Introduction to formal commutative algebra

Peter Schneider

Introduction

These pages constitute the notes of a course I gave at Köln in the academic year 1991/92. They presume some basic knowledge of commutative algebra. Although it may not be apparent to the reader the material is selected with an eye towards the connections between formal commutative algebra and rigid analysis. The following references are explicitly referred to in the text:

[Bou] Bourbaki, N.: Commutative Algebra. Paris: Hermann 1972.

[EGA] Dieudonné, J., Grothendieck, A.: Eléments de Géométrie Algébriques, vol. IV 1, Publ. Math. IHES 20 (1964)

[Mat] Matsumura, H.: Commutative ring theory. Cambridge Univ. Press 1994

List of contents

§1 Adic noetherian rings §2 Adic algebras §3 Complete tensor products §4 Complete localization §5 Complete blowing up §6 Weierstraß theory §7 Discrete valuation rings §8 Affinoid spectra §1 Adic noetherian rings

Rings always are commutative and have a unit element, and homomorphisms of rings always respect the unit elements.

Now let A be a fixed ring and let I ⊆ A be an ideal. A subset U ⊆ A is called open if

for any a ∈ U there is a n ∈ IN such that a + I^{n} ⊆ U .

In this way A is equipped with a topology with respect to which it is a topological ring. We call this topology the \underline{I-adic topology} on A. The following properties are immediate.

– The ideals I^{n} for n ≥ 1 form a fundamental system of open neighbourhoods of the zero element 0 ∈ A.

– A is Hausdorff if and only if ⋂_{n∈IN} I^{n} = {0}.

– Any open ideal in A also is closed.

– The J-adic topology on A for some other ideal J ⊆ A coincides with the I-adic topology if and only if there are m, n ∈ IN such that I^{m} ⊆ J and J^{n} ⊆ I.

Definition:

An element a ∈ A is called \underline{topologically nilpotent} if its residue class in A/I^{n} is nilpotent for any n ∈ IN. Let t_{I}(A) denote the subset of topologically nilpotent elements in A.

Lemma 1:

i. t_{I}(A) = {a ∈ A : a^{m} ∈ I for some m ∈ IN}; ii. t_{I}(A) is an ideal; iii. t_{I}(A) is equal to the intersection of all open prime ideals in A; iv. t_{I}(A) is equal to the union of all ideals J ⊆ A for which the J-adic topology coincides with the I-adic topology.

Proof: i. is obvious. ii. is a consequence of i. For iii. we note that any open prime ideal ℘ ⊆ A contains some I^{n}. Therefore t_{I}(A) is contained in ℘. On the other hand let a be an element in the intersection of all those ℘. The residue class of a then lies in the intersection of all prime ideals in A/I and consequently is nilpotent. This shows that a ∈ t_{I}(A). iv. It follows from i. that any such ideal J is contained in t_{I}(A). Fix an a ∈ t_{I}(A) and put J := I + Aa. If a^{m} ∈ I then also J^{m} ⊆ I. This implies that the J-adic topology coincides with the I-adic topology.

Corollary 2:

Any open prime ideal in A contains I.

Corollary 3:

Let A be noetherian; then t_{I}(A) is the largest ideal among all ideals J ⊆ A for which the J-adic topology coincides with the I-adic topology.

Proof: By Lemma 1.iv any such J is contained in t_{I}(A). Furthermore by Lemma 1.i the nilradical of A/I is t_{I}(A)/I. Since A/I is noetherian we have t_{I}(A)^{m} ⊆ I for some m ∈ IN. Therefore t_{I}(A) itself occurs among those J’s.

Obviously we have a canonical homomorphism of rings

φ_{A} : A ⟶ lim_{⟵_{n∈IN}} A/I^{n}

which is injective if and only if A is Hausdorff.

Definition:

The ring A is called \underline{(I-)adic} if φ_{A} is bijective.

Lemma 4:

If A is I-adic then we have: i. I is contained in the Jacobson radical of A, i.e., any maximal ideal of A contains I and, in particular, is open; ii. 1 + I ⊆ A^{×}; iii. the natural map A^{×} ⟶ (A/I)^{×} is surjective.

Proof: ii. Given a ∈ I the element b := ∑_{m≥0}(−a)^{m} is well-defined modulo I^{n}

with (1 + a)b ≡ 1 mod I^{n} for any n ∈ IN. Because of the bijectivity of φ_{A} the same holds true in A. i. Assume that the maximal ideal 𝔪 ⊆ A does not contain I. We then have I + 𝔪 = A so that (I + 1) ∩ 𝔪 ≠ ∅; this is a contradiction to ii. iii. If a is a unit mod I then it follows from i. that a is not contained in any maximal ideal of A. This means that a ∈ A^{×}.

Proposition 5:

Let A be I-adic; then A is noetherian if and only if A/I is noetherian and I/I^{2} is finitely generated as an A/I-module.

The proof relies on the following technical fact which will be used also later on repeatedly.

Lemma 6:

Let α : G → E be a homomorphism of filtered abelian groups G = F^{0} ⊇ F^{1}G ⊇ … and E = F^{0}E ⊇ F^{1}E ⊇ …; we assume that a. the associated graded homomorphism gr α : gr G → gr E is surjective, b. the natural map G → lim_{⟵_{n∈IN}} G/F^{n}G is surjective, and

c. ⋂_{n∈IN} F^{n}E = {0};

then α is surjective.

Proof: Fix an element e ∈ E. We construct inductively a sequence (g_{n})_{n≥0} in G such that, for any n ≥ 0,

– g_{n+1} ≡ g_{n} mod F^{n}G and

– α(g_{n}) ≡ e mod F^{n}E.

Put g_{0} := 0 and assume that g_{n} already is constructed. Then e − α(g_{n}) ∈ F^{n}E so that by a. we find an element h ∈ F^{n}G such that

α(h) ≡ e − α(g_{n}) mod F^{n+1}E .

Define g_{n+1} := g_{n} + h.

Because of b. there is an element g ∈ G such that g ≡ g_{n} mod F^{n}G for any n ≥ 0. We then have α(g) ≡ α(g_{n}) ≡ e mod F^{n}E for any n ≥ 0. The assumption c. now implies that α(g) = e.

Coming to the proof of Proposition 5 we first note that the assumption that I/I^{2} is a finitely generated A/I-module implies that gr A := ⊕_{n≥0} I^{n}/I^{n+1} is a finitely

generated (graded) A/I-algebra. With A/I therefore also gr A is noetherian. We now fix an ideal 𝔞 ⊆ A and we have to show that 𝔞 is finitely generated. So far we know that the (homogeneous) ideal gr 𝔞 := ⊕_{n≥0} 𝔞 ∩ I^{n}/𝔞 ∩ I^{n+1} =

⊕_{n≥0} 𝔞 ∩ I^{n} + I^{n+1}/I^{n+1} in gr A is finitely generated. We choose homogeneous

generators of gr 𝔞, i.e., we fix elements a_{1} ∈ 𝔞 ∩ I^{n(1)}, …, a_{r} ∈ 𝔞 ∩ I^{n(r)} with appropriate integers n(1), …, n(r) ≥ 0 such that gr 𝔞 is generated by the cosets a_{1} + 𝔞 ∩ I^{n(1)+1}, …, a_{r} + 𝔞 ∩ I^{n(r)+1}. We claim that then 𝔞 is generated by a_{1}, …, a_{r} or equivalently that the homomorphism

\begin{aligned}α : & A^{r} & ⟶ & 𝔞 \\ & (b_{1}, …, b_{r}) & ⟼ & ∑_{i=1}^{r} b_{i}a_{i} \end{aligned}

is surjective. Defining

F^{n}A^{r} := I^{n−n(1)} × … × I^{n−n(r)}

(with the convention that I^{n} := A if n ≤ 0) and

F^{n}𝔞 := 𝔞 ∩ I^{n}

the map α is a homomorphism of filtered A-modules. We clearly have

⋂_{n∈IN} 𝔞 ∩ I^{n} ⊆ ⋂_{n∈IN} I^{n} = {0}

since A is Hausdorff as an adic ring. For the same reason that A is adic the natural map

A^{r} ⟶^{≅} lim_{⟵_{n∈IN}} A^{r}/F^{n}A^{r}

is bijective. Our claim therefore follows from Lemma 6 once we show that the maps induced by α

F^{n}A^{r}/F^{n+1}A^{r} ⟶ 𝔞 ∩ I^{n}/𝔞 ∩ I^{n+1} for n ≥ 0

are surjective. Fixing an element a ∈ 𝔞 ∩ I^{n} we find elements \overline{b}_{1}, …, \overline{b}_{r} ∈ gr A such that

∑_{i=1}^{r} \overline{b}_{i} · (a_{i} + 𝔞 ∩ I^{n(i)+1}) = a + 𝔞 ∩ I^{n+1} in gr A .

By passing to homogeneous components we may assume that \overline{b}_{i} is homogeneous of degree n − n(i), i.e., that \overline{b}_{i} is the coset of an element b_{i} ∈ I^{n−n(i)}. We then have

∑_{i=1}^{r} b_{i}a_{i} ≡ a mod 𝔞 ∩ I^{n+1} .

In general

Â := lim_{⟵_{n∈IN}} A/I^{n}

equipped with the projective limit topology with respect to the discrete topology on the rings A/I^{n} is a topological ring and the homomorphism φ_{A} is continuous. Â is called the \underline{I-adic completion} of A. Let 𝔟 ⊆ A be an open ideal. Then I^{m} ⊆ 𝔟 for some m ∈ IN and by construction

𝔟̂ := lim_{⟵_{n≥m}} 𝔟/I^{n}

is an open ideal in Â (which is independent of the choice of m). In particular the ideals (I^{m})ˆ ⊆ Â for m ∈ IN form a fundamental system of open neighbourhoods of the zero element 0 ∈ Â.

Lemma 7:

i. φ_{A}(A) is dense in Â; more generally we have 𝔟̂ = \overline{φ_{A}(𝔟)} for any open ideal 𝔟 ⊆ A; ii. the maps

open ideals in A \overrightarrow{⟵} open ideals in Â
𝔟 ⟼ 𝔟̂
φ_{A}^{−1}(𝔠) ⟵ 𝔠

are inverse to each other; iii. φ_{A} induces, for any open ideal 𝔟 ⊆ A, an isomorphism

A/𝔟 ⟶^{≅} Â/𝔟̂

Proof: Since 𝔟̂ is open and therefore closed we clearly have \overline{φ_{A}(𝔟)} ⊆ 𝔟̂. If I^{m} ⊆ 𝔟 then φ_{A}(𝔟) + (I^{m})ˆ = 𝔟̂. This implies that φ_{A}(𝔟) is dense in 𝔟̂ and proves i. The surjectivity of the map in iii. follows from the density of φ_{A}(A) in Â. Its injectivity is a consequence of the obvious equality φ_{A}^{−1}(𝔟̂) = 𝔟. For ii. it remains to consider an open ideal 𝔠 ⊆ Â. By the continuity of φ_{A} then 𝔟 := φ_{A}^{−1}(𝔠) is an open ideal in A. Since 𝔠 also is closed it follows from i. that 𝔟̂ ⊆ 𝔠. In the natural commutative diagram

\xymatrix{A/𝔟 \ar[rr]^{≅} \ar[dr] &  & Â/𝔟̂ \ar[dl] \\  & A/𝔠}

the left, resp. the right, oblique arrow therefore is injective, resp. surjective. But according to iii. the horizontal arrow is bijective. This means 𝔟̂ = 𝔠.

We see that in case A is Hausdorff the topology on Â induces via the injective map φ_{A} the I-adic topology on A.

Warning: In general the ideal 𝔟̂ is strictly larger than the ideal φ_{A}(𝔟) · Â.

Proposition 8:

If A is noetherian then we have: i. Â is Î-adic and noetherian; ii. 𝔟̂ = φ_{A}(𝔟)Â for any open ideal 𝔟 ⊆ A.

It is convenient to deduce this result from a more general statement about Amodules. Let M be an A-module. The submodules I^{n}M for n ∈ IN form a fundamental system of open neighbourhoods of the zero element 0 ∈ M in a unique topology which we call the \underline{I-adic} topology on M. In this way M becomes a topological A-module. Any homomorphism of A-modules is continuous with respect to the I-adic topologies. The \underline{I-adic completion}

M̂ := lim_{⟵_{n∈IN}} M/I^{n}M

in a natural way is an Â-module so that we have the functor

A-modules ⟶ Â-modules
M ⟼ M̂ .

The module M is called \underline{complete} if the obvious homomorphism

φ_{M} : M ⟶ M̂

of A-modules is bijective. We equip M̂ with the projective limit topology with respect to the discrete topology on the modules M/I^{n}M. It then is a topological Â-module and the homomorphism φ_{M} is continuous. For any open submodule N ⊆ M we have I^{m}M ⊆ N for some m ∈ IN so that

N̂ := lim_{⟵_{n≥m}} N/I^{n}M

is an open Â-submodule in M̂ (which is independent of the choice of m). Note that this N̂ also is the I-adic completion of N as an A-module in its own right: We have I^{n}N ⊆ I^{n}M ⊆ I^{n−m}N for n ≥ m. In particular the submodules (I^{m}M)ˆ for m ∈ IN form a fundamental system of open neighbourhoods of the zero element 0 ∈ M̂.

