ARfiCTERFIJA‘YEON OF gm»... EL mam: Gamma? was @536 TEE C W: 5532;:5:55:’51,;35.5:22;: I In H- n‘.‘ 1| . ,2: £121 (K s 3.51.33 This is to certify that the thesis entitled ON THE CHARACTERIZATION OF INERTIAL COEFFICIENT RINGS presented by Ellen Elizabeth Kirkman has been accepted towards fulfillment of the requirements for Ph.D. degreein Mathematics { C Jm CW Maj professor Date Julv 25. 1975 0-7639 ABSTRACT ON THE CHARACTERIZATION OF INERTIAL COEFFICIENT RINGS BY Ellen Elizabeth Kirkman Let R be a commutative ring and A be an R—algebra which is finitely generated as an R—module and has Jacobson radical N. Ingraham.defined R to be an inertial coefficient ring if when A/N is a separable R—algebra there exists a separable R—subalgebra S of A such that A = S + N. (A,N) is called an L.I.ypair (lifting idempotent pair) if every idempotent in .A/N is of the form 5 where e is an idempotent of A. Ingraham has conjectured that if for every finitely generated R-algebra' A. (A,N) is an L.I. pair,then R is an inertial coefficient ring. The main result of Chapter II is that the converse of this conjecture is true: If A is a finitely generated algebra over an inertial coefficient ring R then (A.N) is an L.I. pair. Let X(R) denote the Pierce decomposition space of R and Rx denote the stalk of the sheaf over X(R) at the point x E X(R). In Chapter III it is shown that R is an inertial coefficient ring if and only if R.x is an inertial coefficient ring for all x E X(R). This result is used to show several rings are inertial coefficient rings. ON THE CHARACTERIZATION OF INERTIAL COEFFICIENT RINGS BY Ellen Elizabeth Kirkman A DISSERTATION Submitted to .Michigan State university in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1975 ACKNOWLEDGMENTS I am deeply indebted to Professor E.C. Ingraham for introducing me to interesting topics in ring theory and for his thoughtful guidance and encouragement in the preparation of this dissertation. I would also like to thank Professor W.C. Brown for several useful conversa- tions. Finally, I am very appreciative of the careful and efficient typing of Mary Reynolds. ii TABLE OF CONTENTS Chapter Page I. PRELIMINARIES 1 §1. Separable Algebras and Inertial Coefficient Rings . . . . . . . . . l §2. Hensel Rings and the Idempotent Lifting Property . . . . . . . . . 9 §3. The Decomposition Space . . . . . . 14 II. INERTIAL COEFFICIENT RINGS AND THE IDEMPOTENT LIFTING PROPERTY 18 III. NEW INERTIAL COEFFICIENT RINGS 31 BIBLIOGRAPHY 45 iii CHAPTER I PRELIMI NARI ES Chapter I contains historical material and a description of the tools used in later chapters. we shall also fix the notation which will be used throughout. §l. Separable Algebras and Inertial Coefficient Rings A11 rings we shall consider contain an identity element: all subrings contain the identity of the overring: all homomorphisms preserve the identity. Throughout R denotes a commutative ring and A an R-algebra, where by an R-algebra A we mean a ring A along with a ring homomor- phism e of R into the center of A. This homomorphism induces a natural R-module structure on A by defining r-a = 9(r)-a for r E R and a E A. If e is.a monomorphism, A is called a faithful R-algebra. An R-algebra is called projective or finitelyggenerated if it is projective or finitely generated as an R—module. All R—algebras we shall consider are finitely generated. If A is a commutative ring, A is called a commutative R-algebra. Example 1.1. _Let f(x) be a monic polynomial with coefficients in R. Then A = <§ : > is a finitely generated. faithful, free (as an R-module), commutative R-algebra. If the degree of f(x) is n. then {l,x,...,xn-l] is a free basis of A over R. The notation rad B 'with be used to denote the Jacobson radical of a ring B. Throughout N = rad A. The following proposition relates the maximal ideals of R and the radical of