Algebraic Geometry Lecture 19 – Affine Recappage
Andrew Potter
What is Algebraic Geometry?
It is the study of...
•affine varieties
•projective varieties
•quasi-projective varieties
•varieties
•schemes.
Affine Varieties
Let kbe an algebraically closed field (we’ll always assume this unless explicitly
stated otherwise). Affine n-space is
An={(a1, . . . , an)∈kn}.
We want to define affine varieties.
Anecdote: Andrew’s commutative algebra lecturer once said
Algebra ⇔Geometry.
1) Algebra ⇒Geometry.
Let f1, . . . , fm∈k[x] = k[x1, . . . , xn]. An affine algebraic set Vassociated with
these polynomials is
V(f1, . . . , fm) = {x∈An|f1(x) = f2(x) = . . . =fm(x) = 0}.
We could’ve started with the ideal (f1, . . . , fm)⊂k[x]. In fact we could start with
any ideal I⊂k[x] and construct V(I). This is true by Hilbert’s basis theorem,
which says any ideal I⊂k[x] is finitely generated.
2) Geometry ⇒Algebra.
Suppose Vis an algebraic set – a geometric object in n-space. Define the ideal
of Vto be
I(V) = {f∈k[x]|f(x) = 0 for all x∈V}.
What’s the correspondence betwixt Vand I? Certainly V(I(U)) = U, however
in general I(V(J)) 6=J. We get a “nice” correspondence when we look at affine
varieties.
Suppose Vis an algebraic set and V=W1∪W2where W1, W2are proper
algebraic subsets of V. Then Vis called reducible. If Visn’t reducible then it’s
called irreducible. Equivalently, Vis irreducible if I(V) is a prime ideal. We’ll call
an irreducible algebraic set an (affine algebraic) variety.
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