Lemma 9:

i. M is Hausdorff if and only if φ_{M} is injective; ii. φ_{M}(M) is dense in M̂; more generally we have N̂ = \overline{φ_{M}(N)} for any open submodule N ⊆ M; iii. the maps

open A-submodules in M \overrightarrow{⟵} open Â-submodules in M̂
N ⟼ N̂
φ_{M}^{−1}(L) ⟵ L

are inverse to each other; iv. φ_{M} induces, for any open submodule N ⊆ M, an isomorphism

M/N ⟶^{≅} M̂/N̂ .

We omit the proof since it is entirely analogous to the proofs of the corresponding facts for ideals above.

Lemma 10:

If α : L → M is a surjective homomorphism of A-modules then the homomorphism α̂ : L̂ → M̂ is surjective, too.

Proof: We define filtrations F^{n}L := I^{n}L and F^{n}L̂ := (I^{n}L)ˆ and similarly for M and M̂. Then α and α̂ are filtered homomorphisms. More precisely we have α(I^{n}L) = I^{n}M for any n ≥ 0 which means that gr α is surjective. Since gr α̂ = gr α by Lemma 9.iii we obtain that gr α̂ is surjective. On the other hand it is trivial that ⋂_{n∈IN}(I^{n}M)ˆ = {0} (i.e., that M̂ is Hausdorff). The natural map

L̂ ⟶ lim_{⟵_{n∈IN}} L̂/(I^{n}L)ˆ

is equal to the map

lim_{⟵_{n∈IN}} L/I^{n}L ⟶ lim_{⟵_{n∈IN}} L̂/(I^{n}L)ˆ

induced by φ_{M} which is an isomorphism by Lemma 9.iii. Therefore the assertion is a consequence of Lemma 6.

Proposition 11:

If M is a finitely generated A-module then we have: i. (I^{m}M)ˆ = (I^{m})ˆ · M̂ = (I^{m})ˆ · φ_{M}(M) for any m ≥ 0; ii. M̂ is finitely generated as an Â-module; iii. if A is adic then φ_{M} is surjective; if in addition the ideal I is finitely generated then we have: iv. N̂ = Â · φ_{M}(N) for any open submodule N ⊆ M; v. the topology on M̂ is the Î-adic topology (and M̂ is complete).

Proof: i. For trivial reasons we have

(I^{m})ˆ · φ_{M}(M) ⊆ (I^{m})ˆ · M̂ ⊆ (I^{m}M)ˆ

where the second inclusion is a consequence of the continuity of the multiplication by scalars in M̂. Also the asserted equalities are obvious for a finitely generated free A-module. But by assumption we have a surjective homomorphism of A-modules

α : A^{r} ⟶ M

which induces surjective A-module homomorphisms

α : I^{m}A^{r} ⟶ I^{m}M

for any m ≥ 0. By Lemma 10 the homomorphism

α̂ : (I^{m}A^{r})ˆ ⟶ (I^{m}M)ˆ

is surjective, too. We obtain

\begin{aligned}(I^{m}M) & ˆ = α̂((I^{m}A^{r})ˆ ) = α̂((I^{m})ˆ · φ_{A^{r}}(A^{r})) = (I^{m})ˆ · α̂(φ_{A^{r}}(A^{r})) \\ & = (I^{m})ˆ · φ_{M}(α(A^{r})) = (I^{m})ˆ · φ_{M}(M) . \end{aligned}

The assertions ii. and iii. are immediate consequences; for iii., e.g., observe that

M̂ = Â · φ_{M}(M) = φ_{A}(A) · φ_{M}(M) = φ_{M}(M) .

iv. We have I^{n}M ⊆ N for some n ∈ IN. Applying i. to m = 0 and the finitely generated A-module I^{n}M we obtain

(I^{n}M)ˆ = Â · φ_{M}(I^{n}M) ⊆ Â · φ_{M}(N) .

On the other hand Lemma 9.iii implies that

N̂ = (I^{n}M)ˆ + φ_{M}(N) .

v. Applying iv. to N := I^{m}N we obtain

(I^{m}M)ˆ = Â · φ_{M}(I^{m}M) = φ_{A}(I)^{m} · M̂ = (Î)^{m} · M̂ .

The completeness of M̂ was shown already in the proof of Lemma 10.

\underline{Proof of Proposition 8}: Everything except the noetherian property of Â is a special case of Prop. 11. But Î/Î^{2} = Î/(I^{2})ˆ = I/I^{2} is a finitely generated module for the noetherian ring Â/Î = A/I. It therefore follows from Prop. 5 that Â is noetherian.

Corollary 12:

If A is noetherian and M is a finitely generated A-module then we have

(I^{m}M)ˆ = I^{m} · M̂ for any m ∈ IN .

Proof: The argument was given already in the proof of Prop. 11.v.

As in the previous statement we will in the following usually omit the maps φ_{A} and φ_{M} in the formulas if the meaning is clear from the context.

Lemma of Artin-Rees:

Let A be noetherian and let N be a submodule of the finitely generated A-module M; then there is a m_{0} ∈ IN such that

I(I^{m}M ∩ N) = I^{m+1}M ∩ N for any m ≥ m_{0} .

Proof: Obviously the left hand side is contained in the right hand side for any m ≥ 0. In order to establish the reverse inclusion we consider the subring

B := ∑_{m≥0} I^{m}T^{m} ⊆ A[T]

of the ring of polynomials over A and the B-submodule

N′ := ∑_{m≥0}(I^{m}M ∩ N) ⊗_{A} AT^{m} ⊆ ∑_{m≥0} I^{m}M ⊗_{A} AT^{m} = M ⊗_{A} B

in M ⊗_{A} B. If a_{1}, …, a_{s} are generators of the ideal I then a_{1}T, …, a_{s}T are

generators of B as an A-algebra. Therefore B is noetherian and N′ is a finitely generated B-module. Let

x_{1} ⊗ T^{m(1)}, …, x_{r} ⊗ T^{m(r)} with x_{i} ∈ I^{m(i)}M ∩ N

be generators of N′ as a B-module and put

m_{0} := max(m(1), …, m(r)) .

Fixing an integer m ≥ m_{0} and an element y ∈ I^{m}M ∩ N we have

y ⊗ T^{m} = ∑_{i=1} t_{i}(x_{i} ⊗ T^{m(i)})

with appropriate elements t_{i} ∈ B. By passing to homogeneous components we may assume that t_{i} is of the form

t_{i} = b_{i}T^{m−m(i)} with b_{i} ∈ I^{m−m(i)} .

Then

y ⊗ T^{m} = (∑_{i=1}^{r} b_{i}x_{i}) ⊗ T^{m}

and hence

y = ∑_{i=1}^{r} b_{i}x_{i} .

This shows that

I^{m}M ∩ N ⊆ ∑_{i=1}^{r} I^{m−m(i)}(I^{m(i)}M ∩ N) ⊆ I^{m−m_{0}}(I^{m_{0}}M ∩ N)

and proves the assertion in the equivalent form

I^{m}M ∩ N = I^{m−m_{0}}(I^{m_{0}}M ∩ N) for any m ≥ m_{0} .

Proposition 13: (Krull)

Let A be noetherian and let N be a submodule of the finitely generated A-module M; then the topology on N induced by the I-adic topology of M is equal to the I-adic topology of N.

Proof: With m_{0} being the constant in the Lemma of Artin-Rees we have

I^{m}M ∩ N = I^{m−m_{0}}(I^{m_{0}}M ∩ N) ⊆ I^{m−m_{0}}N

for m ≥ m_{0}.

Corollary 14: (Krull)

Let A be noetherian; for any ideal 𝔞 ⊆ A and any finitely generated A-module M we have: i. ⋂_{n∈IN} 𝔞^{n}M = {x ∈ M : (1 − a)x = 0 for some a ∈ 𝔞};

ii. if 1 + 𝔞 contains no zero divisors then ⋂_{n∈IN} 𝔞^{n} = {0}.

Proof: i. If we have x = ax for some a ∈ 𝔞 then clearly x = a^{n}x lies in 𝔞^{n}M for any n ∈ IN. On the other hand let x be contained in the intersection ⋂_{n∈IN} 𝔞^{n}M.

The 𝔞-adic topology on M then induces the trivial topology on the submodule Ax. By the above Prop. 13 the 𝔞-adic topology on Ax therefore is equal to the trivial topology. Since 𝔞x is open in Ax with respect to the 𝔞-adic topology we obtain 𝔞x = Ax. This means that we find an element a ∈ 𝔞 such that ax = x. ii. This is an obvious consequence of the first assertion.

Proposition 15:

Let A be noetherian and let M be a finitely generated A-module equipped with the I-adic topology; assume that I is contained in the Jacobson radical of A; then we have: i. M is Hausdorff; ii. every submodule of M is closed; iii. A is Hausdorff and every ideal in A is closed.

Proof: Our assumption implies that 1 + I consists of units in A. Therefore M is Hausdorff as a consequence of Cor. 14.i. Moreover it is clear that, for any submodule N ⊆ M, the I-adic topology on M/N is the quotient topology of the I-adic topology on M. Hence M/N is Hausdorff in the quotient topology which means that N is closed in M.

Corollary 16:

Let A be I-adic and noetherian; then every finitely generated A-module M is complete and any submodule in M is closed (w.r.t. the I-adic topology).

Proof: According to Lemma 4.i the ideal I is contained in the Jacobson radical of A so that Prop. 15 applies: The canonical homomorphism φ_{M} is injective and every submodule of M is closed. The surjectivity of φ_{M} was shown already in Prop. 11.iii.

If A is noetherian the completion functor M ↦ M̂ (always w.r.t. the I-adic topologies) has very nice properties. Let

\begin{matrix}α_{M} : Â ⊗_{A} M & ⟶ M̂ \\ â ⊗ x & ⟼ âx \end{matrix}

be the obvious homomorphism of Â-modules.

Proposition 17:

If A is noetherian then we have: i. The functor M ↦ M̂ is exact for finitely generated A-modules M; ii. if M is a finitely generated A-module then α_{M} is bijective; iii. Â is a flat A-algebra.

Proof: i. Let

0 ⟶ L ⟶^{β} M ⟶^{γ} N ⟶ 0

be a short exact sequence of finitely generated A-modules. It was shown in Lemma 10 that γ̂ is surjective. On the other hand we have

ker(L/I^{n}L ⟶ M/I^{n}M) = I^{n}M ∩ L/I^{n}L .

The Lemma of Artin-Rees implies that, for n large, the natural map

I^{n}(I^{n}M ∩ L)/I^{2n}L = I^{2n}M ∩ L/I^{2n}L
↓
I^{n}M ∩ L/I^{n}L

is the zero map. Hence β̂ is injective. By functoriality we have im β̂ ⊆ ker γ̂. Fix now an element x̂ = (x_{n} + I^{n}M)_{n∈IN} ∈ ker γ̂ so that γ(x_{n}) ∈ I^{n}N for any n ∈ IN. Because of γ(I^{n}M) = I^{n}N we may assume that γ(x_{n}) = 0 for any n ∈ IN. Let y_{n} ∈ L be the unique element such that β(y_{n}) = x_{n}. Then

(y_{n} + I^{n}M ∩ L)_{n∈IN} ∈ lim_{⟵_{n∈IN}} L/I^{n}M ∩ L .

But according to the Lemma of Artin-Rees (compare the proof of Prop. 13) there is a m_{0} ∈ IN such that

ŷ:= (y_{n+m_{0}} + I^{n}L)_{n∈IN} ∈ L̂ .

Obviously β̂(ŷ) = x̂. ii. The assertion is trivial for M = A and therefore for any finitely generated free A-module. For general M we then consider a presentation

A^{r} ⟶ A^{s} ⟶ M ⟶ 0 .

It gives rise to the commutative exact diagram

Â ⊗_{A} A^{r} ⟶ Â ⊗_{A} A^{s} ⟶ Â ⊗_{A} M ⟶ 0
≅↓α_{Ar}  ≅↓α_{As}  ↓α_{M}
(A^{r})ˆ ⟶ (A^{s})ˆ ⟶ M̂  ⟶ 0

where the exactness of the lower row is a consequence of the previous assertion. With α_{A^{s}} also α_{M} is bijective. iii. Since any A-module is the filtered direct limit of its finitely generated submodules and since the tensor product commutes with filtered direct limits the exactness of Â ⊗_{A} . follows from the first two assertions.

Proposition 18:

Let A be noetherian and assume that I is contained in the Jacobson radical of A; then we have: i. Â is a faithfully flat A-algebra; ii. if M is a finitely generated A-module then M̂ is free over Â if and only if M is free over A.

