研究生一年级Geometry & Topology计划中
Algebraic geometry
Affine and projective varieties, the Zariski topology, coordinate rings, morphisms and rational maps, dimension, and a first look at sheaves and schemes.
math.geom.algebraic-geometry
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前置知识
46 项必需的前置知识（一直追溯到底）
你将学会
- 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.
定义
- 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.
定理
- 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.
常见误解
- 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.
可选发现
- 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
延伸阅读
- Robin Hartshorne, Algebraic Geometry (1977)
- David Cox, John Little and Donal O'Shea, Ideals, Varieties, and Algorithms
- Igor R. Shafarevich, Basic Algebraic Geometry