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 , with group operation written multiplicatively. Equip 0 with the ordering from together with 0 < for every . A (non-Archimedean) Krull seminormt{ } on a ring A is now given by a function : A 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 = 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. First define orda : A⇥ Z by orda(a) = max n N a a ;thenset a = " for some fixed constant " (0, 1).! See Exercise 1.2.16 for{ some2 properties| 2 } of this seminorm.k k 2 Example 1.2.13. A complex Banach algebra is an example of a Banach ring. More precisely, a Banach ring A is a complex Banach algebra if A is a C-algebra, and a = a for all a A and C.See 2.3 for more information. k k | |1 ·k k 2 2 § 1.2.5. Invertibility criteria and closed ideals. A key feature of Banach rings is that we have nice criteria for elements to be invertible. Most of them rely on the following simple result. Lemma 1.2.14. If A is a Banach ring and a A is an element with a < 1, then 1 a is invertible in A. 2 k k
i n 1 i Proof. The inverse is given by i1=0 a . More precisely, let bn := i=0 a for n 1. Then a N (b ) is a Cauchy sequence, since b b k k for n, m N.Setb =lim b . Now n 1 Pn m 1 a P n n n k n k k k (1 a)bn =1 a ,so 1 (1 a)bn a ,whichimplies(1 a)b =limn (1 a)bn = 1. ⇤ k kk k !1 Corollary 1.2.15. If a, b are elements in a Banach ring A such that a is invertible and 1 1 a b < a , then a b is invertible. As a consequence, the set of invertible elements in A is open.k k k k Proof. We have
1 1 1 1 1 ba = aa ba a a b < 1, k k k kk kk k 1 so Lemma 1.2.14 implies that ba , and hence also b,isinvertible. ⇤ With these results in hand, we can study closures of ideals. It is easy to see that the closure of an ideal in a seminormed ring is an ideal, see Exercise 1.2.22, but not necessarily a proper ideal. Proposition 1.2.16. Let A be a Banach ring. Then: (i) the closure in A of any proper ideal in A is a proper ideal in A; (ii) any maximal ideal in A is closed; (iii) if a A is a multiplicative element, then the principal ideal (a) A is closed; (iii) if the2 norm on A is multiplicative, then every principal ideal of ⇢A is closed. 32 1.SEMINORMEDCOMMUTATIVEALGEBRA
Proof. We may assume A = 0. By Corollary 1.2.15, the set of non-invertible elements in A is open and of course nonempty. This6 implies that if a is an ideal of A, then the closure a,whichis an ideal of A by Exercise 1.2.22, is a proper ideal. Thus (i) holds, and (ii) is a direct consequence. Clearly (iii) implies (iv), so it only remains to prove (iii). We may assume a = 0. Let (abn)n 1 be asequencein(a) converging to some element c A.Then ab ab 06 as m, n .Since 2 k m nk! !1 the norm is multiplicative and a = 0, we have bm bn 0 as m, n ,so(bn)n is a Cauchy sequence converging to some elementk k6 b A. It isk now straightforwardk! to!1 check that c = ab (a), 2 2 completing the proof. ⇤ Remark 1.2.17. In general, not every ideal in a Banach ring need be closed, see Exercise 1.2.46. However, in a noid algebras, which are the building blocks for k-analytic spaces, any ideal is indeed closed, see Corollary 4.3.9 and Corollary 5.1.14. 1.2.6. Spectral radius and uniformization. If A is a seminormed ring, then the spectral radius of an element a A is defined as 2 ⇢(a):= lim an 1/n; n !1 k k the existence of the limit follows from the following version of Fekete’s Lemma:
Lemma 1.2.18. Let (rn)n1=1 be a sequence of nonnegative real numbers satisfying rm+n rmrn 1/n 1/n for m, n 1. Then the limit limn rn exists and equals infn rn R+. !1 2 This result is classical, but the reader is encouraged to prove it: see Exercise 1.2.28.
