Algebraic geometry

Se basa en

Requisitos previos obligatorios, hasta el fondo: 46

Aprenderás a

• Pass between ideals and affine varieties using the Nullstellensatz.
• Work with projective varieties, homogeneous coordinates and Bézout's theorem for plane curves.
• Explain why schemes generalise varieties, using Spec of a ring and its structure sheaf.

Definición

• An affine algebraic set is the common zero set in kⁿ of a set of polynomials; a variety is often required in addition to be irreducible.
• The Zariski topology has the algebraic sets as its closed sets.
• Spec R, with the Zariski topology and its structure sheaf, is an affine scheme.

Teorema

• Over an algebraically closed field, radical ideals of k[x₁, …, xₙ] correspond bijectively, reversing inclusion, to algebraic subsets of kⁿ.
• Bézout: over an algebraically closed field, two projective plane curves of degrees d and e with no common component meet in exactly de points counted with multiplicity.
• Over an algebraically closed field, the category of affine varieties is equivalent to the opposite of the category of finitely generated reduced k-algebras.

Errores frecuentes

• Thinking the Zariski topology is Hausdorff; on any infinite irreducible variety it is not.
• Treating a variety as just a set of points; its coordinate ring carries the structure.

Descubrimientos opcionales

• Derived categories Derived categories arose in Grothendieck and Verdier's work on duality theory in algebraic geometry.
• ← Complex analysis
• ← Representable functors
• ← Model theory
• ← Algebraic number theory

Lecturas complementarias

• Robin Hartshorne, Algebraic Geometry (1977)
• David Cox, John Little and Donal O'Shea, Ideals, Varieties, and Algorithms
• Igor R. Shafarevich, Basic Algebraic Geometry