A. Proposition 1.2. [11, Lemma 1.1, p.78] Let A be a finitely generated R-algebra and let rlmA denote the intersection of the ideals mA as m runs over all maximal ideals of R. (i) rad(R) -A E N. (ii) There exists a positive integer n such that N“ g. n mA. (iii) If A is projective, rad(R)-A = n mA. The following proposition due to Azumaya relates the radical of A to the radical of a subalgebra. Proposition 1.3. [2, Corollary, p.126] Let A be a finitely generated algebra over R and B be an R-subalgebra of A. Then NntradB. Our interest is in the relationship between properties of R and the structure of finitely generated R—algebras. Our technique is to take data from A/N and "pull it back" to A. Our work here is concerned with pulling back two types of structure-idempotents and separability - and the relationship between them. A finite dimensional algebra A over a field F is called a separable F-algebra if and only if A is isomorphic to a direct sum of full matrix rings over division rings having centers which are separable field extensions of F. In 1908 wedderburn proved (in the case F has characteristic 0) that if A/N is separable over F then this separability can be "pulled badk" to a subalgebra of A: wedderburn Principal Theorem [24]. If A is a finite dimensional F—algebra such that A/N is F-separable then there exists a separable F-subalgebra S of A such that A = S + N. The wedderburn Principal Theorem decomposes A into two parts: a separable algebra which is a direct sum of matrix algebras and the radical which is nilpotent. Azumaya (1951) obtained a generalization of wedderburn's Theorem for a class of rings called Hensel local rings (which will be defined in the following section). Azumaya's Theorem [2]. If A is a finitely generated algebra over a Hensel local ring R 'with maximal ideal m and if A/N is separable over R/m, then there exists an R-subalgebra S of A such that A = S + N and S/mS is separable over R/m. Auslander and Goldman (1960) [l] generalized the notion of a separable algebra to algebras over an arbitrary commutative ring R. For finitely generated R-algebras their definition is equivalent to the following: Definition. A finitely generated R—algebra A is a separable R-algebra if A/mA is a separable R/m-algebra for every maximal ideal m of R. Example 1.4. Let A = Rnxn be the ring of nxn matrices over a commutative ring R. A/mA e:(R/m)nxn. and thus A is separable. Ingraham [11] has considered rings all of Whose finitely generated algebras satisfy the analogue of the wedderburn Principal Theorem under Auslander and Goldman's generalized notion of separability: Definition. A ring R is called an inertial coefficient ring if for every finitely generated R-algebra A such that A/N is a separable R-algebra, there exists a separable R- subalgebra S of A such that A = S + N. S is called an inertial subalgebra of A. Known inertial coefficient rings include the following: a field, a Hensel local ring, a von Neumann regular ring [4, Theorem 1, p.370] (R is von Neumann regular if and only if for every r 6 R there exists an s 6 R such that r28 = r), and a Noetherian Hilbert ring [13, Corollary 2, p.553] (R is a Hilbert ring if and only if every prime ideal is the intersection of the maximal ideals containing it). In Chapter III we shall give some new examples of inertial coefficient rings. The following theorem of Ingraham motivated the work in Chapter II. Theorem 1.5. [13, Theorem 2, p.554] If for any finitely generated, commutative R—algebra A each idempotent in A/N is the image of an idempotent in A, then if B is a finitely generated, commutative R-algebra such that B/rad B is R- separable then B contains an R-inertial subalgebra. we shall next describe and relate two numerical tools, rankR(A) and uh(A), which will be used in Chapter II. For any prime ideal p of R, let Rp denote the