Proposition 1.2.19. The function ⇢: A R+ is a power-multiplicative seminorm on A.Ifthe seminorm on A is non-Archimedean, so is ⇢.! Proof. It is elementary to see that ⇢(an)=⇢(a)n, ⇢( a)=⇢(a), and ⇢(ab) ⇢(a)⇢(b) for all a, b A. The main di culty is to show that ⇢(a + b) ⇢(a )+⇢(b). To this end, pick ">0. There exists2 C 1 such that an C(⇢(a)+")n and bn C(⇢(b)+")n k k k k for all n 0. Now use the binomial theorem: n n j n j n j (a + b) ( 1) a b k k j k k·k k j=0 ✓ ◆ X n 2 n j n j 2 n C (⇢(a)+") (⇢(b)+") = C (⇢(a)+⇢(b)+2") . j j=0 X ✓ ◆ Taking nth roots and letting n yields ⇢(a + b) ⇢(a)+⇢(b)+2",soletting" 0 we see that ⇢ is a seminorm on A. If the seminorm!1 is non-Archimedean, then ⇢(n) n !1 for all n Z ; k·k k k 2 + hence ⇢ is non-Archimedean by Lemma 1.2.2. ⇤ The proofs of the next two results are left as exercises: see Exercises 1.2.29 and 1.2.30. Proposition 1.2.20. We have ⇢(a) a for every a A. Further, ⇢(a)= a i↵ a is power-multiplicative, i.e. an = a n for nk1.k 2 k k k k k k Proposition 1.2.21. Equivalent seminorms on A give rise to the same spectral radius seminorm. The spectral radius seminorm is the largest multiplicative seminorm dominated by the given norm. Remark 1.2.22. The name “spectral radius” comes from the theory of complex Banach algebras, where ⇢(a) is equal to the radius of the largest disc in C centered at the origin and containing the spectrum of a,see 2.3. § 1.2.SEMINORMEDRINGS 33
A Banach ring is called uniform if its norm is power-multiplicative. This terminology comes from the theory of complex Banach algebras, see 2.3.4. To any seminormed ring A we can associate its uniformization Au. This is defined as the separated§ completion of A with respect to the spectral radius seminorm. The map A Au satisfies a natural universal property, see Exercise 1.2.34. In practice, the uniformization! can often be characterized as in the following lemma, the proof of which is left to the reader (Exercise 1.2.35).
Lemma 1.2.23. Let A be a seminormed ring, A0 a uniform Banach ring, and ': A A0 a morphism of seminormed rings. Assume that '(a) = ⇢(a) for all a A,andthat'(A) is! dense k k 2 in A0. Then ': A A0 is the uniformization of A. ! 1.2.7. Classical reduction ring. Let A be a non-Archimedean seminormed ring. The classical reduction ring (or simply reduction ring) of A is defined as class A˜ := A /A , where A = a A a 1 and A = a A a < 1 . { 2 |k k } { 2 |k k } The graded reduction is functorial in the sense that any contractive morphism ': A B of non-Archimedean seminormed rings induces a homomorphism ! '˜: A˜class B˜class. ! of classical reduction rings. Indeed, ' being contractive implies '(A ) B and '(A ) B . ⇢ ⇢ Remark 1.2.24. The classical reduction ring is often defined using the spectral radius seminorm. In this case, any morphism ': A B of seminormed rings (i.e. bounded ring homomorphism) ! satisfies ⇢A ' ⇢B, see Exercise 1.2.39, and therefore induces a homomorphism' ˜: A˜ B˜.The classical reduction plays a key role in the study of strictly k-a noid algebras, see 4.6. ! § 1.2.8. Graded reduction ring. Now fix a subgroup H R⇥. We can then discuss H-graded ⇢ + rings and modules, as reviewedTemkinLocalII in Appendix A. Following Temkin [Tem04] we associate to any non-Archimedean seminormed ring A an H- graded ring, that generalizes the classical reduction ring. For r H,set 2 A˜ := r / A˜H := A˜r. r H M2 Note that the classical reduction ring A˜class can be identified with the with the 1 -graded ˜ { } reduction ring A 1 of A. We have a graded{ } reduction map A A˜ ! H defined as follows. Pick any element a A and set r := a .Ifr H,thenwesenda to its image 2 k k 2 in A˜r; otherwise we send a to 0. The graded reduction is also functorial: any contractive morphism ': A B of non-Archimedean ! seminormed rings induces a homomorphism' ˜ : A˜ B˜ of H-graded rings, defined as follows. H H ! H Givena ˜ A˜ , r H, pick a preimage a A ofa ˜ under the graded reduction map. If '(a) = r, 