p-Archimedean Divisibilities - p-adic Representations of Commutative Rings

Remark 3.1.8 In all the results above, we can change the hypothesis [∀ |^v∈Vp( | ) :g |^v f]

[∀ |^v∈V_p^max( | ) :f p -^v g], and we obtain the same conclusions, because

[∀ |^v∈V_p^max( | ) :f p -^v g] ⇒[∀ |^v∈V_p( | ) :f p-^v g]

⇒[∀ |^v∈V_p( | ) :gp|^v pf]

⇒[∀ |^v∈Vp( | ) :g |^v f]

3.2 p-Archimedean Divisibilities

We establish here the main results about p-Archimedean divisibilities and discuss them.

p-Archimedean divisibilities have very good properties. For example, we can prove thatV_p(|)6=∅, if|is a p-Archimedean divisibility, what in general is not true, or need some stronger conditions to holds, like the hypothesis that | is a strong p-divisibility.

They have also better local-global principles under more reasonable con-ditions than strong p-divisibilities.

This section will be mainly devoted to two problems: to prove the asser-tion V_p(|)6=∅and to find local-global principles, for a given p-Archimedean divisibility | .

We start remembering the definition of a p-Archimedean divisibility.

Definition 3.2.1 A p-divisibility | of A is p-Archimedean if it satisfies (∀a∈A)(∃n ∈N) : 1|pⁿa (3.2) To verify that V_p(| ) is not empty for a p-Archimedean divisibility | , we will prove that each p-Archimedean divisibility has an extension |^∗ being a strong p-divisibility. Thereafter, we can use the fact seen in the last section, saying that V_p( |^∗)6=∅ when|^∗ is a strong p-divisibility.

To do this, we need to understand the behavior of the elements c ∈ A satisfying cp|c.

If each such c is an infinitesimal with respect to |, then we are done, because | is then a strong p-divisibility. However, in general they do not belong to I( | ), but have a property also satisfied by the infinitesimals elements. This is the content of the next lemma.

Lemma 3.2.2 Let | be a p-Archimedean divisibility of A. Then, for all c∈A

(cp|c)⇒(∀a∈A)(∀n∈N) : (pⁿ|ac). (3.3) Proof : Let m be minimal such that 1|p^m(ac). For alln∈N,

pⁿ |p^n+mca| · · · |pca|ca, because pc |c.

This is enough to prove the next lemma, which will imply the existence of strong p-divisibility extensions of p-Archimedean divisibilities.

Lemma 3.2.3 Let | be a p-Archimedean divisibility ofA. Ifcp |cfor some c∈A, then p-0,c 1.

Proof : Suppose that p|_0,c1. Then there are there arer∈N⁺ andc₁, ..., c_r ∈ A satisfying a defining sequence forp|_0,c 1. We can write the last relation as

1∈S(p, c₁c, ..., c_rc) Ifcp |c, then

p|cci for all 1≤i≤r,

by Lemma(3.2.2), and hence 1 ∈S(p) = M(p), a contradiction.

Theorem 3.2.4 If | is a p-Archimedean divisibility of A, then V_p^max( | )6=∅

Proof : Define

X :={ |^∗ ∈E( |) :p-^∗ 1}

This is a non-empty partially ordered set and each element of X is p-Archimedean.

If ∅ 6= ( |_i)_i∈I ⊆ X is a chain, then |_I is p-Archimedean and p -^I 1, because otherwise p|_i 1 for somei∈I. Thus, by Zorn’s lemma, we can take some maximal element |^∗ ∈X.

By Lemma(3.2.3), if cp |^∗ c, then |^∗ ⊆ ( |^∗)_0,c ∈ X, and therefore, by maximality of |^∗ , ( |^∗)_0,c =|^∗ , which implies 0|^∗ c.

Thus |^∗ ∈E_p^str( | ), and hence Theorem(3.1.2) implies V_p( | )6=∅.

Now we pass to the other goal of this section, the search for local-global principles.

We will prove a local-global principle for a class of p-divisibilities, which are called LG (local-global). An example of a LG p-Archimedean divisibility is the canonical divisibility |₀ of C_X.

The first local-global principle is indeed a weak version of what we hope, because it just says that, for each n ∈N and b ∈A,

pⁿ|^v b for all |^v ∈V_p^max(| )⇒pⁿ|b, and in general we want that, for all a, b∈A,

a|^v b for all |^v ∈V_p^max(| )⇒a|b.