localization of R at p. R.p is a local ring (a commutative ring with a unique maximal ideal). Let M: be a finitely generated, pro- jective R-module. Since projective modules over a local ring are free, the RP-module M GR RP 2:Mb is free of finite rank rankp(M). If there is a fixed integer n such that rankP(M) = n for all prime ideals p of R, ‘we say the rank 9: M i§ defined and equals n and we write rankR(M) = n. For reasons which will become clear in section 3, in much of our work we can assume the ring R has no idempotents but 0 and 1. (Such a ring is called a connected ring). If R is a connected ring and M is a finitely generated, projective R-module, then rankR(M) is defined [6, Theorem 4.12, p.32]. Let A be a finitely generated, faithful, commutative R-algebra and let m be a maximal ideal of R. Then m has only finitely many maximal ideals M of A such that M n R = m, so we can define Hm(A) = E. [A/M:R/m], where [A/M:R/m] is the dimension of the field extension A/M over the field R/m. um(A) is easily evaluated for algebras A of the type in Example 1 . 1 : _ R x . . Lemma 1.6. Let A — (f(x > where f(x) is a monic polynomial contained in R[x]. Let f(x) denote the polynomial in R/m[x] obtained by reducing the coefficients of f(x) n e. modulo m. Suppose f(x) = [I (pi(x)) 1 is the factorization i=1 of f(x) into positive integer powers of monic polynomials pi(x) which are irreducible in R/m[x]. Then um(A) 1: n = 23 degree (pi(x)). Proof: Any maximal ideal M of A such that M n R = m contains the ideal mA. Since $7- 2 319135.]. . then by the n Chinese Remainder Theorem £5 a- e Z R/m[:] . Each i—l (mi x 1) Bi - R/ml':] is a local ring with unique maximal ideal <53")? 1> M. of the form ’m'iT = Bl one Bi_1 (9 Mi’ 9 Bi+1 s...e an. a A mA 9‘ R m x : = . . A/Mi .. Kjfi _ _LLJ-(p. x > , and [A/Mi R/m] degree (_yp1(x ) l Ingraham and W.C. Brown computed um(S) for inertial subalgebras of certain algebras. Proposition 1.7. [5, see proof of Lemma 1, p.11] Let A be a finitely generated, projective, faithful, commutative R-algebra. If A has an inertial subalgebra S, then for each maximal ideal m of R, um(A) = um(s) = rankm[s]. The following prOperties will be used in Chapters II . and III and are listed here for reference: (I) Let :1 be an ideal of R. An R/SI-algebra A has an R—algebra structure induced by the homomorphism of R onto R/fl. .A is a separable Realgebra if and only if it is a separable R/u-algebra [6, p.45]. (II) If S is a separable R-algebra and I is a two—sided ideal of S, then 8/1 is a separable R—algebra [6, Preposition 1.11, p.46]. (III) If S1 and S are separable R-algebras then 2 S1 6’82 is a separable Realgebra [6, Proposition 1.13, p.47]. (IV) Projective Lifting Property. If S is a separable R-algebra, then every S-module which is R—projective is S-projective [6, PrOposition 2.3, p.48]. (v) (VI) (VII) (VIII) (Ix) (x) Inertial subalgebras of finitely generated algebras are finitely generated [21, Theorem 5, p.5]. Inertial subalgebras of finitely generated, commutative, projective R-algebras are pro- jective [11, Preposition 2.8, p.80]. Let R be an inertial coefficient ring. If R’ is a finitely generated, commutative, R-algebra, where R’/rad R’ is a separable R-algebra, then R’ is an inertial coefficient ring [11, see proof of Proposition 3.3, p.85]. If R is an inertial coefficient ring and u is an ideal of R, then R/fl is an inertial coefficient ring [11, Corollary 3.4, p.86]. If rankR(M and rankR(M2) are defined for l) finitely generated, projective Remodules M1 and M2, then rankR(M1 @lM2) is defined and equals rankR(M1) + rankR(M2). If A is a commutative, finitely generated, projective R-algebra and M is a finitely generated, projective A-module such that rankA(M) and rankR(A) are defined, then rankR(M) is defined and equals rankR(A)-rankA(M) [6, Exercise 2. p.35]. §2. Hensel Rings and the Idempgtent Lifting Property Let f(x) be a polynomial in R[x] and fTiT. be the polynomial in R/u[x], where u is an ideal of R. obtained by reducing the coefficients of f(x) modulo u. Suppose foT.