2 r 2 2 k k define' ˜ (˜a) A˜ as the the image of '(a) under the graded reduction map. If '(a) Remark 1.2.25. As mentioned above, the classical reduction functor is a very useful tool in the study of strictly k-a noid algebras, but not as well behaved for general k-a noid algebras as defined in 5.1. It was in order to study the latter that Temkin introduced the graded reduction functor, see§ 5.3. § Exercises for Section 1.2 EX70003 (1) Prove that a seminorm on a ring is the zero seminorm i↵ 1 = 1. k·k k k6 EX70019 (2) Suppose A is a seminormed ring such that n 1 for all n Z. Is it necessarily true that is non-Archimedean? k k 2 k·k EX70125 (3) Prove that is the largest possible seminorm on the ring Z. |·|1 EX70004 (4) Let A be a ring, and let be a nonzero seminorm on A as an abelian group for which there exists a constant C>0k such·k that ab C a b for all a, b A. Define a new function k k k k·k k 2 0 : A R by k·k ! + a 0 := sup ab / b b A, b =0 . k k {k k k k| 2 k k6 } Prove that 0 is a ring seminorm on A equivalent to . k·k k·k EX70001 (5) Let A be a seminormed ring of characteristic p. Characterize when the Frobenius map F : A A, given by F (a)=ap for a A, a morphism of seminormed rings. ! 2 EX70110 (6) Let ': A B be a ring homomorphism, and let be a seminorm (resp. non-Archimedean ! k·k seminorm) on B. Prove that the pullback 0 := '⇤( ), defined by a 0 := '(a) ,isa seminorm (resp. non-Archimedean seminorm)k· onk A. k·k k k k k EX70035 (7) Prove that if is a valuation on a ring A,thenA must be an integral domain, and that the valuation extendsk·k uniquely to the fraction field of A. EX70113 (8) Give an example of a normed field where the norm is not multiplicative. 1 1 EX70114 (9) Let (k, ) be a normed field. Prove that the norm is multiplicative i↵ a = a for k·k k k k k a k⇥. 2 EX70036 (10) Let A be a seminormed ring, and a A an ideal. Prove that A/a, equipped with the quotient seminorm, is a seminormed ring. ⇢ EX70117 (11) Let be a seminorm on a ring A. (a) kProve·k that t is a seminorm on A for any 0 EX70040 (16) Let A be a ring, and a A an ideal. Verify that seminorm = "orda in Example 1.2.12 really is a non-Archimedean ring⇢ seminorm. Is always a norm?k·k k·k EX70119 (17) Let A be a ring. (a) Prove that the maximum of a finite collection of seminorms on A is a seminorm on A. (b) Prove that the maximum of a finite collection of non-Archimedean seminorms on A is a non-Archimedean seminorm on A. (c) Is the sum of of a finite collection of seminorms on A necessarily a seminorm on A? EX70120 (18) Let A be any ring equipped with the trivial norm. Characterize algebraically what it means for the norm to be (a) multiplicative; (b) power-multiplicative. EX70207 (19) Let ': A B be a morphism of seminormed rings, and that the seminorm on B is power- multiplicative.! Prove that ' is contractive. EX70121 (20) Let ': A B be an admissible injective morphism of seminormed rings. Assume that the seminorms! on A and B are power-multiplicative. Prove that ' is an isometry. EX70122 (21) Let ': A B be a contractive map between nonzero seminormed rings. Assume that A is a valued field.! Prove that ' is an injective isometry. EX70081 (22) Let A be a seminormed ring, and a A an ideal. Prove that the closure of a is an ideal of A. ⇢ EX70213 (23) Give an example of a normed ring where the set of invertible elements is not open. EX70214 (24) Give an example of a dense proper ideal in a normed ring. EX70123 (25) Let n>1 be an integer and let " (0, 1). Define the n-adic norm on Q by 2 d d a := min " d Z,an Z , k kn { | 2 2 (n)} where Z Q is the set of rational numbers with denominator relatively prime to n. (n) ⇢ (a) Prove that n," is a non-Archimedean norm on the ring Q. Also verify that this construction generalizesk the·k p-adic norm defined in Example 1.2.8. (b) Give necessary and su cient conditions on n and " for the norm n," to be multiplicative and power-multiplicative, respectively. k·k (c) Prove