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Seminormed Rings Jan 23 2020.Pdf 1.2.SEMINORMEDRINGS 29 January 23, 2020, 3:12pm 1.2. Seminormed rings All rings will be commutative with identity, denoted by 1. All ring homomorphisms take the identity to the identity. If R is a ring, we write R⇥ := R 0 . \{ } 1.2.1. Seminormed rings. A seminorm on a ring A is a seminorm on the underlying abelian group that satisfies 1 1 and is submultiplicative in the sense thatk·kab a b for all a, b A.Thisimplies 1k k=1unless is the zero seminorm. A seminormedk kk ringkis·k ak pair (A, 2), where A is a ringk andk is a seminormk·k on A.Anormed ring is a seminormed ring for whichk·k the seminorm is a norm.k A·kBanach ring is a complete normed ring. A normed field (resp. Banach field) is a normed (resp. Banach) ring that is also a field. Lemma 1.2.1. If is a nonzero seminorm on a ring A, then a =0for all invertible elements a A. As a consequence,k·k a nonzero seminorm on a field is a norm.k k6 2 1 Proof. If a A is invertible, then 1 = 1 a a− ,so a = 0. 2 k kk k·k k k k6 ⇤ A morphism of seminormed rings is a bounded ring homomorphism. As in the case of semigroups, there are two natural categories of seminormed rings, using bounded homomorphisms and contractive homomorphisms as morphisms, respectively. Each category has several full subcategories in which the objects are normed or even Banach rings, and/or are non-Archimedean. Some properties of these categories are developed in the exercises. 1.2.2. Power-multiplicative norms and valuations. We often require seminorms to satisfy stronger conditions than being submultiplicative. For instance, we say that a seminorm on A is power-multiplicative if an = a n for n 1. For such norms, we have the following usefulk· criterion.k k k k k ≥ Lemma 1.2.2. A power-multiplicative seminorm on a ring A is non-Archimedean i↵ n 1 for all n Z. k·k k k 2 The inequality n 1 can be interpreted as the failure of Archimedes’ axiom. k k Proof. The direct implication is trivial. Now suppose n 1 for all n and pick a, b A. For any n 1wehave k k 2 ≥ 1/n n n 1/n n j n j a + b = (a + b) = a b − k k k k j j=0 ✓ ◆ X 1/n n n j n j 1/n a b − (n + 1) max a , b . 0 k j kk k k k 1 {k k k k} j=0 X ✓ ◆ 1/n @ A Since limn (n + 1) = 1, this proves that a + b max a , b . ⇤ !1 k k {k k k k} A seminorm on a ring A is multiplicative if 1 = 1 and ab = a b for a, b A. More generally, ank· elementk a A is multiplicative if abk k= a b kfork all bk kA·.k Ak multiplicative2 seminorm will be called a semivaluation2 .3 k k k k·k k 2 A multiplicative norm on a ring A is called a valuation; we then call (A, )avalued ring. In this case, A must be an integralk·k domain, and extends uniquely to a valuationk·k on the fraction field Frac(A), see Exercise 1.2.7, so (Frac(A), k·)k is therefore a valued field. Valued fields will be studied in 1.3. We shall often denote a valuationk·k on a field by instead of . § |·| k·k 3When is the zero seminorm, the terminology is a bit inconsistent: every element is multiplicative but the k·k seminorm is not multiplicative. should address this at some point! 30 1.SEMINORMEDCOMMUTATIVEALGEBRA Remark 1.2.3. In the literature, valuations are usually assumed non-Archimedean and are often written additively, so that a valuation on a ring A becomes a function v : A⇥ R satisfying v(1) = 0, ! v(a+b) min v(a),v(b) , and v(ab)=v(a)+v(b) for a, b A⇥. To such a function we can associate a valuation≥ (in the{ sense above)} by setting = "v,where"2 (0, 1) is an arbitrary constant. Following Berkovich we shall work multiplicatively,|·| even though, in2 some situations, the additive convention may seem more natural. Remark 1.2.4. It is also possible to consider more general norms and seminorms. Consider an ordered abelian group Γ, with group operation written multiplicatively. Equip Γ 0 with the ordering from Γtogether with 0 <γfor every γ Γ. A (non-Archimedean) Krull seminormt{ } on a ring A is now given by a function : A Γ 2 0 satisfying 0 = 0, 1 1, a = a for a A, a + b max a , b k,· andk ab! ta{ }b for a, b kAk. A Krullk k seminormk isk a Krullk k norm2if a k=0ik↵a = 0,{ andk k itk isk} a Krullk valuationkk kkif kab = a2 b for all a, b A. In the theory of adic spaces,k k general Krull valuations (there simply calledk k valuations)k kk k play a key2 role, whereas for Berkovich spaces the focus is on the case Γ= R+⇥. 