Proof: i. Since Â is flat over A according to the previous Prop. 17 it remains to show that the natural map

M ⟶^{1⊗id} Â ⊗_{A} M

is injective for any A-module M. By a direct limit argument we again may assume that M is finitely generated. We have the commutative diagram

\xymatrix{ & Â ⊗_{A} M \ar[dd]_{≅}^{α_{M}} \\ M \ar@{-->}[ur]^{1⊗id} \ar[dr]_{φ_{M}} \\  & M̂}

The map α_{M} being an isomorphism by Prop. 17 we are reduced to prove that φ_{M} is injective, i.e., that M is Hausdorff. This was established in Prop. 15. ii. By assumption M̂/ÎM̂ is free over Â/Î. But using Lemma 9 and Prop. 11 we obtain

M/IM = M̂/(IM)ˆ = M̂/ÎM̂ .

Therefore M/IM is free over A/I. Let x_{1} + IM, …, x_{r} + IM be an A/I-basis of M/IM and consider the homomorphism

\begin{aligned}α : & L := A^{r} & ⟶ & M \\ & (a_{1}, …, a_{r}) & ⟼ & ∑_{i=1}^{r} a_{i}x_{i} . \end{aligned}

The Nakayama lemma immediately implies that α is surjective. For the injectivity we look at the commutative exact diagram

I ⊗_{A} ker α ⟶ I ⊗_{A} L ⟶ I ⊗_{A} M ⟶ 0
↓  ↓  ↓
0 ⟶ ker α ⟶ L ⟶^{α} M ⟶ 0
↓  ↓
A/I ⊗_{A} L ⟶^{≅} A/I ⊗_{A} M
↓  ↓
0  0  .

We claim that the map I ⊗_{A} M → M is injective. Then the map I ⊗_{A} ker α → ker α

is surjective so that ker α = I · ker α. Applying again the Nakayama lemma we obtain that ker α = 0, i.e., that α is injective. Since Â is faithfully flat over A the claimed injectivity can be checked after tensoring by Â. But in the commutative diagram

Â ⊗_{A}(I ⊗_{A} M) ⟶ Â ⊗_{A} M
≅↓α_{I}⊗α_{M}  ≅↓α_{M}
Î⊗_{Â} M̂  ⟶ M̂

the perpendicular maps are isomorphisms by Prop. 17 and the lower horizontal map is injective since M̂ is free over Â. Therefore the upper horizontal map is injective, too.

§2 Adic algebras

Throughout this section we fix an I-adic and noetherian ring A. Let B be an A-algebra. Because of I^{n}B = (IB)^{n} the I-adic topology of B as an A-module coincides with the IB-adic topology on the ring B. Therefore B equipped with the I-adic topology is a topological ring such that the ring homomorphism A → B which defines the algebra structure is continuous.

Definition:

The A-algebra B is called \underline{adic} if B as an A-module is complete.

The following properties of this notion are immediate.

– If B is an adic A-algebra then B is IB-adic as a ring.

– If B is noetherian then its completion B̂ (with respect to the IB-adic topology) is a noetherian adic A-algebra (Prop. 1.8).

– If B is finitely generated as an A-module then B is an adic A-algebra (Cor. 1.16).

– If B is an adic A-algebra then B is noetherian if and only if B/IB is noetherian (Prop. 1.5).

Applying the second property to the ring of polynomials A[T_{1}, …, T_{r}] we see that its completion

A{T_{1}, …, T_{r}} := A[T_{1}, …, T_{r}]ˆ

with respect to the I-adic topology is a noetherian adic A-algebra such that

A{T_{1}, …, T_{r}}/I^{n} · A{T_{1}, …, T_{r}} ≅ (A/I^{n})[T_{1}, …, T_{r}] .

Moreover, since the ring of polynomials is flat over A we obtain from Prop. 1.17 that A{T_{1}, …, T_{r}} is flat over A. There is the following explicit description in terms of power series.

Definition:

A formal power series

∑_{m_{1},...,m_{r}≥0} a_{m_{1},...,m_{r}}T_{1}^{m_{1}} · . . . · T_{r}^{m_{r}} ∈ A[[T_{1}, …, T_{r}]]

is called \underline{restricted} if for any n ∈ IN there is an m ≥ 0 such that a_{m_{1},...,m_{r}} ∈ I^{n} whenever max(m_{1}, …, m_{r}) ≥ m.

It is easily seen that

A[[T_{1}, …, T_{r}]]^{res} := set of all restricted formal power series

is a subring of A[[T_{1}, …, T_{r}]]. Reducing the coefficients of a power series modulo I^{n} defines a homomorphism of rings

A[[T_{1}, …, T_{r}]]^{res} ⟶ (A/I^{n})[T_{1}, …, T_{r}] .

In the projective limit with respect to n these homomorphisms give rise to a natural isomorphism of rings

A[[T_{1}, …, T_{r}]]^{res} ⟶^{≅} lim_{⟵_{n∈IN}}(A/I^{n})[T_{1}, …, T_{r}] ≅ A{T_{1}, …, T_{r}}

which we always will view as an identification. For this reason A{T_{1}, …, T_{r}} is called the \underline{ring of restricted formal power series} in T_{1}, …, T_{r} over A. We have

I^{n} · A{T_{1}, …, T_{r}} = set of all restricted formal power series

whose coefficients are contained in I^{n}.

Proposition 1: (Universal property)

Let C be any adic ring and let φ : A → C be a continuous homomorphism of rings; for any choice of elements c_{1}, …, c_{r} ∈ C there is a unique continuous homomorphism of rings φ̃ : A{T_{1}, …, T_{r}} → C such that

φ̃ | A = φ and φ̃(T_{i}) = c_{i} for 1 ≤ i ≤ r .

Proof: Certainly there is a unique homomorphism of rings φ′ : A[T_{1}, …, T_{r}] → C such that φ′ | A = φ and φ′(T_{i}) = c_{i}. We claim that φ′ is continuous with respect to the I-adic topology on A[T_{1}, …, T_{r}]. If the topology on C is the J-adic one for some ideal J ⊆ C then the continuity of φ implies that φ(I^{m}) ⊆ J for some m ∈ IN. It follows that φ′(I^{mn}A[T_{1}, …, T_{r}]) ⊆ J^{n} for any n ∈ IN. Therefore φ′ induces a continuous homomorphism of rings

A{T_{1}, …, T_{r}} = lim_{⟵_{n∈IN}} A[T_{1}, …, T_{r}]/I^{mn}A[T_{1}, …, T_{r}]
↓φ̃
C = lim_{⟵_{n∈IN}} C/J^{n}

such that φ̃ | A[T_{1}, …, T_{r}] = φ′. Since C is Hausdorff the unicity of φ̃ follows from the fact that A[T_{1}, …, T_{r}] is dense in A{T_{1}, …, T_{r}}.

In the situation of the above Proposition we usually will write

F (c_{1}, …, c_{r}) := φ̃(F)

for any restricted formal power series F ∈ A{T_{1}, …, T_{r}}. One useful application of this universal property is the following. First note that for s ≥ r we have

(A{T_{1}, …, T_{r}}){T_{r+1}, …, T_{s}} = A{T_{1}, …, T_{s}} .

Therefore any F ∈ A{T_{1}, …, T_{s}} can be written as

F = ∑_{m_{r+1},...,m_{s}≥0} G_{m_{r+1},...,m_{s}}(T_{1}, …, T_{r}) · T_{r+1}^{m_{r+1}} · . . . · T_{s}^{m_{s}}

with G_{m_{r+1},...,m_{s}} ∈ A{T_{1}, …, T_{r}}. Applying the universal property to the natural homomorphism A → A{T_{r+1}, …, T_{s}} and the s elements a_{1}, …, a_{r} ∈ A and T_{r+1}, …, T_{s} we obtain the identity

F (a_{1}, …, a_{r}, T_{r+1}, …, T_{s})= ∑_{m_{r+1},...,m_{s}≥0} G_{m_{r+1},...,m_{s}}(a_{1}, …, a_{r})·T_{r+1}^{m_{r+1}}·. . .·T_{s}^{m_{s}}

in A{T_{r+1}, …, T_{s}}.

Of basic importance is the following characterization of the residue class algebras of restricted formal power series rings.

Proposition 2:

For any A-algebra B the following assertions are equivalent: i. B is a (noetherian) adic A-algebra such that B/IB is an A/I-algebra of finite type; ii. B (with the I-adic topology) is as an A-algebra (topologically) isomorphic to a residue class algebra of A{T_{1}, …, T_{r}} for some r ≥ 0.

Proof: First let us assume that ii. holds true. Then B is of the form

B ≅ A{T_{1}, …, T_{r}}/𝔞

for some ideal 𝔞 ⊆ A{T_{1}, …, T_{r}}. Obviously B is noetherian. By Cor. 1.16 the ideal 𝔞 is closed so that B is Hausdorff. Consider now the commutative diagram

\xymatrix{ & B \ar[dd]^{φ_{B}} \\ A\{T_{1}, …, T_{r}\} \ar@{-->}[ur]^{φ} \ar@{-->}[dr]_{φ̂} \\  & B̂}

where φ denotes the obvious projection map and where the completions are formed with respect to the I-adic topologies. We have seen already that φ_{B} is injective. But by Lemma 1.10 with φ also φ̂ is surjective. Therefore φ_{B} has to be bijective, i.e., B is an adic A-algebra. Moreover φ induces a surjection

A/I[T_{1}, …, T_{r}] ≅ A{T_{1}, …, T_{r}}/IA{T_{1}, …, T_{r}} ⟶ B/IB

which shows that B/IB is an A/I-algebra of finite type. This proves i. Now assume that i. holds true. We then find elements b_{1}, …, b_{r} ∈ B such that the homomorphism

\begin{matrix}A/I[T_{1}, …, T_{r}] & ⟶ B/IB \\ T_{i} & ⟼ b_{i} + IB \end{matrix}

is surjective. By the universal property in Prop. 1 this homomorphism lifts to a continuous homomorphism of adic A-algebras

φ : A{T_{1}, …, T_{r}} ⟶ B such that φ(T_{i}) = b_{i} for 1 ≤ i ≤ r .

We claim that φ is surjective. For that it suffices by Lemma 1.6 to show that the induced maps

gr^{n}φ : I^{n}A{T_{1}, …, T_{r}}/I^{n+1}A{T_{1}, …, T_{r}} ⟶ I^{n}B/I^{n+1}B

are surjective. This is the case for n = 0 by construction and then follows for any n ≥ 0 since gr B is generated by gr^{0}B = B/IB as a gr A-module. The surjectivity implies that

φ(I^{n}A{T_{1}, …, T_{r}}) = I^{n}B for any n ≥ 0

which means that φ is open. Therefore φ induces the topological isomorphism of A-algebras

A{T_{1}, …, T_{r}}/ ker φ ⟶^{≅} B

required in ii.

Definition:

The A-algebra B is called \underline{topologically of finite type} if B is a noetherian adic A-algebra such that B/IB is an A/I-algebra of finite type. Let Alg_{tft}(A) denote the category of all A-algebras topologically of finite type.

Note that any A-algebra homomorphism between two algebras in Alg_{tft}(A) is continuous. Also it is clear that if A → B and B → C are topologically of finite type then A → C is topologically of finite type, too.

Proposition 3:

Let B be a noetherian adic A-algebra; then B is flat over A if and only if B/I^{n}B is flat over A/I^{n} for all n ∈ IN.

Proof: The other implication being obvious we assume that B/I^{n}B is flat over A/I^{n} for all n ∈ IN. We have to show that for any injective homomorphism α : N ↪ M of A-modules the homomorphism

α ⊗ id : N ⊗_{A} B ⟶ M ⊗_{A} B

is injective, too. A direct limit argument reduces us to the case where M and N are finitely generated A-modules. In a first step we now claim that

ker(α ⊗ id) ⊆ I^{n}(N ⊗_{A} B)

for any n ∈ IN. By the Lemma of Artin-Rees we find a m_{0} ∈ IN such that

I^{n+m_{0}}M ∩ N ⊆ I^{n}N for any n ∈ IN .

Our assumption implies that the map

(N/I^{n+m_{0}}M ∩ N) ⊗_{A} B = (N/I^{n+m_{0}}M ∩ N) ⊗_{A/I^{n+m_{0}}} (B/I^{n+m_{0}}B)
↓
(M/I^{n+m_{0}}M) ⊗_{A} B = (M/I^{n+m_{0}}M) ⊗_{A/I^{n+m_{0}}} (B/I^{n+m_{0}}B)

is injective. Considering the commutative diagram

N ⊗_{A} B ⟶ (N/I^{n+m_{0}}M ∩ N) ⊗_{A} B
↓α⊗id  ↓
M ⊗_{A} B ⟶ (M/I^{n+m_{0}}M) ⊗_{A} B

we then see that

ker(α ⊗ id) ⊆ ker(N ⊗_{A} B ⟶ (N/I^{n+m_{0}}M ∩ N) ⊗_{A} B)

⊆ ker(N ⊗_{A} B ⟶ (N/I^{n}N) ⊗_{A} B)

= I^{n}(N ⊗_{A} B)

holds true as we have claimed. But by Cor. 1.16 the finitely generated B-module N ⊗_{A} B is Hausdorff so that ⋂_{n∈IN} I^{n}(N ⊗_{A} B) = 0. Therefore α ⊗ id is injective.