that the topology on Q induced by n," does not depend on ". Also prove that and define the same topologyk·k on Q i↵ n and n have the same prime k·kn1,"1 k·kn2,"2 1 2 factors. In this case, can we always choose "1 and "2 such that n1,"1 and n2,"2 are equivalent norms on Q? k·k k·k (d) Let Q be the (separated) completion of (Q, ), and let p ,1 i l be the prime n," k·kn," i factors of n. Prove that there exist "i R+⇥,1 i l, such that Qn," is isomorphic, as a l 2 Banach ring, to i=1 Qpi,"i . (e) Conclude that the (separated) completion operation on a normed ring does not preserve the property of beingQ an integral domain. (f) Compute the spectral radius seminorm ⇢n," induced by n," on Q. Are ⇢n," and n," equivalent norms on Q? k·k k·k EX70118 (26) Consider the ring A = k[T ] provided with the norm in Example 1.2.10. (a) Verify that A is really a normed ring, and that A is not non-Archimedean. (b) Compute the (separated) completion of A. (c) Compute the spectral radius seminorm ⇢ on A. (d) Compute the uniformization of A. EX70124 (27) Let (A, ) be a non-Archimedean Banach ring. Let (a ) be a sequence in A such that 1 + a k·k n n n is not a zero divisor in A for any n. Prove that the product n1=1(1 + an) converges to a nonzero element of A as soon as an 0. Also prove the converse when the norm on A is multiplicative. Is the converse true in general?! Q EX70126 (28) Prove Fekete’s Lemma (see Lemma 1.2.18). 36 1.SEMINORMEDCOMMUTATIVEALGEBRA EX70127 (29) Prove Proposition 1.2.20. EX70128 (30) Prove Proposition 1.2.21. EX70058 (31) Let A be a seminormed ring. Prove that an element a A is topologically nilpotent (i.e. an 0 in the topology induced by the seminorm) i↵ ⇢(a) < 1.2 ! EX70129 (32) Let A be a Banach ring. (a) Let a A be an element with ⇢(a) < 1. Prove that 1 a is invertible in A. (b) Assume2 that A is non-Archimedean, and let M be an n n matrix with coe cients in A ⇥ such that ⇢(mij) < 1 for all i, j. Prove that I M is an invertible matrix. (c) Give an example showing that (b) may fail when A is not non-Archimedean. EX70130 (33) Give an example of a Banach ring (A, ) such that the norm is not non-Archimedean, but the spectral radius ⇢ is. k·k k·k EX70132 (34) Prove that the uniformization map A Au satisfies the following universal property: for any morphism of seminormed rings ': A !B,withB a uniform Banach ring, there exists a unique morphism 'u : Au B making the following! diagram commutative ! A Au ' 'u B Further, we have 'u ' . k kk k EX70140 (35) Prove Lemma 1.2.23. EX70133 (36) Let A be any ring equipped with the trivial norm. Give an algebraic description of the uni- formization Au. EX70215 (37) Let A , i I be a family of seminormed rings. Prove that there is a canonical isometric i 2 isomorphism ( A )u ⇠ Au. i i ! i i EX70134 (38) Let A be a non-Archimedean Banach ring. Prove that the subsets ⇢ 1 and ⇢<1 of A are Banach groupsQ that are openQ and closed in A. { } { } EX70051 (39) Let ': A B be a morphism of seminormed rings. Prove that ! ⇢ ' ⇢ , B A where ⇢A and ⇢B are the spectral radius seminorms on A and B,respectively. EX70135 (40) * Let A be a Noetherian normed ring, and let p1,...,pn be its minimal prime ideals. Equip A/p with the quotient seminorm for 1 i n, and equip B := n A/p ,withtheproduct i i=1 i seminorm. Prove that the canonical map A B induces an isometric isomorphism Au ⇠ Bu Q between the uniformizations of A and B. ! ! EX70007 (41) Let k be a field and set A = k(( T )). Let (rn)n Z be an decreasing sequence in R+⇥ such that 2 r0 = 1 and rn+m rn rm for m, n Z. (a) Prove that there exists· a unique2 non-Archimedean norm on A whose restriction to k n k·k is the trivial norm and such that T = rn for all n Z. Also prove that A becomes a Banach field in this norm. k k 2 (b) Give a characterization, in terms on (rn)n, of the norm being multiplicative and power- multiplicative, respectively. (c) Compute the spectral radius seminorm ⇢ on A. u (d) Give a characterization, in terms on (rn)n, of the uniformization A of A being a non- Archimedean field. (e) Use this construction to give an example of a Banach ring A and an element a A such that a is quasinilpotent but not nilpotent (i.e. an = 0 for all n 1but⇢(a) = 0). 