= go(x)-ho(x) in R/fl[x]. A problem histori- cally of interest to algebraists is the existence of polynomials g(x),h(x) 6 R[x] such that f(x) = g(x)-h(x) and §T§T'= go(x), 3757 = h0(x). In 1902 Hensel proved a lemma [10] stating that if R is the p-adic numbers and if u is its unique maximal ideal then certain factorizations, ITET = ‘ = go(x)-ho(x), can be "lifted" to corresponding factorizations of f(x) in R[x] In 1951 Azumaya [2] called a local ring satisfying Hensel's lemma a Hensel ring — i.e. a local ring R ‘with maximal ideal m is a Hensel ring if for every monic polynomial f(x) 6 R[x] such that ETET = go(x)h0(x) where go(x),ho(x) are monic, relatively prime polynomials in R/m[x], there exist monic polynomials g(x),h(x) in R[x] such that f(x) = g(x)-h(x), §Y§T = go(x), and h7§7'= ho(x). Azumaya considered algebras over Hensel local rings and showed that the Hensel pr0perty of the ring was entirely reflected in the algebras over the ring: that is a local ring is Hensel if and only if idemp potents can be lifted from A/I to A in all finitely generated R-algebras A, for any two-sided ideal I of A. This allows one to lift families of pairwise orthogonal idempotents and matrix ring decompositions. Azumaya used these 10 results to determine the structure of finitely generated algebras over a Hensel local ring. In 1963 Lafon [15] extended the definition of a Hensel ring to nonlocal rings. Greco (1968) [7,8,9] used Lafon's definition to generalize certain results Azumaya had Obtained for Hensel local rings. Definition. A ring R is a Hensel ring if for all monic polynomials f(x) 6 R[x] and every decomposition f = gO-hO in EaTg-R [x] with go and ho monic and (go) + (ho) = Egg—R [x], there exists a pair of monic polynomials g,h E R[x] such that f = g-h and g = go, 5 a ho. (It can be proved that g and h are unique [15, Proposition 1, p.80]). Any ring R having rad(R) = 0 is trivially a Hensel ring. we now give another example. Example 1.8. A commutative ring which is Hausdorff and complete with respect to a linear t0pology in which rad R is a closed ideal with every element tOpologically nilpotent is a Hensel ring [3, Theorem 1, p.215-6]: e.g. rings with nil- potent radical, the p-adic numbers, and the formal power series rings over a Noetherian ring. Definition. Let I be a two-sided ideal of a ring A. we call (A,I) and L.I. pair (lifting idempotent pair) if every idempotent of A/I is of the form. 5 where e is an idempotent of A. we say A has the idempotent liftinggproperty if (A,N) 11 is an L.I. pair. (In a commutative ring A, ‘if an idempotent in A/N can be lifted to A there is exactly one lift in A [7, Lemma 1.2, p.46].) Example 1.9. Jacobson proved that if N is a nil ideal of A. A has the idempotent lifting pr0perty [14, Proposition 4, p.54]. Lemma 1.10.. Let u be a two-sided ideal of A, (i) If (A,N) and (A/sl,N/21) are L.I. pairs. then A has the idempotent lifting pr0perty. (ii) If A is a commutative ring having the idempotent lifting property, then (A,fl) and (A/fl,N/fl) are L.I. pairs. Proof: (i) is an immediate consequence of the definition of L.I. pairs. See [7, Corollary 1.3, p.46] for a proof of (ii). The following theorem relates Hensel rings and L.I. pairs. Theorem 1.11. [7, Theorem 4.1, p.55 and 8, Theorem 2.2, p.51] The following are equivalent prOperties of a commutative ring R: (i) R is a Hensel ring. (ii) For every finitely generated Realgebra A, (A,rad(R)-A) is an L.I. pair. (iii) For every finitely generated, commutative, free R-algebra A, (A,rad(R)-A) is an L.I. pair. 12 Corollary 1.12. R is a Hensel ring if and only if every finitely generated, projective R-algebra A has the idempotent lifting