§3 Complete tensor products

We fix a continuous homomorphism φ : A → C of noetherian adic rings. Let A, resp. C, be I-adic, resp. J-adic. Then there is a m_{0} ∈ IN such that φ(I^{m_{0}}) ⊆ J. Let now B be an A-algebra topologically of finite type. Equipping the tensor product B ⊗_{A} C considered as a C-algebra with the J-adic topology the natural

homomorphisms

B ⟶ B ⊗_{A} C
↑
C

are continuous. The completion

B ˆ⊗_{A} C := (B ⊗_{A} C)ˆ

is a topological ring called the \underline{complete tensor product}. Composing with φ_{B ⊗_{A} C} gives rise to the commutative diagram of continuous homomorphisms of rings

B ⟶ B ˆ⊗_{A} C
↑ ↑
A ⟶^{φ} C  .

Because of

B ⊗_{A} C/J^{n}(B ⊗_{A} C) = B ⊗_{A}(C/J^{n}) = (B/I^{m_{0}n}B) ⊗_{A}(C/J^{n})

we have

B ˆ⊗_{A} C = lim_{⟵_{n∈IN}}(B/I^{m_{0}n}B) ⊗_{A}(C/J^{n}) .

Lemma 1:

A{T_{1}, …, T_{r}} ˆ⊗_{A} C = C{T_{1}, …, T_{r}}; moreover, for any m ∈ IN, we have

(J^{m}(A{T_{1}, …, T_{r}} ⊗_{A} C))ˆ = J^{m}C{T_{1}, …, T_{r}} .

Proof: Using that A{T_{1}, …, T_{r}} is flat over A we have, for any m ≥ 0, that

(J^{m}(A{T_{1}, …, T_{r}} ⊗_{A} C))ˆ

= lim_{⟵_{n≥m}} J^{m}(A{T_{1}, …, T_{r}} ⊗_{A} C)/J^{n}(A{T_{1}, …, T_{r}} ⊗_{A} C)

= lim_{⟵_{n≥m}} A{T_{1}, …, T_{r}} ⊗_{A}(J^{m}/J^{n})

= lim_{⟵_{n≥m}}(A{T_{1}, …, T_{r}}/I^{m_{0}n}A{T_{1}, …, T_{r}}) ⊗_{A}(J^{m}/J^{n})

= lim_{⟵_{n≥m}} A/I^{m_{0}n}[T_{1}, …, T_{r}] ⊗_{A}(J^{m}/J^{n})

= lim_{⟵_{n≥m}} J^{m}C[T_{1}, …, T_{r}]/J^{n}C[T_{1}, …, T_{r}]

= J^{m}C{T_{1}, …, T_{r}} .

Proposition 2: B ˆ⊗_{A} C is a C-algebra topologically of finite type. Proof: Let ψ : A{T_{1}, …, T_{r}} ⟶ B be a surjective homomorphism of Aalgebras. According to Lemma 1.10

(ψ ⊗ id)ˆ : A{T_{1}, …, T_{r}} ˆ⊗_{A} C ⟶ B ˆ⊗_{A} C

then is surjective, too. The same argument more precisely shows that

(ψ ⊗ id)ˆ ((J^{m}(A{T_{1}, …, T_{r}} ⊗_{A} C))ˆ ) = (J^{m}(B ⊗_{A} C))ˆ for any m ≥ 0 .

Using Lemma 1 we obtain

(J^{m}(B ⊗_{A} C))ˆ = (ψ ⊗ id)ˆ (J^{m}C{T_{1}, …, T_{r}}) = J^{m}(B ˆ⊗_{A} C)

which means that the topology on B ˆ⊗_{A} C is the J-adic one. Hence B ˆ⊗_{A} C is topologically isomorphic to a residue class algebra of C{T_{1}, …, T_{r}}.

Corollary 3: With B and B′ also B ˆ⊗_{A} B′ is an A-algebra topologically of finite type.

Proposition 4: (Universal property) Let D be any adic ring and let

\begin{matrix}B & ⟶ & D \\ ↑ & & ↑ \\ A & ⟶^{φ} & C \end{matrix}

be any commutative diagram of continuous homomorphisms of rings; then there exists a unique continuous homomorphism of rings ψ : B ˆ⊗_{A} C → D such that the diagram

\xymatrix{B \ar[dr] \ar[r] & B ˆ⊗_{A} C \ar[d]^{ψ} & C \ar[l] \ar[dl] \\  & D}

is commutative.

Proof: Since the proof is entirely analogous to the proof of Prop. 2.1 we only indicate the argument. The universal property of the usual tensor product gives rise to a homomorphism B ⊗_{A} C → D which is easily seen to be continuous with

respect to the J-adic topology on B ⊗_{A} C. The wanted homomorphism ψ then is

obtained by completion.

The complete tensor product of course is functorial. In particular

\begin{matrix}Alg_{tft}(A) & ⟶ Alg_{tft}(C) \\ B & ⟼ B ˆ⊗_{A} C \end{matrix}

is a functor. It follows from the formula before Lemma 1 that the complete tensor product is commutative and associative and that we have natural topological isomorphisms

B′ ˆ⊗_{B}(B ˆ⊗_{A} C) = B′ ˆ⊗_{A} C

for any B′ in Alg_{tft}(B) and

(B ˆ⊗_{A} C) ˆ⊗_{C} C′ = B ˆ⊗_{A} C′

for any continuous homomorphism C → C′ of noetherian adic rings.

Lemma 5:

If B is finitely generated as an A-module then B ˆ⊗_{A} C = B ⊗_{A} C.

Proof: B ⊗_{A} C is a finitely generated C-module and therefore is complete for the

J-adic topology by Cor. 1.16.

Proposition 6:

If B is flat over A then B ˆ⊗_{A} C is flat over C.

Proof: By assumption B/I^{m_{0}n}B is flat over A/I^{m_{0}n} for any n ∈ IN. By tensoring with C/J^{n} we obtain that B ˆ⊗_{A} C/J^{n}(B ˆ⊗_{A} C) is flat over C/J^{n} for all n ∈ IN.

Now apply Prop. 2 and Prop. 2.3.

§4 Complete localization

Let A be a noetherian I-adic ring and let S ⊆ A be a multiplicative subset. The localization S^{−1}A again is noetherian. Therefore its completion

A{S^{−1}} := (S^{−1}A)ˆ

with respect to the I-adic topology is a noetherian adic A-algebra called the \underline{complete localization} of A in S. Because of

I^{n} · S^{−1}A = S^{−1}I^{n}

we have

A{S^{−1}} = lim_{⟵_{n∈IN}} S^{−1}(A/I^{n}) .

Proposition 1: (Universal property)

Let C be any adic ring and let φ : A → C be a continuous homomorphism of rings such that φ(S) ⊆ C^{×}; then there exists a unique continuous homomorphism of rings φ̃ : A{S^{−1}} → C such that φ̃ | A = φ.

Corollary 2:

Let φ : A → C be a continuous homomorphism of noetherian adic rings and let T ⊆ C be a multiplicative subset such that φ(S) ⊆ T; then there exists a unique continuous homomorphism of rings φ̃ : A{S^{−1}} → C{T^{−1}} such that the diagram

A  ⟶^{φ}  C
↓  ↓
A{S^{−1}} ⟶^{φ̃} C{T^{−1}}

is commutative.

Remark 3:

A\{S^{−1}\} = 0 if and only if the zero element 0 is contained in the closure \overline{S} of S in A.

Proof: If 0 ∈ \overline{S} then I^{n} ∩ S ≠ ∅ for any n ∈ IN which implies S^{−1}(A/I^{n}) = 0 for any n and therefore A{S^{−1}} = 0. Now assume that A{S^{−1}} = 0. Then {0} is a dense subset in S^{−1}A which means I^{n} · S^{−1}A = S^{−1}A for any n ∈ IN. We therefore have 1 ∈ I^{n} · S^{−1}A and hence I^{n} ∩ S ≠ ∅ for any n ∈ IN. The latter implies 0 ∈ \overline{S}.

Lemma 4:

A{S^{−1}} is flat over A.

Proof: Since S^{−1}A is flat over A this follows from Prop. 1.17.

Lemma 5: If S is finitely generated then A{S^{−1}} is an A-algebra topologically of finite type. Proof: We have

A{S^{−1}}/I · A{S^{−1}} = S^{−1}A/I · S^{−1}A = S^{−1}(A/I) .

This is an A/I-algebra of finite type if S is finitely generated.

For any element b ∈ A put S_{b} := {1, b, b^{2}, …} and

A_{{b}} := A{S_{b}^{−1}} .

This is an A-algebra topologically of finite type.

Proposition 6: The maps

set of all open prime ⟶_{⟵} set of all open prime
ideals in A disjoint from S  ideals in A{S^{−1}}
𝔭 ⟼ 𝔭 · A{S^{−1}}
preimage of 𝔮 in A ⟵ 𝔮

are inverse to each other; moreover if 𝔭 ⊆ A is an open prime ideal such that 𝔭 ∩ S = ∅ then we have the isomorphism

Quot(A/𝔭) ⟶^{≅} Quot(A{S^{−1}}/𝔭A{S^{−1}})

between the corresponding quotient fields. Proof: Since the analogous maps for S^{−1}A are inverse to each other the first assertion follows from Lemma 1.7 and Prop. 1.8.ii. It also follows that

A{S^{−1}}/𝔭A{S^{−1}} = S^{−1}A/𝔭 · S^{−1}A = S^{−1}(A/𝔭)

which implies the second assertion.

§5 Complete blowing up

First let A be an arbitrary ring. The localization A_{b} := S_{b}^{−1}A of A in an element b ∈ A solves the universal problem of making the element b invertible. It is natural to pose the more general universal problem of making the element b generate a given ideal in A. Let J ⊆ A be an ideal which contains b. In the localization A_{b} we have the subring

A_{J,b} := \{u ∈ A_{b} : u = \frac{a}{b^{m}} for some m ≥ 0 and some a ∈ J^{m}\} .

It is called a generalized localization or \underline{blowing up} of A. The natural homomorphism

\begin{matrix}A & ⟶ & A_{J,b} & ⊆ & A_{b} \\ a & ⟼ & \frac{a}{1} \end{matrix}

into the localization factorizes through A_{J,b}. Moreover:

– J · A_{J,b} = \frac{b}{1} · A_{J,b}, and

– \frac{b}{1} is not a zero divisor in A_{J,b} (since it is invertible in A_{b}).

Lemma 1: (Universal property)

Let φ : A → A′ be any homomorphism of rings such that

− φ(J)A′ = φ(b)A′ , and

− φ(b) is not a zero divisor in A′ ;

then there is a unique homomorphism of rings φ̃ : A_{J,b} → A′ such that the diagram

\xymatrix{ & A_{J,b} \ar[dr]^{φ̃} \\ A \ar[ur] \ar[rr]_{φ} &  & A′}

is commutative.

Proof: Define φ̃ (\frac{a}{b^{m}} ) := a′ if φ(a) = φ(b^{m}) · a′.

Lemma 2:

If A is noetherian then A_{J,b} is an A-algebra of finite type and, in particular, is noetherian.

Proof: If a_{1}, …, a_{r} generate the ideal J then \frac{a_{1}}{b} , …, \frac{a_{r}}{b} generate A_{J,b} as an A-algebra.

Let φ : A → A′ be a homomorphism of rings and put J′ := φ(J)A′ and b′ := φ(b) ∈ J′. Then φ induces the homomorphism of rings

\begin{matrix}A_{J,b} & ⟶ A′_{J′,b′} \\ \frac{a}{b^{m}} & ⟼ \frac{φ(a)}{b^{′m}} \end{matrix}

which we usually again denote by φ.

Lemma 3:

If φ is flat then

\begin{matrix}A_{J,b} ⊗_{A} A′ & ⟶^{≅} A′_{J′,b′} \\ u ⊗ a′ & ⟼ φ(u) · \frac{a′}{1} \end{matrix}

is an isomorphism.

Proof: The surjectivity is obvious. For the injectivity consider the commutative diagram

A_{b} ⊗_{A} A′ ⟶^{≅} A′_{b′}
↑  ↑⊆
A_{J,b} ⊗_{A} A′ ⟶ A′_{J′,b′} .

The upper horizontal arrow is an isomorphism by one of the basic properties of localization. Our assumption that φ is flat guarantees that the left perpendicular arrow is injective.

If we apply this Lemma to the natural homomorphism A → A_{c} for some c ∈ J we obtain that

(A_{J,b})\frac{c}{1} = A_{bc} .

Now we assume again that A is a noetherian I-adic ring. By Lemma 2 the completion

A_{{J,b}} := (A_{J.b})ˆ

with respect to the I-adic topology is an A-algebra topologically of finite type called a \underline{complete blowing up} of A. The natural homomorphisms

A ⟶ A_{J,b} ⟶^{⊆} A_{b}

by completion induce (continuous) homomorphisms of rings

A ⟶ A_{{J,b}} ⟶ A_{{b}}

but the second one in general is no longer injective.

Remark 4:

We have J · A_{\{J,b\}} = \frac{b}{1} · A_{\{J,b\}}, and \frac{b}{1} is not a zero divisor in A_{\{J,b\}}.

