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:

Definition 1 Let \(X\) be a set and \(A\) be a \(k\)-subalgebra of functions \(X \to k\). We call the pair \((X, A)\) an affine variety if \(A\) is a nontrivial finitely generated \(k\)-subalgebra with a bijective evaluation map \(\operatorname{ev}: X \to \operatorname{Hom}_{k\text{-alg}}(A, k)\). More explicitly:

1. Finiteness: \(A\) is finitely generated as a \(k\)-algebra and contains the constant functions.

2. Separation: The functions in \(A\) separate the points of \(X\).

   That is, if \(x \neq y\), there is some \(f \in A\) such that \(f(x) \neq f(y)\).

   That is, the evaluation map \(\operatorname{ev}\) is injective.

3. Completeness: Every \(k\)-algebra homomorphism \(\lambda: A \to k\) can be realized as an evaluation map \(\operatorname{ev}_x\) for some \(x \in X\).

   That is, the evaluation map \(\operatorname{ev}\) is surjective.

Definition 2 A morphism between two affine varieties \(\varphi: (X, A) \to (Y, B)\) can be defined in two equivalent ways:

• a function \(\varphi: X \to Y\) such that for every \(g \in B\), the pullback \(g \circ \varphi \in A\).
• a \(k\)-algebra homomorphism \(B \to A\).

The equivalence is given by the following correspondences:

• Given \(\varphi: X \to Y\), we get a homomorphism \(\varphi^*: B \to A\) by sending \(g\) to \(g \circ \varphi\).
• Given a homomorphism \(\psi: B \to A\), for every \(x \in X\), \(\psi \circ \operatorname{ev}_x \in B\), hence by condition (3) there is a \(y \in Y\) such that \(\psi \circ \operatorname{ev}_x = \operatorname{ev}_y\). By condition (2) this \(y\) is unique, so we can define \(\varphi: X \to Y\) by sending \(x\) to this \(y\).

2 Strengths

This definition is elegant because:

• intrinsic: it does not explicitly rely on embedding \(X\) into some ambient affine space \(k^n\). The existence of such an embedding is encoded in the finiteness of \(k\)-algebra \(A\).

• 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”. 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).
  • For a compact Hausdorff space \(X\), the evaluation map gives a homeomorphism between \(X\) and the maximal ideals of continuous real-valued functions.

Next, we connect this definition to other definitions of affine varieties.

3 Connection to the Concrete Definition

In a standard introductory course, an affine variety is defined extrinsically: you start with \(k^n\), pick a set of polynomials \(f_1, \dots, f_m\), and define \(V\) as their common zero set.

• Condition (1) gives the existence of a projection \(\pi: k[x_1, \dots, x_n] \to A\), which gives an embedding of \(X\) into \(k^n\): 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 really an embedding, i.e. injective.

• Condition (3) is essentially the weak form of Hilbert’s Nullstellensatz: Every \(k\)-algebra homomorphism \(A \to k\) corresponds to an maximal ideal in the coordinate ring, which corresponds to a point in \(X\).

4 Connection to the Scheme-Theoretic Definition

TODO

References

[1]

M. Geck, An introduction to algebraic geometry and algebraic groups. in Oxford graduate texts in mathematics. Oxford, New York: Oxford University Press, 2013.