property. Proof: If R is a Hensel ring and A is a finitely generated, projective R-algebra then (A,rad(R)-A) is an L.I. pair by Theorem 1.11 (ii). By Pr0position 1.2 (ii) and (iii) E53%%TTX' is a nilpotent ideal of E35€§TTX’ and by Example 1.9, (Fa—$75.. , ENE?) 18 an L.I. pair. Thus by Lemma 1.10 (i), (A,N) is an L.I. pair. Conversely, we shall show R is Hensel , by proving that (iii) of Theorem 1.11 holds. Let A be a finitely generated, commutative, free R-algebra. By the hypothesis (A,N) is an L.I. pair. Since A is commutative, (A,rad(R)-A) is an L.I. pair by Lemma 1.10 (ii). All finitely generated algebras over a Hensel ring need not have the idempotent lifting prOperty, e.g. let R = Z(p)[x]. where 2(p) is the subring of the rational numbers having denominators relatively prime to a fixed prime integer p. R has rad(R) = 0 and therefore is a Hensel ring. Let Z [x] _ A = ——éEl—-—-. Then x is an idempotent in A/N, but A has no idempotents other than 0 and l [18, Theorem 43.14, p.184]. If A is a finitely generated algebra over a Hensel local ring, a von Neumann regular ring, or a Noetherian Hilbert ring, then idempotents can be lifted from A/N to A [13, see proof of Corollary 2, p.553]. It is of interest to find necessary and sufficient conditions on R such that every l3 finitely generated R-algebra has the idempotent lifting property. we have seen [Corollary 1.12] that such a ring must necessarily be a Hensel ring, but that R being a Hensel ring is not sufficient to guarantee that every finitely generated algebra has the idempotent lifting preperty: in Chapter II we shall see that R. being an inertial coefficient ring is a sufficient condition. we conclude this section with a new example of a ring all of whose finitely generated algebras have the idempotent lifting prOperty. Example 1.13. Let U be the ring of formal series in one indeterminate over the rational numbers and S be the subring of formal power series over the integers. Since U and S are NOetherian rings, their radical topologies satisfy the topological criteria of Example 1.8, and thus U and S are Hensel rings. Let T and R be the subrings of U and S, respectively, consisting of power series which are con- vergent at zero (g(x) e U is called convergent at zero if there exists an Open interval N of the real line such that 0 6 N and g(r) is an absolutely convergent series for every r e N [25, p.142]). It can be shown that the radicals of U,S,T, and R are each generated by x, and it follows that the canonical homomorphisms U 4 U/rad U, S 4 S/rad S, T 4 T/rad T, and R a R/rad R all can be described as evalua- tion at x = o, T is a Hensel ring [18, Theorem 45.5, p.193]. 14 we shall show that R is a Hensel ring. Let f(y) E R[y] be a monic polynomial such that 'fF;T== WW where 90(Y).hO(Y) €R[y] are such that W, W are monic polynomials in (R/rad R)[y] with (W) + W> = (R/rad R)[y1- /U[y]\ Since R[y] s S[y] S[y] T[y] \Rlyl/ and T[y], and since S and T are both Hensel rings, it is easily seen that the factorization of nyT' in (R/radIU[y] can be lifted to a corresponding factorization of f(y) in both S[y] and T[y]. Since S[y] and T[y] are both con- tained in U[y] and since U is a Hensel ring and thus the lift of a factorization is unique, we therefore have a factor- ization f(y) = g(Y)-h(y) in S[y] n T[y] = R[y] where W = W and W = W. Thus R is a Hensel ring and by Theorem 1.11 (ii) for any finitely generated R-algebra A, (A,rad(R)-A) is an L.I. pair. Since R/rad R is isomorphic to the integers and the integers form a NOetherian Hilbert ring, (A/rad(R)-A,N/rad(R)-A) is an L.I. pair. Hence by Lemma 1.10 (i) for any finitely generated R-algebra A, (A,N) is an L.I. pair. §3. The Decomposition Space The Pierce decomposition space X(R) of a commutative ring R ‘was first described by Pierce [20] using sheaf-theoretic methods. The description of