Proof: The first assertion is clear. Concerning the second one we observe that \frac{b}{1} being not a zero divisor in A_{J,b} means that the map

\frac{b}{1}· : A_{J,b} ⟶ A_{J,b}

is injective. By Prop. 1.17.i the map

φ_{A_{J,b}} (\frac{b}{1}) · : A_{\{J,b\}} ⟶ A_{\{J,b\}}

then is injective, too. This amounts to \frac{b}{1} (or more precisely φ_{A_{J,b}} (\frac{b}{1})) not being a zero divisor in A_{{J,b}}.

Proposition 5: (Universal property)

Let C be any adic ring and let φ : A → C be a continuous homomorphism of rings such that

− φ(J)C = φ(b)C , and

− φ(b) is not a zero divisor in C ;

then there exists a unique continuous homomorphism of rings φ̃ : A_{{J,b}} → C such that the diagram

\xymatrix{ & A_{\{J,b\}} \ar[dr]^{φ̃} \\ A \ar[ur] \ar[rr]_{φ} &  & C}

is commutative.

Lemma 6:

Let φ : A → C be a flat continuous homomorphism of noetherian adic rings and put J′ := φ(J)C and b′ := φ(b); we then have

A_{{J,b}} ˆ⊗_{A} C = C_{{J′,b′}} .

Proof: Let C be Ĩ-adic and let m_{0} ∈ IN be such that φ(I^{m_{0}}) ⊆ Ĩ. The isomorphism

A_{J,b} ⊗_{A} C = C_{J′,b′}

in Lemma 3 induces by completion an isomorphism

C_{{J′,b′}} = lim_{⟵_{n∈IN}} A_{J,b} ⊗_{A} C/Ĩ^{n}(A_{J,b} ⊗_{A} C)

= lim_{⟵_{n∈IN}} A_{J,b} ⊗_{A}(C/Ĩ^{n})

= lim_{⟵_{n∈IN}}(A_{J,b}/I^{m_{0}n}A_{J,b}) ⊗_{A}(C/Ĩ^{n})

= A_{{J,b}} ˆ⊗_{A} C .

As an application we obtain

(A_{\{J,b\}})_{\{ \frac{c}{1} \}} = A_{\{bc\}} for any c ∈ J .

§6 Weierstraß theory

Let A be a noetherian I-adic ring and let 𝔞 ⊆ A be an ideal which is contained in the ideal t_{I}(A) of topologically nilpotent elements (e.g., 𝔞 = I). We view A/𝔞 as an adic A-algebra; in particular 𝔞 is closed. Given F ∈ A{T } we write F mod 𝔞 for its canonical image in A/𝔞{T }. Additional results about the structure of restricted formal power series rings can be obtained by using Hensel’s lemma. Since the proof of the latter given in [Bou] III §4.3 is completely elementary we only give the statement without repeating the proof.

Hensel’s lemma:

Let F ∈ A\{T \} and \overline{G} ∈ A/𝔞\{T \} be restricted formal power series and let \overline{P} ∈ A/𝔞[T] be a monic polynomial such that

− F mod 𝔞 = \overline{P} · \overline{G} , and
− \overline{P} and \overline{G} generate A/𝔞\{T \} as an ideal ;

then there exists a uniquely determined restricted formal power series G ∈ A{T } and a uniquely determined monic polynomial P ∈ A[T] such that

− F = P · G , and
− P mod 𝔞 = \overline{P} , G mod 𝔞 = \overline{G} ;

moreover we have: i. P and G generate A{T } as in ideal; ii. if F is a polynomial so, too, is G.

In order to make the proof of the Weierstraß theorems more transparent it is useful to establish some simple technical facts first.

Remark 1:

i. Let P = Q · G be an identity in A{T }; if P and Q are monic polynomials so, too, is G and we have deg G = deg P − deg Q; ii. an element a ∈ A is a unit if and only if a + 𝔞 is a unit in A/𝔞; iii. an element F ∈ A{T } is a unit if and only if F mod 𝔞 is a unit in A/𝔞{T }.

Proof: i. This assertion clearly holds true in A{T }/I^{n}A{T } = A/I^{n}[T] for any n ∈ IN. Since A is Hausdorff it then also holds true in A{T }. ii. This is proved in the same way as Lemma 1.4 using, of course, the assumption that 𝔞 ⊆ t_{I}(A). iii. The kernel of the projection map A{T } → A/𝔞{T } is the ideal 𝔞{T } of all restricted formal power series whose coefficients are contained in 𝔞. By ii. (applied to A{T } and 𝔞{T }) it suffices to show that any power series in 𝔞{T } is topologically nilpotent. Let F (T) = ∑_{m≥0} a_{m}T^{m} be in 𝔞{T } and fix

an n ∈ IN. Since F is a restricted formal power series there is some m_{0} ≥ 0 such that a_{m} ∈ I^{n} for m ≥ m_{0}. Since any a_{m} is topologically nilpotent we find some n_{0} ∈ IN such that a_{m}^{n_{0}} ∈ I^{n} for all 0 ≤ m < m_{0}. We then have F^{m_{0}(n_{0}−1)+1} ∈ I^{n} · A{T }.

Weierstraß preparation theorem:

Let F ∈ A{T } be a restricted formal power series such that F mod 𝔞 is a unitary polynomial of degree d ≥ 0; then there exist a uniquely determined monic polynomial P ∈ A[T] and a uniquely determined unit U ∈ A{T }^{×} such that

F = P · U ;

moreover P has degree d.

Proof: (Recall that a polynomial is called unitary if its highest coefficient is a unit.) Because of Remark 1.ii we may multiply F by a unit in A in such a way that F mod 𝔞 becomes a monic polynomial. We therefore are reduced to prove the assertion under the additional assumption that F mod 𝔞 is monic. Applying Hensel’s lemma to \overline{P} := F mod 𝔞 and \overline{G} := 1 we obtain a uniquely determined monic polynomial P ∈ A[T] and a uniquely determined U ∈ A{T } such that

F = P · U

and a. P mod 𝔞 = F mod 𝔞, and b. U mod 𝔞 = 1. It immediately follows from a. that P has degree d. On the other hand U is a unit in A{T } by b. and Remark 1.iii. In order to establish the unicity let (P′, U′) be another pair with the properties in the assertion. We then have the identity

F mod 𝔞 = P′ mod 𝔞 · U′ mod 𝔞

in A/𝔞{T } where F mod 𝔞 and P′ mod 𝔞 are monic polynomials. The Remark 1.i then says that U′ mod 𝔞 is a monic polynomial, too. On the other hand U′ mod 𝔞 is a unit in A/𝔞{T }. Therefore the same Remark 1.i applied to the identity 1 = U′ mod 𝔞 · U^{′−1} mod 𝔞 implies that U′ mod 𝔞 = 1 and hence that P′ mod 𝔞 = F mod 𝔞. Now the unicity statement in Hensel’s lemma says that P = P′ and U = U′.

Corollary 2:

Let F ∈ A{T_{1}, …, T_{r}} be a restricted formal power series such that after reducing its coefficients mod 𝔞 it becomes a polynomial in A/𝔞{T_{1}, …, T_{r−1}}[T_{r}] unitary of degree d ≥ 0 in T_{r}; then there exist a uniquely determined polynomial P ∈ A{T_{1}, …, T_{r−1}}[T_{r}] monic in T_{r} and a uniquely determined unit U ∈ A{T_{1}, …, T_{r}}^{×} such that

F = P · U ;

moreover P has degree d in T_{r}.

Proof: In the proof of Remark 1.iii we have seen that the ideal 𝔞{T_{1}, …, T_{r−1}} of all restricted formal power series in T_{1}, …, T_{r−1} whose coefficients are contained in 𝔞 consists of topologically nilpotent elements in the noetherian adic ring A{T_{1}, …, T_{r−1}}. We therefore may apply the preparation theorem to A{T_{1}, …, T_{r−1}} and the ideal 𝔞{T_{1}, …, T_{r−1}}.

Weierstraß division theorem:

Let G ∈ A{T } be a restricted formal power series such that G mod 𝔞 is a unitary polynomial of degree d ≥ 0; for any F ∈ A{T } there exist a uniquely determined H ∈ A{T } and a uniquely determined polynomial R ∈ A[T] of degree < d such that

F = G · H + R .

Proof: In a first step we assume that G is a monic polynomial of degree d ≥ 0. By the usual division algorithm we then find, for each n ∈ IN, uniquely determined polynomials H_{n}, R_{n} ∈ A/I^{n}[T] such that

F mod I^{n} = (G mod I^{n}) · H_{n} + R_{n} and deg R_{n} < d .

It follows from the unicity that

H_{n+1} mod I^{n} = H_{n} and R_{n+1} mod I^{n} = R_{n} .

Therefore

H := (H_{n})_{n∈IN} and R := (R_{n})_{n∈IN}

in A{T } = lim_{⟵_{n∈IN}} A/I^{n}[T] have the required properties. For general G we first

apply the preparation theorem which allows us to write G as

G = P · U

with U ∈ A{T }^{×} and a monic polynomial P ∈ A[T] of degree d. According to our first step we have

F = P · H′ + R

with H′ ∈ A{T } and R ∈ A[T] of degree < d. We obtain

F = G · H + R with H := U^{−1} · H′ .

The unicity of H and R follows from the unicity statements in the first step and in the preparation theorem.

Corollary 3:

Let G ∈ A{T_{1}, …, T_{r}} be a restricted formal power series such that after reducing its coefficients mod 𝔞 it becomes a polynomial in A/𝔞{T_{1}, …, T_{r−1}}[T] unitary of degree d ≥ 0 in T_{r}; for any F ∈ A{T_{1}, …, T_{r}} there exist a uniquely determined H ∈ A{T_{1}, …, T_{r}} and a uniquely determined polynomial R ∈ A{T_{1}, …, T_{r−1}}[T_{r}] of degree < d in T_{r} such that

F = G · H + R .

Corollary 4:

Let P = Q · H be an identity in A{T } where P and Q are polynomials; if Q is unitary then H is a polynomial.

Proof: By the division algorithm in A[T] we find polynomials H′ and R in A[T] such that

P = Q · H′ + R and deg R < deg Q .

The division theorem says that this decomposition even is unique in A{T }. We therefore must have H = H′ and R = 0.

In order to make use of these results we have to understand to which power series they actually can be applied. From now on we restrict to the case 𝔞 = I.

Definition:

A restricted formal power series

F (T_{1}, …, T_{r}) = ∑_{m≥0} G_{m}(T_{1}, …, T_{r−1})T_{r}^{m} ∈ A{T_{1}, …, T_{r}}

is called \underline{T_{r}-special}, resp. \underline{T_{r}-distinguished}, of degree d ≥ 0 if

− G_{m} ∈ I · A{T_{1}, …, T_{r}} for any m > d , and
− G_{d} mod I ∈ (A/I)\{0} , resp. ∈ (A/I)^{×} .

Lemma 5:

For any F ∈ A{T_{1}, …, T_{r}}\IA{T_{1}, …, T_{r}} there is an A-algebra automorphism φ of A{T_{1}, …, T_{r}} such that φ(F) is T_{r}-special.

Proof: First of all we observe that for any natural numbers ℓ_{1}, …, ℓ_{r−1} we have the A-algebra automorphism

\begin{matrix}A\{T_{1}, …, T_{r}\} & ⟶^{∼} A\{T_{1}, …, T_{r}\} \\ T_{i} & ⟼ \{ \begin{matrix}T_{i} + T_{r}^{ℓ_{i}} & if i ≠ r , \\ T_{r} & if i = r . \end{matrix} \end{matrix}

It is well-defined by Prop. 2.1 and it is bijective sine T_{i} ↦ T_{i} − T_{r}^{ℓ_{i}} for 1 ≤ i < r and T_{r} ↦ T_{r} defines an inverse. Write now

F = ∑_{\underline{m}≥0} a\underline{m}T_{1}^{m_{1}} · . . . · T_{r}^{m_{r}} .

Let \underline{n} = (n_{1}, …, n_{r}) be the maximal (w.r.t. the lexicographic ordering) tuple such that a\underline{n} ∉ I and put

t := total degree of F mod I .

We recursively define natural numbers

ℓ_{r−1} := 1 + t ,
ℓ_{r−2} := 1 + t · (1 + ℓ_{r−1}) ,
⋮
ℓ_{1} := 1 + t · (1 + ℓ_{r−1} + … + ℓ_{2})

and put

d := ℓ_{1}n_{1} + … + ℓ_{r−1}n_{r−1} + n_{r} .

The reason for these definitions is the following property. The function

\begin{matrix}M := \{\underline{m} : m_{1} + … + m_{r} ≤ t\} & → IN ∪ \{0\} \\ \underline{m} & ↦ d(\underline{m}) := ℓ_{1}m_{1} + … + ℓ_{r−1}m_{r−1} + m_{r} \end{matrix}

is strictly increasing with respect to the lexicographical ordering on the left hand side. In order to see this let \underline{m}, \underline{m}′ ∈ M be tuples such that \underline{m}′ is strictly larger than \underline{m}. We then have

m_{1} = m′_{1}, …, m_{i−1} = m′_{i−1} , and m_{i} < m′_{i} for some 1 ≤ i ≤ r

and we compute

d(\underline{m}) ≤ ℓ_{1}m′_{1} + … + ℓ_{i−1}m′_{i−1} + ℓ_{i}(m′_{i} − 1) + t · (ℓ_{i+1} + … + ℓ_{r−1} + 1)
= ℓ_{1}m′_{1} + … + ℓ_{i−1}m′_{i−1} + ℓ_{i}m′_{i} − 1
< d(\underline{m}′) .