The weak version will serve to classify the elements of the kernel of rep-resentations of commutative rings in the ring of p-adic valued continuous functions CX, with X compact, as well as the set of totally positive p-adic valued elements of C_X, i.e.,

O₀ :={f ∈C_X : (∀x∈X) :f(x)∈Z^p}

At the end of this section, we also observe another property, also satisfied by the canonical p-divisibility |₀ and stronger than the previous one, that guarantees a strong version of the above weak local-global principle for p-bounded pairs of elements ofA (defined later).

We start the work defining an auxiliary function ord_p :A→Z∪ {∞}

Remember that, when working with p-divisibilities in this text, we will always assume that p⁻¹ ∈A.

Definition 3.2.5 Let | be a p-Archimedean divisibility of A. For all a∈A ord_p(a) := sup{n ∈Z:pⁿ|a} ∈Z∪ {∞}

is the order of a with respect to p.

We say that | is LG (Local-Global) if

(∀a∈A)(∀r∈N) :ord_p(a^r) =r·ord_p(a) (3.4) If we want to verify this property for some p-Archimedean divisibility |, it is better to use an easier equivalent sentence, namely

(∀a∈A) :ordp(a²) = 2·ordp(a) (3.5) The equivalence is confirmed by the following lemma collecting some basic properties of the function ord_p.

Lemma 3.2.6 Let | be a p-Archimedean divisibility of A. For all a, b∈A, 1. (∀a, b∈A) :ordp(ab)≥ordp(a) +ordp(b).

2. (∀a∈A)(∀k ∈Z) :ordp(ap^k) =k+ordp(a)

3. (∀a∈A) :ord_p(a²) = 2·ord_p(a) if, and only if, | is LG.

Proof : The first is true because, for alla, b∈A,

p^ord^p^(a) |a and p^ord^p^(b)|b ⇒p^ord^p^(a)+ord^p^(b)|ab,

and, if ord_p(a) = ∞ or ord_p(b) = ∞, then p^k | ab for all k ∈ Z, and hence ord_p(ab) =∞.

For the second, note that, given a∈A and k ∈Z,

p^ord^p^(p^k^a) |p^ka ⇒p^ord^p^(p^k^a)−k |a⇒ord_p(a)≥ord_p(p^ka)−k p^ord^p^(a) |a⇒p^ord^p^(a)+k |p^ka⇒ord_p(p^ka)≥ord_p(a) +k

The third property is actually the only requiring more work. Clearly, only the implication from left to right needs a verification.

Supposing that ord_p(a²) = 2·ord_p(a), for all a ∈ A, we can prove the non-trivial implication above using induction on r ≥2.

If ord_p(a) = ∞, then clearly ord_p(a^r) = ∞ = r·ord_p(a) for all r ≥ 1.

Thus we can restrict our proof to the case ord_p(a)<∞.

The case r = 2 was assumed true. Take r ≥ 2 such that ord_p(a^k) = k·ord_p(a) for all a∈A with 2≤k ≤r.

As p⁻¹ ∈ A, ord_p(a) = s if, and only if, ord_p(ap^k) = s+k for all k ≥0.

Thus, we can suppose, without loss of generality, that ord_p(a) > 0. In this case, by 1., we also have

ord_p(a^r+1)> ord_p(a^r)

Ifr+ 1 is odd andr ≥2, thenr+ 1 = 2q+ 1 with q+ 1 ≤r, and hence (r+ 1)·ord_p(a) = ord_p(a^r) +ord_p(a)

≤ordp(a^r+1) = ordp(a^2q+1)

< ordp(a^2(q+1)) = 2·ordp(a^q+1)

= 2(q+ 1)·ord_p(a) = (r+ 2)·ord_p(a) implying that ord_p(a^r+1) = (r+ 1)·ord_p(a). For r+ 1 = 2q follows that

ord_p(a^r+1) =ord_p(a^2q) = 2·ord_p(a^q) = 2q·ord_p(a) = (r+ 1)·ord_p(a)

The main property of LG p-Archimedean divisibilities, needed for the next results, is contained in the next lemma.

Lemma 3.2.7 If | is a LG p-Archimedean divisibility of A, then ord_p(a)<∞, p|^a,b1⇒ord_p(a)> ord_p(b), for all a, b∈A.