X(R) given here avoids the use 15 of sheaf-theoretic language and is due to Villamayer and Zelinsky [23] (they call X(R) the Boolean spectrum of R). Magid's bodk [17] to which we will frequently refer is an easily accessible, complete source on this approach to X(R). The decomposition space X(R) is a tool useful in proving that certain results which are known to hold for connected, commutative rings also hold for arbitrary commutative rings. The technique is to study certain connected homomorphic images of R and "patch together" these results to Obtain ‘. the result for R. Let Spec(R) denote the set of prime ideals of R endowed with the Zariski tapology (a basis of closed sets of Spec(R) is {V(I)} where I ranges over all ideals of R and where V(I) = [p 6 Spec(R):p 2 1]). We define X(R) to be the quotient space of Spec(R) obtained by identifying connected components of Spec(R). It can be shown that X(R) is a totally disconnected, compact, Hausdorff spaoelor equivalently a pgofinite space (an inverse limit of finite discrete spaces) [17, Corollary II.4, p.26]. The space X(R) has the following useful tOpological property. Preposition 1.14. (The Partition Property) [17, Lemma 1.7, p.3] Any open cover of X(R) has a refinement which is a partition (a finite family of disjoint open subsets of X(R) which covers X(R)). 16 It can be shown that two prime ideals of R belong to the same connected component of Spec(R) if and only if they contain the same idempotents [17, Proposition II.3, p.26]. Thus if R has no idempotents but 0 or 1, Spec(R) is connected (justifying our calling such a ring a connected ring). Let e be an idempotent of R and let N(e) = [x 6 X(R) :x _c_ V(R(1-e))}. The sets N(e) have the following useful prOperties: Proposition 1.15. [17, Proposition II.12, p.30] (1) N(0) {I and N(l) =X(R)- (ii) N(e) ONCE) = N(ef). (iii) N(e) N(f) if and only if e = f. (iv) The sets {N(e)} form a basis of Open, closed sets for the tOpology on X(R). Furthermore, any Open, closed subset of X(R) is of the form N(e) for some idempotent e of R. Let I(x) be the ideal of R generated by the set of idempotents in any prime ideal contained in a point x E X(R) and define Rx = R/I(x). Rx is a connected ring [17. Corollary 11.21, p.34]. When computing X(R) and R.x for a particular ring R it is usually easier to view X(R) in an equivalent formu- lation as the collection of maximal Boolean ideals of R [17, pp.27-28]. A set of idempotents 1c of R is called a 17 maximal Boolean ideal if (i) For every idempotent e of R either e 6 x or l-e e x, but not both: and, (ii) If e and f are idempotents of R then ef E x if and only if e E x or f E x. It can be shown that Rx = R/I(x) where I(x) is the ideal of R generated by the elements of x [17, Proposition II.9, p.28]. Let M. be an R-module and let m 6 M. Throughout let m.x = m + I(x)M. denote the image of m in Mk = M GR Rx 2.M/I(x)M. Proposition 1.16. [17, Proposition II.16, p.32] Let a and b belong to the R-module M, let x G X(R), and suppose ax = bx“ Then there exists a neighborhood N(e) of x in X(R) for some idempotent e e R such that a = b Y Y In My for all y E N(e), ex = 1x, and as = be. Proposition 1.17. [17, Proposition II.17, p.32] Let a and b be elements of the R-module M such that ax = bx for all x 6 X(R). Then a = b. PrOpositions 1.16 and 1.17 along with the Partition Pr0perty will be the principal tools used to "patch together" results from the connected rings Rx to Obtain results for R. CHAPTER II INERTIAL COEFFICIENT RINGS AND THE IDEMPOTENT LIFTING PROPERTY E.C. Ingraham has conjectured that a ring R is an inertial coefficient ring if idempotents can be lifted from A/N to A in all finitely generated R-algebras A. Both I he and Azumaya have used the technique of lifting idempotents to produce inertial subalgebras. The main result of this chapter is that idempotents can be