By construction the set M′ := \{\underline{m} : a\underline{m} ∉ I\} is contained in M so that the function d(.) is strictly increasing on M′ with maximum value d(\underline{n}) = d. Consider now the A-algebra automorphism φ of A{T_{1}, …, T_{r}} given by

φ(T_{i}) := T_{i} + T_{r}^{ℓ_{i}} for 1 ≤ i < r and φ(T_{r}) := T_{r} .

We claim that φ(F) is T_{r}-special of degree d. We have

\begin{matrix}φ(F) = ∑_{\underline{m}≥0} a\underline{m}(T_{1} + T_{r}^{ℓ_{r}})^{m_{1}}·. . .·(T_{r−1} + T_{r}^{ℓ_{r−1}})^{m_{r−1}}·T_{r}^{m_{r}} = \\ ∑_{\underline{m}≥0} a\underline{m} ∑_{\begin{matrix}μ_{1},...,μ_{r−1} \\ 0≤μ_{i}≤m_{i} \end{matrix}} ( \begin{matrix}m_{1} \\ μ_{1} \end{matrix} ) … ( \begin{matrix}m_{r−1} \\ μ_{r−1} \end{matrix} ) T_{1}^{m_{1}−μ_{1}}…T_{r−1}^{m_{r−1}−μ_{r−1}}T_{r}^{d(μ_{1},...,μ_{r−1},m_{r})}. \end{matrix}

Since d(μ_{1}, …, μ_{r−1}, m_{r}) ≤ d(\underline{m}) ≤ d for \underline{m} ∈ M′ it follows that φ(F) mod I is a polynomial of degree ≤ d in T_{r}. It remains to consider those \underline{m} ∈ M′ such that d(μ_{1}, …, μ_{r−1}, m_{r}) = d occurs. Since d is the maximum value of d(.) on M′ it follows that μ_{i} = m_{i} = n_{i} for 1 ≤ i < r and that m_{r} = n_{r}. We conclude that

(coefficient of T_{r}^{d} in φ(F)) mod I = a\underline{n} mod I ≠ 0 .

Let B be a complete discrete valuation ring. We will discuss this notion in more detail in the next section. Here we work with the following definition: B is a noetherian local integral domain such that

— the maximal ideal of B is a nonzero principal ideal πB, and

— B is πB-adic.

Note that we have ⋂_{n∈IN} π^{n}B = {0} by Cor. 1.14.

Lemma 6:

For any nonzero F ∈ B{T_{1}, …, T_{r}} there is a B-algebra automorphism ψ of B{T_{1}, …, T_{r}} such that

F = π^{m} · ψ(P) · U

holds true for some m ≥ 0, U ∈ B{T_{1}, …, T_{r}}^{×}, and some polynomial P ∈ B{T_{1}, …, T_{r−1}}[T_{r}] monic in T_{r}.

Proof: First we find a m ≥ 0 such that

π^{−m}F ∈ B{T_{1}, …, T_{r}}\πB{T_{1}, …, T_{r}} .

Applying Lemma 5 we then find a B-algebra automorphism φ of B{T_{1}, …, T_{r}} such that φ(π^{−m}F) is T_{r}-distinguished. Finally Cor. 2 provides the polynomial P and a unit U′ such that

φ(π^{−m}F) = P · U′ .

Set ψ := φ^{−1} and U := ψ(U′).

Proposition 7:

Let C be an adic B-algebra such that C is torsion free as a B-module and let ξ : B{T_{1}, …, T_{r}} → C be a finite homomorphism of B-algebras; then there exist a B-algebra automorphism ψ of B{T_{1}, …, T_{r}} and an integer 0 ≤ d ≤ r such that the composite homomorphism

B{T_{1}, …, T_{d}} ⟶^{⊆} B{T_{1}, …, T_{r}} ⟶^{ψ} B{T_{1}, …, T_{r}} ⟶^{ξ} C

is finite and injective.

Proof: We will prove the assertion by induction with respect to r. By the torsion freeness of C the homomorphism B → C is injective which settles the case r = 0. Let us assume therefore that r ≥ 1 and that F ≠ 0 is in the kernel of ξ. Applying Lemma 6 we find a B-algebra automorphism ψ of B{T_{1}, …, T_{r}}, an m ≥ 0, and a polynomial P ∈ B{T_{1}, …, T_{r−1}}[T] monic in T_{r} such that

π^{m} · P ∈ ker(ξ ∘ ψ) .

Since C is torsion free it follows that even

P ∈ ker(ξ ∘ ψ) .

On the other hand Cor. 3 implies that the natural homomorphism

B{T_{1}, …, T_{r−1}} ⟶ B{T_{1}, …, T_{r}}/P · B{T_{1}, …, T_{r}}

is finite. Therefore the composite homomorphism

B{T_{1}, …, T_{r−1}} ⟶ B{T_{1}, …, T_{r}}/P · B{T_{1}, …, T_{r}} ⟶^{ξ·ψ} C

is finite, too. Apply now the induction hypothesis.

Normalization lemma:

Let C be a B-algebra topologically of finite type which is torsion free as a Bmodule; then there exists a finite and injective homomorphism

B{T_{1}, …, T_{d}} ↪ C

of B-algebras for some d ≥ 0.

Proof: Apply Prop. 7 to some surjective homomorphism B{T_{1}, …, T_{r}} ⟶ C of B-algebras.

§7 Discrete valuation rings

Let A be a noetherian I-adic integral domain with quotient field K.

Definition:

A \underline{discrete valuation v of A} is a surjective homomorphism v : K^{×} → ZZ such that

1) v(a + b) ≥ min(v(a), v(b)) for all a, b ∈ K^{×} with a + b ≠ 0,

2) v(a) ≥ 0 for all a ∈ A\{0}, and

3) v(a) > 0 for all a ∈ I\{0}.

Fixing a discrete valuation v of A let

o_{(v)} := {a ∈ K^{×} : v(a) ≥ 0} ∪ {0}

be the valuation ring of v and

𝔪_{(v)} := {a ∈ K^{×} : v(a) > 0} ∪ {0}

its maximal ideal. Then 2) says that A ⊆ o_{(v)} and 3) says that the inclusion

map A ↪^{⊆} o_{(v)} is continuous with respect to the I-adic, resp. 𝔪_{(v)}-adic, topology on A, resp. o_{(v)}. Put

o_{v} := 𝔪_{(v)}-adic completion of o_{(v)} and
𝔪_{v} := maximal ideal in o_{v} .

Then o_{v} is a complete discrete valuation ring; the corresponding discrete valuation of o_{v} will again be denoted by v. We have a natural continuous injective homomorphism of noetherian adic rings A ↪ o_{v}. The following easy observation is sometimes useful in proofs.

Remark 1:

The map v : A\{0} → ZZ is locally constant.

Proof: Fix an element a ∈ A\{0} and put m := v(a) ≥ 0. For any b ∈ I^{m+1} we have v(b) > m. From the well-known formula

v(a + b) = min(v(a), v(b)) whenever v(a) ≠ v(b)

we therefore obtain v(a + I^{m+1}) = {m}.

Let A ↪ B be a continuous injective homomorphism of noetherian adic integral domains. A discrete valuation w of B is said to \underline{extend} (or to be an extension of) a discrete valuation v of A if the corresponding valuation rings are mapped into each other and the induced injective homomorphism o_{(v)} ↪ o_{(w)} is continuous. In terms of the maps v and w this means that there is a natural number e ∈ IN such that w | K^{×} = e·v; this number e is called the \underline{ramification index} of w over v. By completion we obtain in this case a commutative diagram of continuous injective homomorphisms of rings

A ⟶ B
↓ ↓
o_{v} ⟶ o_{w} .

A crucial definition for later purposes is the following.

Definition: Let v be a discrete valuation of A; a discrete valuation w of B extending v is called \underline{finite over v} if the homomorphism o_{v} ↪ o_{w} is finite.

If w is finite over v then o_{w} is a finitely generated free o_{v}-module and the 𝔪_{w}-adic and 𝔪_{v}o_{w}-adic topologies on o_{w} coincide (note that 𝔪_{w}^{e} = 𝔪_{v}o_{w} if w | K^{×} = e · v). In this section we want to study the set

V := set of all discrete valuations of A .

Theorem 2: V is nonempty if and only if A ≠ K.

Clearly V is empty if A = K. We therefore assume that A ≠ K. First of all we need to know how to recognize a discrete valuation ring.

Lemma 3: Let o ⊆ K be a subring such that

– K is the quotient field of o,

– o is a noetherian local ring whose maximal ideal 𝔫 is nonzero and principal,

– A ⊆ o, and I ⊆ A ∩ 𝔫; then there exists a (unique) discrete valuation v of A such that o_{(v)} = o. Proof: From Cor. 1.14 we know that ⋂_{m∈IN} 𝔫^{m} = {0}. Since 𝔫 ≠ 0 we furthermore have 𝔫^{m+1} ⊂_{≠} 𝔫^{m} for any m ≥ 0. Therefore there is, for any nonzero element a ∈ o, a unique integer v(a) ≥ 0 such that

a ∈ 𝔫^{v(a)} and a ∉ 𝔫^{v(a)+1} .

If π is a generator of 𝔫 this means that

a = π^{v(a)} · u for some u ∈ o^{×} .

For an arbitrary element \frac{a}{b} ∈ K^{×} with a, b ∈ o we then put v (\frac{a}{b}) := v(a) − v(b). This defines the required discrete valuation.

Lemma 4:

Let o ⊆ K be a subring such that

— K is the quotient field of o,

— o is an integrally closed noetherian local ring of dimension 1,

— A ⊆ o, and I ⊆ A ∩ 𝔫 where 𝔫 denotes the maximal ideal in o; then there exists a (unique) discrete valuation v of A such that o_{(v)} = o.

Proof: All we have to show, by Lemma 3, is that 𝔫 is a principal ideal. Since 𝔫 ≠ 0 we know from Cor. 1.14 that 𝔫^{2} ⊂_{≠} 𝔫. Fix an element π ∈ 𝔫\𝔫^{2}. We want

to show that 𝔫 = πo. In order to do that we consider the o-subalgebra

𝔫^{−1} := {a ∈ K : a · 𝔫 ⊆ o}

in K and the ideal

𝔫^{−1} · 𝔫 ⊆ o .

Let us assume for a moment that

𝔫^{−1} · 𝔫 = o

holds true. Because of π · 𝔫^{−1} · 𝔫 = πo ⊂_{≠} 𝔫^{2} the ideal π · 𝔫^{−1} ⊆ o cannot be

contained in 𝔫 then and we must have π · 𝔫^{−1} = o. This implies

πo = π · 𝔫^{−1} · 𝔫 = 𝔫 .

Concerning our assumption we certainly have o ⊆ 𝔫^{−1} and therefore 𝔫 ⊆ 𝔫^{−1} · 𝔫 which means that there are only the two possibilities 𝔫^{−1} ·𝔫 = 𝔫 or = o. We have to exclude the first possibility. For this we first construct an element d ∈ 𝔫^{−1}\o. For any b ∈ o\πo we have the proper ideal

𝔞_{b} := {a ∈ o : ab ∈ πo}

in o. Since o is noetherian there is a largest ideal 𝔞_{c} among the 𝔞_{b}’s . This 𝔞_{c} has to be a prime ideal: If a′ac ∈ πo but ac ∉ πo then 𝔞_{c} ⊆ 𝔞_{ac} and a′ ∈ 𝔞_{ac}; the maximality of 𝔞_{c} then implies 𝔞_{c} = 𝔞_{ac} and hence a′ ∈ 𝔞_{c}. On the other hand 𝔞_{c} ≠ {0} since π ∈ 𝔞_{c}. We consequently obtain

𝔞_{c} = 𝔫 .

Now define d := cπ^{−1} ∈ K. Then d ∉ o since c ∉ πo but d ∈ 𝔫^{−1} since d𝔫 = d𝔞_{c} ⊆ o. Assuming 𝔫^{−1} · 𝔫 = 𝔫 we would have d · 𝔫 ⊆ 𝔫. It would mean that 𝔫 is a faithful o[d]-module which as an o-module is finitely generated. But this is an equivalent way of saying that d is integral over o. Since o is integrally closed we would be led to the contradiction that d ∈ o.

In addition we need some deeper results from commutative algebra for the proofs of which we have to refer to the standard literature.

Proposition 5: (Krull-Akizuki)

Let o be a noetherian local integral domain of dimension 1 with quotient field K, let L/K be a finite extension of fields, and let o ⊆ C ⊆ L be any intermediate ring; then C is noetherian of dimension ≤ 1 and C/𝔠, for any nonzero ideal 𝔠 ⊆ C, is of finite length as an o-module.