2 6 EX70052 (42) Let H R⇥ be a subgroup. ⇢ + (a) Let A be a non-Archimedean seminormed ring. Is the graded reduction map A A˜H additive? ! EXERCISES FOR SECTION 1.2 37 (b) Assume that the norm on A is multiplicative (resp. power-multiplicative). Prove that the H-graded reduction A˜H is an integral domain (resp. reduced) (as an H-graded ring). Is the converse true? (c) Let ': A B be a contractive morphism of non-Archimedean seminormed rings. Prove ! that the induced map' ˜ : A˜ B˜ is well-defined. Does the diagram H H ! H ' A / B ✏ '˜H ✏ A˜H / B˜H commute? (d) Given contractive morphisms ': A B and : B C of non-Archimedean seminormed ! ! rings, prove that ( ]') = ˜ '˜ . H H H (e) Prove that the separated completion map A Aˆ induces an isomorphism A˜H AˆH on graded reductions. ! ! u ue (f) Let A A be the uniformization map. Describe the associated map A˜H (A )H on H-graded! reductions. ! (g) Let A , i I be a direct system of seminormed rings. Prove that the canonicalf map i 2 lim (A ) (lim^A ) is an isomorphism. i i H ! i H ! ! EX70209 (43) Compute the classical and R+⇥-graded reductions of the valued ring (Z, p.") and the valued field (Q, f ) from Example 1.2.8. |·| |·|p." EX70211 (44) Compute the classical and R+⇥-graded reductions of the polynomial ring k[T1,...,Tn]withthe monomial valuation as in Example 1.2.11 EX70210 (45) Compute the classical and R+⇥-graded reductions of the normed ring (Q, n."), where n," is the n-adic norm from Exercise 1.2.25. k·k k·k EX70005 (46) * Let k be field that is a complete with respect to a nontrivial non-Archimedean valuation. Let A := k T1,T2,... be the Tate algebra in countably many variables. This means that A is the { } I set of formal power series a = I cI T ,whereI ranges over multi-indices (i1,i2,...,in,...)with i Z for all n and i = 0 for n 0, where T I = T i1 T i2 ..., c k, and where c 0 as n 2 + n P 1 2 I 2 I ! 1 ni . The norm on A is given by a = max c . Prove that A is a Banach ring and n=1 n !1 k k | I | that the ideal a = 1 (Ti) is not closed. Hint: look at a = cnTn,wherecn = 0 and cn 0. P i=1 n 6 ! Categorical exercisesP. Consider the following categories. P Name Objects Morphisms SnRingbdd seminormed rings bounded homomorphisms NRingbdd normed rings bounded homomorphisms BRingbdd Banach rings bounded homomorphisms NASnRingbdd non-Archimedean seminormed rings bounded homomorphisms NANRingbdd non-Archimedean normed rings bounded homomorphisms NABRingbdd non-Archimedean Banach rings bounded homomorphisms SnRingcontr seminormed rings contractive homomorphisms NRingcontr normed rings contractive homomorphisms BRingcontr Banach rings contractive homomorphisms NASnRingcontr non-Archimedean seminormed rings contractive homomorphisms NANRingcontr non-Archimedean normed rings contractive homomorphisms NABRingcontr non-Archimedean Banach rings contractive homomorphisms EX70138 (47) Which of the categories above admit products? What about finite products? 38 1.SEMINORMEDCOMMUTATIVEALGEBRA EX70139 (48) Which of the categories above admit coproducts? What about finite coproducts? EX70141 (49) Which of the categories above admit direct limits? EX70142 (50) Which of the categories above admit inverse limits? EX70143 (51) Are the categories above additive? Do they admit kernels and cokernels? Are they abelian? EX70144 (52) * Describe all the monomorphisms and epimorphisms in the categories above. EX70145 (53) Find the initial and terminal objects of the categories above (if they exist). Possible additional exercises EX70137 (54) * Is there a relationship between the Noetherianity of a seminormed ring and its graded reduction? (55) Ring of integers in a number field. (56) Huber rings. (57) Tate rings.