IDENTITY OF DECIDABLE OBJECTS AND DOUBLE-NEGATION SHEAVES

2018 ◽  
Vol 83 (04) ◽  
pp. 1667-1679
Author(s):  
MATÍAS MENNI
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Algebraic Topology ◽  
Full Subcategory ◽  
Sufficient Conditions ◽  
Double Negation ◽  
Synthetic Differential Geometry ◽  
Simplicial Sets ◽  
Image Position

AbstractLet ${\cal E}$ be a topos, ${\rm{Dec}}\left( {\cal E} \right) \to {\cal E}$ be the full subcategory of decidable objects, and ${{\cal E}_{\neg \,\,\neg }} \to {\cal E}$ be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity ${\cal E} \to {\cal S}$ for the two subcategories of ${\cal E}$ above, making them Adjointly Opposite. Typical examples of such ${\cal E}$ include many ‘gros’ toposes in Algebraic Geometry, simplicial sets and other toposes of ‘combinatorial’ spaces in Algebraic Topology, and certain models of Synthetic Differential Geometry.

THE MAIN TRENDS OF ALGEBRAIC TOPOLOGY AND ALGEBRAIC GEOMETRY

1964 ◽  
Vol 19 (6) ◽  
pp. 67-73
Author(s):  
S P Novikov ◽  
I I Pyatetskii-Shapiro ◽  
I R Shafarevich
Keyword(s):  
Algebraic Geometry ◽  
Algebraic Topology

scholarly journals ON FAMILIES IN DIFFERENTIAL GEOMETRY

2013 ◽  
Vol 10 (09) ◽  
pp. 1350042 ◽  
Author(s):  
GIOVANNI MORENO
Keyword(s):  
Differential Geometry ◽  
Algebraic Topology ◽  
Differential Calculus ◽  
Ad Hoc ◽  
Theoretical Physics ◽  
Homotopy Formula ◽  
Commutative Algebras ◽  
Algebraic Framework ◽  
Fundamental Theorems

Families of objects appear in several contexts, like algebraic topology, theory of deformations, theoretical physics, etc. An unified coordinate-free algebraic framework for families of geometrical quantities is presented here, which allows one to work without introducing ad hoc spaces, by using the language of differential calculus over commutative algebras. An advantage of such an approach, based on the notion of sliceable structures on cylinders, is that the fundamental theorems of standard calculus are straightforwardly generalized to the context of families. As an example of that, we prove the universal homotopy formula.

scholarly journals The Hodge conjecture: The complications of understanding the shape of geometric spaces

2018 ◽  
pp. 51
Author(s):  
Vicente Muñoz Velázquez
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Natural Condition ◽  
Mathematical Physics ◽  
Complex Numbers ◽  
Hodge Conjecture ◽  
Real Numbers ◽  
Geometric Objects ◽  
Complex Submanifolds ◽  
Geometry Analysis

The Hodge conjecture is one of the seven millennium problems, and is framed within differential geometry and algebraic geometry. It was proposed by William Hodge in 1950 and is currently a stimulus for the development of several theories based on geometry, analysis, and mathematical physics. It proposes a natural condition for the existence of complex submanifolds within a complex manifold. Manifolds are the spaces in which geometric objects can be considered. In complex manifolds, the structure of the space is based on complex numbers, instead of the most intuitive structure of geometry, based on real numbers.

scholarly journals Algunos tipos especiales de determinantes en extensiones PBW torcidas graduadas

2021 ◽  
Vol 39 (1) ◽  
Author(s):  
Héctor Suárez ◽  
Duban Cáceres ◽  
Armando Reyes
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Regular Algebra ◽  
Noncommutative Algebraic Geometry ◽  
Noncommutative Differential Geometry ◽  
Regular Algebras ◽  
Skew Pbw Extensions ◽  
Finitely Presented

In this paper, we prove that the Nakayama automorphism of a graded skew PBW extension over a finitely presented Koszul Auslander-regular algebra has trivial homological determinant. For A = σ(R)<x1, x2> a graded skew PBW extension over a connected algebra R, we compute its P-determinant and the inverse of σ. In the particular case of quasi-commutative skew PBW extensions over Koszul Artin-Schelter regular algebras, we show explicitly the connection between the Nakayama automorphism of the ring of coefficients and the extension. Finally, we give conditions to guarantee that A is Calabi-Yau. We provide illustrative examples of the theory concerning algebras of interest in noncommutative algebraic geometry and noncommutative differential geometry.

scholarly journals Tōhoku

2015 ◽  
Vol 1 (3) ◽  
Author(s):  
Rick Jardine
Keyword(s):  
Algebraic Geometry ◽  
Algebraic Topology ◽  
Turning Point ◽  
Homological Algebra ◽  
Modern Mathematics

Rick Jardine evaluates Alexander Grothendieck’s 1957 mathematical masterpiece, Some Aspects of Homological Algebra, a turning point in homological algebra, algebraic topology and algebraic geometry, and for modern mathematics as a whole.

Thirteen Papers on Group Theory, Algebraic Geometry and Algebraic Topology

1968 ◽  
Author(s):  
A. Andrianov ◽  
V. Dem′janenko ◽  
S. Demuškin ◽  
N. Efimov ◽  
N. Fel′dman ◽  
...  
Keyword(s):  
Algebraic Geometry ◽  
Group Theory ◽  
Algebraic Topology
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