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Algebraic Geometry

Algebraic Geometry and K-Theory

Algebraic geometry is a branch of mathematics that uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. It chiefly studies algebraic varieties, the geometric shapes that form the solution sets of systems of polynomial equations. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Central Question
What geometric shape does the solution set of a system of polynomial equations form, and what can algebraic techniques reveal about that shape's intrinsic properties that geometry alone cannot? 1
Key Debate
Whether the classical style of algebraic geometry, studying varieties as concrete geometric loci of solutions over the complex numbers in the manner of the Italian school, or Grothendieck's scheme-theoretic reformulation, recasting varieties as functors of points over any commutative ring, is the right foundation for the subject. Grothendieck's schemes won out as the working foundation for research from the 1960s on, prized for their generality and their functorial precision, but the classical, more geometrically visualizable approach remains how the subject is usually first taught and pictured. 1
Classification
Pure or Applied
Pure Mathematics 1
Algebraic Geometry
Filter Results2 entries

Dimension of an Algebraic Variety

Algebraic Geometry

Proj Construction

Algebraic Geometry
Connections

Associated With

Algebra, Branches of Mathematics
Source Algebraic Geometry (Wikipedia)
Geometry, Branches of Mathematics
Source Algebraic Geometry (Wikipedia)

Includes

Alexander Grothendieck, Mathematicians
Source Alexander Grothendieck (Wikipedia)
Algebraic Variety, Concepts
Andre Weil, Mathematicians
Source André Weil (Wikipedia)
Bass Conjecture, Conjectures
Source Bass Conjecture, Wikipedia
Chow's Theorem, Theorems
Dimension of an Algebraic Variety, Mathematical Objects
Source Dimension of an Algebraic Variety (Wikipedia)
Fujita Conjecture, Conjectures
Source Fujita Conjecture, Wikipedia
Oscar Zariski, Mathematicians
Source Oscar Zariski (Wikipedia)
Parshin's Conjecture, Conjectures
Source Parshin's Conjecture (Wikipedia)
Proj Construction, Mathematical Objects
Source Proj Construction (Wikipedia)
Riemann-Roch Theorem, Theorems
Source Riemann-Roch theorem (Wikipedia)
Scheme (Mathematics), Concepts
Section Conjecture, Conjectures
Source Section Conjecture (Wikipedia)
Sheaf (Mathematics), Concepts
Virasoro Conjecture, Conjectures
Source Virasoro Conjecture (Wikipedia)
William Vallance Douglas Hodge, Mathematicians
Sources
1. Algebraic Geometry (Wikipedia)
Wikimedia Foundation
  • lead paragraph
    Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.
  • body text on Grothendieck's schemes
    Most remarkably, in the early 1960s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a scheme.
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Alexander Grothendieck (Wikipedia)
WikipediaIncludes: Alexander Grothendieck, Lead section
Quote, Includes: Alexander Grothendieck, Lead section
His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics.
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Bass Conjecture, Wikipedia
Includes: Bass Conjecture, Lead sentence
Quote, Includes: Bass Conjecture, Lead sentence
In mathematics, especially algebraic geometry, the Bass conjecture says that certain algebraic K-groups are supposed to be finitel
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Parshin's Conjecture (Wikipedia)
Includes: Parshin's Conjecture, Lead sentence
Quote, Includes: Parshin's Conjecture, Lead sentence
In mathematics, more specifically in algebraic geometry, Parshin's conjecture (also referred to as the Beilinson-Parshin conjectur
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Dimension of an Algebraic Variety (Wikipedia)
Includes: Dimension of an Algebraic Variety, Lead sentence
Quote, Includes: Dimension of an Algebraic Variety, Lead sentence
In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent
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Proj Construction (Wikipedia)
Includes: Proj Construction, Lead sentence
Quote, Includes: Proj Construction, Lead sentence
In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces o
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Virasoro Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Virasoro Conjecture, Lead sentenceView the Source
Section Conjecture (Wikipedia)
Includes: Section Conjecture, Lead sentence
Quote, Includes: Section Conjecture, Lead sentence
In anabelian geometry, a branch of algebraic geometry, the section conjecture gives a conjectural description of the splittings of
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Fujita Conjecture, Wikipedia
Includes: Fujita Conjecture, Lead sentence
Quote, Includes: Fujita Conjecture, Lead sentence
jita's conjecture is a problem in the theories of algebraic geometry and complex manifolds.
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Riemann-Roch theorem (Wikipedia)
Includes: Riemann-Roch Theorem, Lead sentence
Quote, Includes: Riemann-Roch Theorem, Lead sentence
mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromor
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André Weil (Wikipedia)
Includes: Andre Weil, Lead paragraph [in-branch 2]
Quote, Includes: Andre Weil, Lead paragraph [in-branch 2]
algebraic geometry
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Oscar Zariski (Wikipedia)
Includes: Oscar Zariski, Lead paragraph
Quote, Includes: Oscar Zariski, Lead paragraph
algebraic geometers
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