CONGRUENT ZETA FUNCTIONS. NO.5

YOSHIFUMI TSUCHIMOTO

5.1. Affine schemes. We define affine schemes as a representable functor.

Definition 5.1. Let R be a ring. Then we denote by Spec(R) the affine scheme with coordinate ring R.

For any affine scheme Spec(R) and for any ring S, we define the S-valued point of Spec(R) by

Spec(R)(S) = Hom_{ring}(R, S)

Lemma 5.2. Let k be a ring. Let {f_{1}, f_{2}, …, f_{m}} be a set of equations in n-variables X_{1}, X_{2}, …, X_{n} over k. Let us put

A = k[X_{1}, X_{2}, …, X_{n}]/(f_{1}, f_{2}, …, f_{m}).

Then we have a natural identification

V (f_{1}, f_{2}, …, f_{m})(K) = Spec(A)(K)

for any algebra K over k.

Corollary 5.3. We employ the assumption as the Lemma. Then:

(1) When the “target algebra” K is given, the set of solutions V (f_{1}, f_{2}, …, f_{m})(K) depends only on the affine coordinate ring A.

(2) For any element P ∈ Spec(A)(K), the “evaluation map”

A ∋ f ↦ eval_{P} (f) ∈ K

is defined in an obvious way. Thus every element of A may be regarded as a K-valued function on Spec(A)(K).

5.2. localization.

Definition 5.4. Let f be an element of a commutative ring A. Then we define the localization A_{f} of A with respect to f as a ring defined by

A_{f} = A[Y]/(Y f − 1)

where Y is a indeterminate.

Lemma 5.5. When K is a field, then we have a canonical identification

Spec(A_{f})(K) = {P ∈ Spec(A)(K); eval_{P} (f) ≠ 0}.