Proof: [Bou] VII §2.5 Prop. 5.

Proposition 6:

Let o be a noetherian local integral domain of dimension 1 with maximal ideal 𝔫 and with quotient field K, let L/K be a finite extension of fields, and let C be the integral closure of o in L; then the map

\{ \begin{matrix}maximal ideals \\ 𝔪 ⊆ C \end{matrix} \} ⟶^{∼} \{ \begin{matrix}discrete valuation rings o_{(v)} ⊆ L \\ such that o ⊆ o_{(v)} and 𝔫 = o ∩ 𝔪_{(v)} \end{matrix} \}
𝔪 ⟼ C_{𝔪}

is a bijection between two finite sets.

Proof: Let 𝔪 ⊆ C be a maximal ideal. We claim that 𝔪 ∩ o is a maximal ideal in o: Let \overline{c} ∈ o/𝔪 ∩ o be any nonzero element; then \overline{c}^{−1} ∈ C/𝔪 is integral over o/𝔪 ∩ o so that we have an equation

\overline{c}^{−n} + \overline{a}_{1}\overline{c}^{−(n−1)} + … + \overline{a}_{n} = 0

with appropriate elements \overline{a}_{i} ∈ o/𝔪 ∩ o; it follows that

\overline{c}^{−1} = −\overline{a}_{1} − … − \overline{a}_{n}\overline{c}^{n−1} ∈ o/𝔪 ∩ o

which means that o/𝔪 ∩ o is a field. We then necessarily have 𝔪 ∩ o = 𝔫, resp. 𝔫C ⊆ 𝔪. By the same argument C cannot be a field since it is integral over o which is not a field. We deduce from Prop. 5 that the integral domain C is noetherian of dimension 1 and that C/𝔫C is a finite dimensional o/𝔫-vector space. The latter implies that the set of maximal ideals in C is finite. And it follows from the former that C_{𝔪} is a noetherian local integral domain of dimension 1 with quotient field L. As a localization of an integrally closed domain C_{𝔪} is integrally closed, too. Therefore, by Lemma 4, C_{𝔪} is a discrete valuation ring in L. Note also that o ⊆ C_{𝔪} and o ∩ 𝔪C_{𝔪} = o ∩ (C ∩ 𝔪C_{𝔪}) = o ∩ 𝔪 = 𝔫. We see that the map in the assertion is well-defined. It clearly is injective. Now let o_{(v)} ⊆ L be a discrete valuation ring such that o ⊆ o_{(v)} and 𝔫 = o ∩ 𝔪_{(v)}. Discrete valuation rings are easily seen to be integrally closed. We obtain C ⊆ o_{(v)} and then also C_{𝔪} ⊆ o_{(v)} for 𝔪 := C ∩ 𝔪_{(v)}. Since 𝔫C ⊆ 𝔪 and C has dimension 1 the prime ideal 𝔪 has to be maximal. We have seen that C_{𝔪} itself is a discrete valuation ring in L. But any discrete valuation ring is maximal among all proper subrings of its quotient field. This shows that C_{𝔪} = o_{(v)} and proves the surjectivity of our map.

Principal ideal theorem: (Krull)

Let o be a noetherian integral domain, let a ∈ o be a nonzero element, and let 𝔭 ⊆ o be a prime ideal which is minimal with respect to containing a; then o_{𝔭} is a noetherian local integral domain of dimension 1.

Proof: [Mat] Thm. 13.5.

\underline{Proof of Theorem 2}: Assume that A ≠ K. Since also I ≠ A we find an open ideal J ⊆ A such that {0} ⊂_{≠} J ⊂_{≠} A. For any nonzero element b ∈ J the blowing

up A′ := A_{J,b} by Lemma 5.2 is a noetherian integral domain with quotient field K such that the ideal J′ := J · A_{J,b} = \frac{b}{1} · A_{J,b} is principal. Let us assume for a moment that we can choose b in such a way that J′ ≠ A′. Fixing then a prime ideal 𝔭′ ⊆ A′ which is minimal with respect to containing J′ the principal ideal theorem says that A′_{𝔭′} is a noetherian local integral domain of dimension 1 with quotient field K. In this situation Prop. 6 ensures the existence of a discrete valuation ring o_{(v)} ⊆ K such that A′_{𝔭′} ⊆ o_{(v)} and 𝔭′A′_{𝔭′} = A′_{𝔭′} ∩ 𝔪_{(v)}. In particular we have A ⊆ o_{(v)} and I ⊆ A ∩ 𝔭′ ⊆ A ∩ 𝔪_{(v)}. This means that v ∈ V. It remains to justify the above choice of b. First of all we note that by Cor. 1.14 we have J^{m+1} ⊂_{≠} J^{m} for all m ∈ IN. We claim that there exists a b ∈ J such that

b^{m−1} ∉ J^{m} for any m ∈ IN: Otherwise we find, for a given system of generators b_{1}, …, b_{r} of the ideal J, natural numbers m_{i} such that b_{i}^{m_{i}−1} ∈ J^{m_{i}} for any 1 ≤ i ≤ r; but this implies J^{r(m−1)+1} = J^{r(m−1)} for m := max(m_{1}, …, m_{r}). If with this choice of b we assume that \frac{b}{1} · A_{J,b} = A_{J,b} then there exist a m ∈ IN and an a ∈ J^{m} such that \frac{b}{1} · \frac{a}{b^{m}} = 1, i.e., such that b^{m−1} = a ∈ J^{m}. This is a contradiction and it therefore follows that b has the property that J′ ≠ A′.

Remark 7:

If I ≠ {0} then the map

V ⟶ set of open prime ideals in A
v ⟼ A ∩ 𝔪_{v}

is surjective.

Proof: In the proof of Theorem 2 the assumption that A is adic was actually not used at all. Therefore if 𝔭 ⊆ A is any open prime ideal we can carry out the constructions in that proof starting from the local ring A_{𝔭} equipped with the 𝔭A_{𝔭}-adic topology in order to obtain a discrete valuation v ∈ V such that A ∩ 𝔪_{v} = A ∩ (A_{𝔭} ∩ 𝔪_{v}) = A ∩ 𝔭A_{𝔭} = 𝔭. Note that A_{𝔭} ≠ K since 𝔭 contains the nonzero ideal I.

The set V carries a natural topology which is defined as follows. For any Asubalgebra A′ ⊆ K which is of finite type over A we put

V (A′) := {v ∈ V : A′ ⊆ o_{(v)}} .

We obviously have V (A) = V and

V (A′) ∩ V (A′′) = V (⟨A′, A′′⟩)

whenever A′, A′′ ⊆ K are two A-subalgebras of finite type over A and ⟨A′, A′′⟩ ⊆ K denotes the A-subalgebra generated by A′ ∪ A′′. Calling a subset of V open if it is the union of subsets of the form V (A′) therefore defines a topology on V which we call the \underline{Zariski topology}.

Remark 8:

i. Any point in V is closed; ii. any A-subalgebra A′ ⊆ K which is of finite type over A is a blowing up of A; iii. the subsets V (A_{J,b}) where J ⊆ A is a nonzero open ideal and b is a nonzero element in J form a basis of the topology of V; iv. if J ⊆ A is a nonzero ideal and b_{1}, …, b_{r} ∈ A\{0} is a system of generators of J then we have

V = V (A_{J,b_{1}}) ∪ … ∪ V (A_{J,b_{r}}) .

Proof: i. Let v′ ≠ v be two different points in V. Since o_{(v)} is not contained in o_{(v′)} (remember that o_{(v)} is maximal among all proper subrings of K) we find an element c ∈ o_{(v)}\o_{(v′)}. Then V (A[c]) is an open neighbourhood of v which does not contain v′. ii. Let c_{1}, …, c_{r} ∈ K be generators of A′ as an A-algebra. We find elements b ∈ A\{0} and a_{1}, …, a_{r} ∈ A such that

c_{i} = \frac{a_{i}}{b} for any 1 ≤ i ≤ r .

If J denotes the ideal in A generated by b, a_{1}, …, a_{r} then we have

A′ = A_{J,b} .

iii. We have

V (A_{J,b}) = ⋃_{v∈V (A_{J,b})} V (A_{J+I^{v(b)},b}) .

iv. Fix a discrete valuation v ∈ V. Since J is nonzero we have J · o_{(v)} = 𝔪_{(v)}^{n} for some n ≥ 0. Of course the exponent n is given as

n = min(v(b_{1}), …, v(b_{r})) ;

in particular there is a 1 ≤ i ≤ r such that v(b_{i}) = n. We claim that A_{J,b_{i}} ⊆ o_{(v)}. Any element in A_{J,b_{i}} is of the form \frac{a}{b_{i}^{m}} for some m ≥ 0 and some a ∈ J^{m}. Since

J^{m} ⊆ 𝔪_{(v)}^{nm} we have v(a) ≥ nm and therefore v (\frac{a}{b_{i}^{m}} ) = v(a) − nm ≥ 0.

There is another important fact about extensions of discrete valuations.

Lemma 9:

Let o be a complete discrete valuation ring with quotient field K and let L/K be a finite extension of fields; then the integral closure C of o in L is a finitely generated o-module.

Proof: Let π be a generator of the maximal ideal in o. As a consequence of Prop. 5 we obtain that C is noetherian and that C/πC is a finitely generated o-module. Since o is integrally closed we have π^{−1} ∉ C and hence πC ≠ C. It then follows from Cor. 1.14 that ⋂_{n∈IN} π^{n}C = {0}. Moreover grC := ⊕_{n≥0} π^{n}C/π^{n+1}C as a

module over gr o := ⊕_{n≥0} π^{n}o/π^{n+1}o is generated by gr^{0}C = C/πC and therefore

is finitely generated. Since o is complete we have together all ingredients needed to apply Lemma 1.6 in the same manner as we did in the proof of Prop. 1.5 and to conclude that C is a finitely generated o-module.

Lemma 10:

Let o be a noetherian local integral domain which is adic with respect to its maximal ideal; let o → C be a finite homomorphism of integral domains; then C is a local ring, too.

Proof: Let 𝔫 ⊆ o be the maximal ideal. Then C is 𝔫C-adic. By the same argument as at the beginning of the proof of Prop. 6 we see that any maximal ideal in C lies above 𝔫. Therefore assuming that C is not local the artinian o/𝔫algebra C/𝔫C is not local so that we find a nontrivial idempotent \overline{c} ∈ C/𝔫C. Consider now the polynomial factorization

T^{2} − T mod 𝔫C = (T − \overline{c})(T − 1 + \overline{c}) .

Because of

(1 − 2\overline{c})(T − \overline{c}) + (2\overline{c} − 1)(T − 1 + \overline{c}) = 1

the two factors generate C/𝔫C{T } as an ideal. We therefore may apply Hensel’s lemma in order to obtain a monic polynomial P ∈ C[T] which divides T^{2} − T and such that P mod 𝔫C = T − \overline{c}. But this means that P (T) = T − c where c ∈ C is an element such that c^{2} = c and c + 𝔫C = \overline{c}. Hence c is a nontrivial idempotent in C. This is a contradiction since C was assumed to be an integral domain.

Proposition 11:

Let o = o_{(v)} be a complete discrete valuation ring and let o ↪ C be an injective and finite homomorphism of integral domains so that, in particular, C is an o-algebra topologically of finite type; then there exists precisely one discrete valuation w of C which extends v; moreover w is finite over v.

Proof: According to Lemma 9 the integral closure of C still is finite over o. We therefore may assume that C is integrally closed. Lemma 10 implies that C is local. In view of Prop. 6 this means that C is the valuation ring of the unique discrete valuation w extending v.

The assertion of Lemma 9 should be seen in the following context.

Definition:

A noetherian integral domain o with quotient field K is called \underline{Japanese} if for any extension of fields L/K the integral closure of o in L is a finitely generated o-module.

We can now rephrase Lemma 9 by saying that any complete discrete valuation ring is Japanese. Actually a much stronger result holds true.

Theorem 12: (Nagata)

Any noetherian local integral domain which is adic with respect to its maximal ideal is Japanese.

Proof: [EGA] IV 1 (0.23.1.5).

§8 Affinoid spectra

As a base ring we fix a noetherian 𝔪-adic integral domain o where 𝔪 is some nonzero proper ideal in o. In particular o is not a field; we denote by K the quotient field of o. Moreover we let V = V (o) be the nonempty (Thm. 7.2) set of all discrete valuations of o.

For any o-algebra A topologically of finite type we define

Sp_{o}(A) := set of all pairs (𝔭, w) where 𝔭 ⊆ A is

a prime ideal such that o → A/𝔭 is

injective and w is a discrete valuation

of A/𝔭 which is finite over some v ∈ V.

The set Sp_{o}(A) is called the \underline{affinoid spectrum} of A. If it is clear from the context which base ring we refer to we simply write Sp(A).

Remark 1:

The prime ideal 𝔭, for any point (𝔭, w) ∈ Sp(A), is not open in A.

Proof: If 𝔭 is open in A the injectivity of o ↪ A/𝔭 implies that the zero ideal {0} is open in o. This means that 𝔪 is nilpotent and therefore that 𝔪 = 0 which is a contradiction.