Proof : Suppose thatp|^a,b 1 andord_p(a)≤ord_p(b). The last implies p^ord^p^(a) |p^ord^p^(b) |b and p^ord^p^(a) |a

Moreover, the first hypothesis implies the existence ofr ∈N satisfying a^r ∈S(p· {aⁱb^r−i}0≤i≤r)

Thus

a^r ∈S(p· {aⁱb^r−i}0≤i≤r) ⊆S(p· {p^i·ord^p^(a)p^(r−i)·ord^p^(b)}0≤i≤r)

⊆S(p· {p^i·ord^p^(a)p^(r−i)·ord^p^(a)}0≤i≤r)

=S(p·p^r·ord^p^(a)) =S(p^r·ord^p^(a)+1), and then

r·ord_p(a) =ord_p(a^r)≥r·ord_p(a) + 1 > r·ord_p(a), a contradiction.

Herewith, we find a weak local-global principle.

Theorem 3.2.8 (Weak LGP) Let | be a LG p-Archimedean divisibility of A. For all f ∈A

[∀ |^v∈V_p^max( | ) :p|^v f]⇔p|f, and hence, for all n ∈Z,

[∀ |^v∈V_p^max( | ) :pⁿ|^v f]⇔pⁿ|f Proof : (⇐) : Trivial.

(⇒) : We have

p-f ⇒ord(f)≤ord(1) <∞Lem.(3.2.7)

⇒ p-^f,1 1⇒ ∃ |^v∈V_p^max( |) :f |^v 1, and hence p-^v f for such|^v ∈V_p^max( |).

One way of getting stronger versions of the theorem above is to sup-pose stronger conditions for | . For example, we say that a p-Archimedean divisibility | isSLG (strong LG)⁵ if

(∀n, r ∈N)(∀g, f ∈A) : (g^r |pⁿf^r)∧(r 6= 0)⇒g |p^bn/rcf (3.6) where b·c:Q→Zis the floor function, i.e., for all q∈Q,

bqc:= max{n ∈Z:n≤q}

Observe that the canonical p-Archimedean divisibility |₀ is also SLG p-Archimedean divisibility.

Each SLG p-Archimedean divisibility | satisfies a stronger local-global principle for p-bounded pairs of A, with respect to | , which are defined below.

Definition 3.2.9 Let| be a divisibility of A. An ordered pair (g, f)∈A×A is called a p-bounded pair of A (with respect to | ), if

(∃k∈N) :g |p^kf, The principle is the following.

if (g, f) is p-bounded pair of A.

Proof : We observe two cases: f ∈p

I(| ) and f /∈p I(| ).

If 0 | f^k for some k ≥ 0, then g^k | f^k for all g ∈ A. Thus the SLG property, g |f for all g ∈A.

Takef ∈A\p

I( |) and g ∈Awithg |p^kf. Ifk = 0, then triviallyg |f.

Suppose that k ≥1 and k is minimal with the property g |p^kf.

Ifh:=gp⁻¹, then

∀ |^v∈V_p^max(| ) :hp|^v f ⇒ ∀ |^v∈V_p^max(| ) :f -^v h

f /∈√

I( |)

⇒ p|^f,h1

(p - 1)

⇒ ∃r >0 :f^r ∈S(p· {fⁱh^r−i}0≤i≤r)

⇒ ∃r >0 :f^rp^rk−1 ∈S(p^rk· {fⁱh^r−i}0≤i≤r)

⇒ ∃r >0 :f^rp^rk−1 ∈S({(f p^k)ⁱ(hp^k)^r−i}0≤i≤r)

⇒ ∃r >0 :f^rp^rk−1 ∈S({(hp)ⁱ(hp^k)^r−i}0≤i≤r)

⇒ ∃r >0 :f^rp^rk−1 ∈S(h^rp^r)

⇒ ∃r >0 :f^rp^rk−1 ∈S(g^r)

5Note that each SLG p-Archimedean divisibility is also a LG p-Archimedean divisibility.

Therefore,g^r |p^rk−1f^r with r > 0, and henceg |p^b(rk−1)/rcf =p^k−1f, by the SLG property, contradicting the minimal choice of k. Thus g |f.

Im Dokument p-adic Representations of Commutative Rings (Seite 93-99)