An Intrinsic, Boilerplate-Free Definition of Affine Varieties
1 The new definition
[1, definition 2.1.6] gives an intrinsic, boilerplate-free definition of affine varieties as follows (note our “variety” is not necessarily irreducible):
2 Strengths
This definition is elegant because:
intrinsic: it does not explicitly rely on an closed embedding \(X\) into some ambient affine space \(k^n\). The existence of such an closed embedding is encoded in the finiteness of \(k\)-algebra \(A\).
This is extremely helpful for those quasi-affine varieties that arise more naturally as an principal open subset of an affine variety, while still being able to be embedded into some affine space. For example, the algebraic group \(\operatorname{GL}_n\).
boilerplate-free: it is abstract enough for all use cases withing affine settings, while we don’t have to invoke the machinery of schemes or sheaves.
function-oriented: it emphasizes the algebra of functions on the variety, which is also the same take adopted by the theory of differential manifolds.
We note that condition (b) and (c) together says that \(\operatorname{ev}: X \to \operatorname{Hom}_{k\text{-alg}}(A, k)\) gives a bijection between \(X\) and the set of \(k\)-algebra homomorphisms from \(A\) to \(k\). Somehow this may be interpreted as a canonical isomorphism between \(X\) and its “double dual space”. This enables us to precisely reconstruct \(X\) from those \(k\)-valued functions it admits (that is, \(A\)).
Similar statements take place everywhere in mathematics:
- For a finite-dimensional vector space \(V\), the evaluation map \(V \to V^{**}\) is an isomorphism.
- For a Hilbert space \(H\), the evaluation map gives an isomorphism \(H\) and its double dual space \(H^{**}\) (dual being the continuous dual).
Note that Hilbert’s Nullstellensatz says that all maximal ideals has residue field \(k\) when \(k\) is algebraically closed. This extends the equivalence to \(X \cong \operatorname{Hom}_{k\text{-alg}}(A, k) \cong \operatorname{Specmax}(A)\).
3 Equivalence with the extrinsic definition
An affine variety can be defined extrinsically: start with the affine space \(k^n\) and take the zero set of some polynomials.
Condition (1) gives the existence of a projection \(\pi: k[x_1, \dots, x_n] \to A\), which ensures a \(X \to k^n\) map: say \(X\) is generated by \(f_1, \dots, f_n\) (correspond to \(x_1, \dots, x_n\)), then we have a map \(X \to k^n\) sending \(x\) to \((f_1(x), \dots, f_n(x))\).
Condition (2) ensures that \(X \to k^n\) is injective.
[1, proposition 2.1.10] Shows that this embedding is indeed a closed embedding of affine varieties when \(k\) is algebraically closed.
4 Operations
New affine varieties can be constructed from old ones by the following operations.
4.1 Closed subsets
For any affine variety \((X, A)\) and a closed subset \(V \subseteq X\), by restricting the functions in \(A\) to \(Y\), we get a new \(k\)-algebra \(A \vert_V \subseteq \operatorname{Hom}_{k\text{-alg}}(V, k)\). We check that \((V, A \vert_V)\) is indeed an affine variety:
Analyzing the restriction map \(A \to A \vert_V\) by the first isomorphism theorem gives \(A \vert_V \cong A / I\). This shows that \(A \vert_V\) is finitely generated as an \(k\)-algebra.
Seperation and completeness property is inherited from \(A\) by restriction.
4.2 Principal open subsets
For any affine variety \((X, A)\) and a basic open set \(X_f\) for some \(f \in A\), by restricting the functions in \(A\) to \(X_f\) and adding the function \(1/f : X_f \to k\), we get a new \(k\)-algebra \(A_f := A \vert_{X_f} [1/f] \subseteq \operatorname{Hom}_{k\text{-alg}}(X_f, k)\). We check that \((X_f, A_f)\) is indeed an affine variety:
\(A_f\) is generated by the image of \(A\) (under the restriction map) and \(1/f\), hence is finitely generated as a \(k\)-algebra.
Seperation: inherited from \(A\) by restriction.
Completeness: Say \(\varphi: A_f \to k\). Note that \(\varphi(1/f)\) is completely determined by \(\varphi(f)\): \(1 = \varphi(1) = \varphi(1/f \cdot f) = \varphi(1/f) \varphi(f)\). So \(\varphi(1/f) = 1 / \varphi(f)\) and we only need to worry about \(A \vert_{X_f}\) part, which always pulls back to \(A \to k\) and hence has a realization of \(\operatorname{ev}_x : A \to k\) where \(x \in X\). The validity of \(\varphi(1/f)\) requires \(f(x) = \varphi(f) \neq 0\), i.e. hence \(x \in X_f\). So \(\varphi\) can be realized as \(\operatorname{ev}_x : X_f \to k\) for some \(x \in X_f\).
In fact, \(A_f\) is isomorphic to the localization of \(A\) at \(f\).
Remark. For general open subsets, you may not be able to realize them as affine varieties. For example, the regular functions over \(\mathbb C^2 \setminus \{(0,0)\}\) are in one-to-one correspondence with those over \(\mathbb C^2\), but you can’t realize the evaluation at \((0,0)\) by any point in \(\mathbb C^2 \setminus \{(0,0)\}\), so the completeness condition fails.
These degenerate cases are called quasi-affine varieties, which we shall not discuss here.