Forming the affinoid spectrum of course is functorial. For any homomorphism φ : A → B in Alg_{tft}(o) we have the induced map

\begin{matrix}Sp(φ) : Sp(B) & ⟶ Sp(A) \\ (𝔮, w) & ⟼ (φ^{−1}(𝔮), w | (A/φ^{−1}(𝔮))) ; \end{matrix}

here and in the following the restriction of a discrete valuation to a subring has to be understood in the sense that the restricted map is renormalized again (by dividing it by the corresponding ramification index) to be onto ZZ (so possible). In particular, under the obvious identification

\begin{matrix}Sp(o) & ⟷^{∼} V \\ (\{0\}, v) & ⟼ v \end{matrix}

the natural map Sp(A) → Sp(o) becomes the map

Sp(A) ⟶ V
(𝔭, w) ⟼ v if w extends v .

Let us fix for a moment a discrete valuation v ∈ V. It was shown in Prop. 3.2 that A ˆ⊗_{o} o_{v} is an o_{v}-algebra topologically of finite type. Let

ψ_{v} : A ⟶ A ˆ⊗_{o} o_{v}

be the natural continuous homomorphism of rings. It induces the map

\begin{matrix}ψ_{v}^{∗} : Sp_{o_{v}}(A ˆ⊗_{o} o_{v}) & ⟶ Sp_{o}(A) \\ (𝔮, w) & ⟼ (ψ_{v}^{−1}(𝔮), w | (A/ψ_{v}^{−1}(𝔮))) . \end{matrix}

Proposition 2:

The map ψ_{v}^{∗} induces a bijection

Sp_{o_{v}}(A ˆ⊗_{o} o_{v}) ⟶^{∼} {(𝔭, w) ∈ Sp_{o}(A) : w extends v} .

Proof: Let (𝔭, w) be a point in Sp_{o}(A) such that w extends v. We then have the commutative diagram of continuous injective homomorphisms of rings

o ⟶ A/𝔭
↓ ↓
o_{v} ⟶ o_{w} .

By the universal property of the complete tensor product (Prop. 3.4) it extends uniquely to a commutative diagram of continuous homomorphisms

\xymatrix{o_{v} \ar[dr] \ar[r] & (A/𝔭) ˆ⊗_{o} o_{v} \ar[d] & A/𝔭 \ar[l] \ar[dl] \\  & o_{w} & .}

On the other hand we have the commutative diagram

A  ⟶  (A/𝔭)
ψ_{v} ↓  ↓
A ˆ⊗_{o} o_{v} ⟶ (A/𝔭) ˆ⊗_{o} o_{v}

where the lower horizontal arrow is surjective by Lemma 1.10. Consider the prime ideal

𝔮 := ker(A ˆ⊗_{o} o_{v} ⟶ (A/𝔭) ˆ⊗_{o} o_{v} ⟶ o_{w}) .

It is now a simple diagram chase to see that (𝔮, w) ∈ Sp_{o_{v}}(A ˆ⊗_{o} o_{v}) is the unique

preimage of (𝔭, w) under the map ψ_{v}^{∗}.

Corollary 3:

Sp_{o}(A) = ⋃_{v∈V}^{·} im ψ_{v}^{∗} where the union is disjoint.

Lemma 4:

Let B be an o_{v}-algebra topologically of finite type; then the map

Sp_{o_{v}}(B) ⟶^{∼} {𝔭 ∈ Spec(B) : o_{v} ↪ B/𝔭 is injective and finite}
(𝔭, w) ⟼ 𝔭

is a bijection.

Proof: Given the point (𝔭, w) ∈ Sp_{o_{v}}(B) we have the injective homomorphisms o_{v} ↪ B/𝔭 ↪ o_{w} such that the composition o_{v} ↪ o_{w} is finite. Therefore o_{v} ↪ B/𝔭 is finite, too. This shows that the map is well-defined. Its bijectivity follows from Prop. 7.11.

Proposition 5:

Let B be an o_{v}-algebra topologically of finite type and let K_{v} denote the quotient field of o_{v}; then the map

Sp_{o_{v}}(B) ⟶^{∼} set of maximal ideals in B ⊗_{o_{v}} K_{v}
(𝔭, w) ⟼ 𝔭 ⊗_{o_{v}} K_{v}

is a bijection.

Proof: Because of Lemma 4 we may replace the left hand side in the assertion by the set

{𝔭 ∈ Spec(B) : o_{v} ↪ B/𝔭 is injective and finite} .

For each prime ideal 𝔭 in this set

(B/𝔭) ⊗_{o_{v}} K_{v} = B ⊗_{o_{v}} K_{v}/𝔭 ⊗_{o_{v}} K_{v}

is an integral domain finite dimensional over K_{v} and therefore is a field. This shows that 𝔭 ⊗_{o_{v}} K_{v} is a maximal ideal in B ⊗_{o_{v}} K_{v}. Now let 𝔮 ⊆ B ⊗_{o_{v}} K_{v} be any

prime ideal. Then 𝔭 := 𝔮∩B is the unique prime ideal in B such that 𝔭 ⊗_{o_{v}} K_{v} = 𝔮;

moreover o_{v} ↪ B/𝔭 clearly is injective. It remains to show that if 𝔮 is maximal then o_{v} ↪ B/𝔭 is finite. According to the Normalization lemma in §6 there is a finite and injective homomorphism of o_{v}-algebras

o_{v}{T_{1}, …, T_{d}} ↪ B/𝔭

for some d ≥ 0. Then

o_{v}{T_{1}, …, T_{d}} ⊗_{o_{v}} K_{v} ↪ (B/𝔭) ⊗_{o_{v}} K_{v} = B ⊗_{o_{v}} K_{v}/𝔮

is injective and finite, too, and the right hand side is a field. Therefore (compare the beginning of the proof of Prop. 7.6) the left hand side also has to be a field which only can happen if d = 0.

Remark 6:

If A ≠ 0 then there exists a v ∈ V such that A ˆ⊗_{o} o_{v} ≠ 0.

Proof: It suffices to show that A ˆ⊗_{o} o_{v} ≠ 𝔪_{v} · (A ˆ⊗_{o} o_{v}) for some v ∈ V. We fix an

open prime ideal in A (e.g., a maximal ideal by Lemma 1.4) and we let 𝔭 denote its preimage in o. According to Remark 7.7 there exists a discrete valuation v ∈ V such that 𝔭 = o ∩ 𝔪_{v}. We then obtain

(A ˆ⊗_{o} o_{v})/𝔪_{v}(A ˆ⊗_{o} o_{v}) = (A/𝔭A) ⊗_{o/𝔭}(o_{v}/𝔪_{v}) .

Since o/𝔭 ↪ A/𝔭A is injective and since o_{v}/𝔪_{v} being a field extension of the quotient field of o/𝔭 is flat over o/𝔭 the right hand side certainly is nonzero.

Proposition 7:

If A ≠ 0 is flat over o then Sp_{o}(A) ≠ ∅.

Proof: The Prop. 3.6 implies that A ˆ⊗_{o} o_{v} is flat over o_{v} for any v ∈ V. If we

choose v ∈ V in such a way that A ˆ⊗_{o} o_{v} ≠ 0 which is possible by the previous

Remark 6 then

o_{v} ↪ A ˆ⊗_{o} o_{v}

is injective. For this v we then obtain (A ˆ⊗_{o} o_{v}) ⊗_{o_{v}} K_{v} ≠ 0 where K_{v} again denotes

the quotient field of o_{v}. It follows now from Prop. 5 that Sp_{o_{v}}(A ˆ⊗_{o_{v}} o_{v}) ≠ ∅ and

therefore that Sp_{o}(A) ≠ ∅.

As a next step we will see that our affinoid spectra Sp_{o}(A) carry a natural topology. This comes from the following crucial observation.

Lemma 8:

For any open ideal J ⊆ A and any element b ∈ J the map

Sp_{o}(A_{{J,b}}) ↪ Sp_{o}(A)

induced by the complete blowing up A → A_{{J,b}} is injective; its image is

D_{J,b} := {(𝔭, w) ∈ Sp_{o}(A) : b ∉ 𝔭 and w(b + 𝔭) = min_{a∈J} w(a + 𝔭)} .

Proof: (We adopt from now on the usual convention that w(0) := ∞.) First we fix a point (𝔮, w̃) ∈ Sp(A_{{J,b}}). Let 𝔭 ⊆ A denote the preimage of 𝔮, set w := w̃ | (A/𝔭), and let e ≥ 1 be the ramification index of w̃ over w. By Remark 1 the prime ideal 𝔮 is not open in A_{{J,b}} and therefore cannot contain the open ideal JA_{\{J,b\}} = \frac{b}{1} · A_{\{J,b\}}; hence b ∉ 𝔭. On the other hand, for any a ∈ J, we have

\begin{aligned}e · w(a + 𝔭) & = w̃ (\frac{a}{1} + 𝔮) = w̃ (\frac{b}{1} · \frac{a}{b} + 𝔮) = w̃ (\frac{b}{1} + 𝔮) + w̃ (\frac{a}{b} + 𝔮) \\ & ≥ w̃ (\frac{b}{1} + 𝔮) = e · w(b + 𝔭) . \end{aligned}

This shows that the image of the map under consideration is contained in D_{J,b}. Now we fix a point (𝔭, w) ∈ D_{J,b}. Since b ∉ 𝔭 the image b̃ ∈ o_{w} of b under the associated continuous homomorphism A ⟶ A/𝔭 ↪ o_{w} is not a zero divisor in o_{w}. The second requirement in the definition of D_{J,b} guarantees that the ideal J̃ ⊆ o_{w} generated by the image of J is the principal ideal J̃ = b̃ · o_{w}. By the universal property of a complete blowing up the homomorphism A → o_{w} extends therefore in a unique way to a continuous homomorphism A_{{J,b}} → o_{w}. If 𝔮 denotes the kernel of the latter then (𝔮, w) ∈ Sp(A_{{J,b}}) is the unique preimage of (𝔭, w).

Remark 9:

The proof of Lemma 8 shows that for any point (𝔮, w̃) ∈ Sp_{o}(A_{{J,b}}) with image (𝔭, w) ∈ Sp_{o}(A) we have ow̃ = o_{w}.

In the situation of Lemma 8 let a_{1}, …, a_{r} ∈ J be a system of generators of that ideal. We then of course have

D_{J,b} = {(𝔭, w) ∈ Sp_{o}(A) : b ∉ 𝔭 and w(b + 𝔭) ≤ min_{1≤i≤r} w(a_{i} + 𝔭)} .

Lemma 10:

Let J, J′ ⊆ A be open ideals and let b and b′ be elements in J and J′, respectively; we then have

i. D_{J,b} ∩ D_{J′,b′} = D_{JJ′,bb′}; ii. Sp(φ)^{−1}(D_{J,b})=D_{φ(J),B,φ(b)} for any homomorphism φ:A→B in Alg_{tft}(o);

iii. (ψ_{v}^{∗})^{−1}(D_{J,b}) = D_{ψ(J)A ˆ⊗_{o} o_{v},ψ_{v}(b)} for any v ∈ V.

Proof: Easy exercise. (Observe that with J and J′ also JJ′ is an open ideal.)

The subsets D_{J,b} ⊆ Sp_{o}(A) for J ⊆ A an open ideal and b ∈ J are called \underline{rational subsets}. According to Lemma 10.i they form a basis for the open sets in a unique topology on Sp_{o}(A) which we call the \underline{canonical topology}. The other two assertions of Lemma 10 then say that the maps Sp(φ) for any φ in Alg_{tft}(o) and ψ_{v}^{∗} for any v ∈ V (o) are continuous. On Sp_{o}(o) = V (o) the canonical topology coincides, by Remark 7.8, with the Zariski topology.

Remark 11:

i. The image of ψ_{v}^{∗}, for any v ∈ V (o), is a closed subset in Sp_{o}(A);

ii. for any surjective homomorphism φ : A ⟶ B in Alg_{tft}(o) the map

Sp(φ) : Sp_{o}(B) ↪ Sp_{o}(A)

is a closed immersion.

Proof: i. Remark 7.8.i and Prop. 2. ii. Set 𝔞 := ker φ. The map Sp(φ) clearly is injective and its image is {(𝔭, w) ∈ Sp(A) : 𝔞 ⊆ 𝔭}. For any open ideal J′ ⊆ B and any element b′ ∈ J′ we put J := φ^{−1}(J′) and we choose an element b ∈ J such that φ(b) = b′. We then have

D_{J′,b′} = Sp(φ)^{−1}(D_{J,b}) .

This shows that the canonical topology on Sp(A) induces via the injection Sp(φ) the canonical topology on Sp(B). It remains to prove that the image of Sp(φ) is closed in Sp(A). Consider a point (𝔭, w) ∈ Sp(A) not in this image, i.e., such that 𝔞 ⊈ 𝔭. We choose an element b ∈ 𝔞\𝔭 and we put J := b · A + 𝔪^{w(b)}A. The rational subset D_{J,b} then is an open neighbourhood of (𝔭, w) which is disjoint from the image of Sp(φ).