1.2.3. Operations. Many of the operations on semigroups extend to semirings. The proofs of the following simple properties are relegated to the exercises. A subring of a seminormed ring becomes a seminormed ring in the subspace seminorm. The same is true for normed rings and Banach rings. If a A is an ideal, then the quotient ring A/a is a seminormed ring in the quotient seminorm. If A is a seminormed⇢ ring, then the separation (resp. completion, resp. separated completion) of A is a normed (resp. complete seminormed, resp. Banach) ring. The product of a family of seminormed rings is a seminormed ring. Similarly, the direct limit (resp. complete direct limit) of a direct system of seminormed rings is a seminormed (resp. Banach) ring. There is also a localization procedure for seminormed ring. This is discussed in the more general context of seminormed modules, see 1.4.3. § 1.2.4. Examples. Examples of seminormed rings abound. Here is a small sample. Other examples can be built from these using operations such as products and relative polydisc algebras, see 1.8. § Example 1.2.5. Any ring A equipped with the trivial norm,definedby 0 = 0 and a =1 for a = 0 is a non-Archimedean Banach ring. This example, which at first mayk appeark irrelevant,k k in fact has6 interesting applications. Example 1.2.6. The field C is a Banach field with respect to the usual valuation ,see Example 1.1.5. The valued subfield (R, ) is also a Banach field, whereas the further|·| subfield1 (Q, ) is not a Banach field: its completion|·|1 is of course (R, ). The valued subring (Z, ) is a|·| Banach1 ring. |·|1 |·|1 Example 1.2.7. The field C can be equipped with the hybrid norm,definedby := k·khyb max 0, ,where 0 and are the trivial and usual absolute value on C,respectively. The{| hybrid · | |·| norm1} is power-multiplicative,|·| |·|1 but not multiplicative. Example 1.2.8. Given a prime p and a real number " (0, 1), define a p-adic valuation p," on n b 2 n |·| Q as follows: for a Q⇥,writea = p ,whereb, c Z and p - bc;then a p," := " . The (separated) 2 c 2 | | completion Qp," of Q with respect to this valuation is a the field of p-adic numbers. Arithmetic considerations sometimes dictate a natural choice of ", such as " =1/p. Example 1.2.9. Let k be any field, and r R+⇥. As in Example 1.1.10, the polynomial ring k[T ] 2 i min i ai=0 can be equipped with the following norm: i aiT = r { | 6 }. This norm is non-Archimedean, multiplicative, and restricts to the trivialk norm onkk.Whenr 1, k[T ] is a Banach ring. When r<1, the completion of k[T ]istheringkP[[ T ]] of formal power≥ series, equipped with a valuation 1.2.SEMINORMEDRINGS 31 satisfying the same formula as above. In this case, the fraction field of k[[ T ]] i s t h e fi e l d k(( T )) of formal Laurent series. The valuation on k(( T )) again satisfies the same formula and is complete, so k(( T )) is a complete valued field. Example 1.2.10. As a variant of Example 1.2.9, let k be a field and r R+⇥.Providek[T ]with i i 2 the following norm: aiT = r . This restricts to the trivial norm on k,butisneither k i k i ai=0 non-Archimedean nor multiplicative,{ nor| 6 complete} for any r. See Exercise 1.2.26. P P Example 1.2.11. We can give a multi-variable generalization of Example 1.2.9 as follows. Given r ,...,r R⇥, we can define a monomial valuation on the polynomial ring k[T ,...,T ]by 1 n 2 + k·kr 1 n a T ⌫ := max r⌫ a =0 , k ⌫ kr { | ⌫ 6 } ⌫ Zn X2 + ⌫ n ⌫i ⌫ n ⌫i where we use the multi-index notation T = i=1 Ti and r = i=1 ri for ⌫ =(⌫1,...,⌫n). We will further generalize this construction, as well as the multivariable version of Example 1.2.10, in 1.8. Q Q § Example 1.2.12. If A is a ring and a A is an ideal, then we define an a-adic seminorm on A ⇢ n orda(a) as follows.
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