lifted from A/N to A in all finitely generated algebras A over an inertial co- efficient ring, and thus the converse of Ingraham's conjecture is true. The result is proved in three steps. First it is shown that if R is a connected inertial coefficient ring, idempotents can be lifted from R/rad R to R. Next the decomposition space is used to show idempotents can be lifted from R/rad R to R in any inertial coefficient ring R. Finally we show that idempotents can be lifted from A/N’ to A in any finitely generated R—algebra A. The first step‘will be proved by contradiction; we will assume R has a nonliftable idempotent and produce a finitely generated R-algebra A such that g is R-separable but A contains no inertial subalgebra. 18 19 Lemma 2.1. Let f(x) be a monic polynomial in R[x] such that fTRT = xr(x-1)s for r,s positive integers, where f(x)-6 (R/rad R)[x]. Suppose there do not exist monic polynomials go(x),h0(x) in R[x] such that f(x) = go(x)-ho(x) with 337;) = xr and h37§7 = (x-l)s. Then 'EITET = xr+1(x-l)s, and there do not exist monic polynomials g(x),h(x) in R[x] ‘with xf(x) = g(x)h(x), g(x) = xr+l, and h(x) = (x-l)s. r+s r+s-l i Proof: Let f(x) = x + Z) aix with ai e R. - i=0 r+s-1 . Then xf(x) = xr+s+l + Z) aix1+1. Suppose xf(x) = g(x)h(x) i=0 with g(x) = xr+1 and h(x) = (x-l)s. Then g(x) = x1"”. + E} nkxk with 6 rad R and h(x) = x8 + s£31H8)(-l)s-k+ ’]xk k=0 nk k=0 k nk O with nk E rad R. Equating constant terms of xf(x) = g(x)h(x) gives no((-1)S+n6) = 0. (-1)8 + n6 is a unit of R, and r thus n = 0. Therefore xf(x) = x(xr + Z) nkxk'1)-h(x). Since 0 k=l x is not a zero divisor in R[x] ‘we have f(x) = r (Xr + Z} nka-1)'h(x). Thus k=1 r 90(x) = xr + k2: nkik’l and ho(x) = h(x) contradict the hypotheses. Theorem 2.2. Let R be a connected ring. If R/rad R has an idempotent not equal to 5 or I then R is not an inertial coefficient ring. 20 Proof: By [14, Proposition 4, p.54] there exists a p 6 rad R such that x2 - x + p E R[x] has no root in rad R. Then x2 - x + p a x(x-l) modulo rad R. If there exist monic polynomials g(x),h(x) contained in R[x] such that x2 — x + p = g(x)-h(x) 'with g(x) = x and h(x) = x - 1 then g(x) = x + n for some n e rad R and -n is a root of x2 - x + p in Rad R. Let f(x) = x3 - x2 + px. Then f(x) = x2(x-l), but by Lemma 2.1 there do not exist monic polynomials g(x) and h(x) contained in R[x] such that f(x) = g(x)-h(x) and g(x) = x2, h(x) = x - 1. Let A be the finitely generated, faithful, free, commutative R-algebra A = —%B{';]>——' Then A/ (rad (R) -A) = (x -x +px> W; since and are comaximal ideals, the / (11! Chinese Remainder Theorem gives A/ (rad (R) 'A) a Q: r: lxl O (x > W. Thus 5 a: W e.- R/rad R e R/rad R. There- N _'N rad R fore by Chapter 1, properties II and III, p.7, A/N is R- separable. Furthermore, fer any maximal ideal m of R, Hm(A) = 2 [Lemma 1.6]. we will show that the assumption that A has an inertial subalgebra 8 leads to a contradiction. By Chapter 1, property VI, p.8, if such an S exists, it must be a pro- jective R-module. 21 Case 1. Assume S is connected. Since R is connected, rankR(S) is defined and by PrOposition 1.7 um(A) = rankR(S). By the "projective lifting property" (Chapter 1, property IV, p.7), A a projective R-module and S a separable R-algebra imply that A is a projective S- module. Since S is connected, rankS(A) is defined. But then by Chapter I, property X, p.8, rank is multiplicative and = rankR(A) = rankR(S)-rankS(A) = 2-rankS(A). Thus 2 divides 3 3, a contradiction. To do the case when S is not connected, we need the following lemmas: Lemma 2.3. Let R be a connected ring and A = R[x] for p e rad R. If A has an idempotent e, 3 2 (x -x +px> e # 0,1, then e or 1 - e is of the form alx + a2x2 where a1 6 rad R and 33' is an idempotent in R/rad R. _ ' 2 Proof. Let e — a0 + alx + a2x , ai E R, represent e in the free Rebasis for A, {l,x,x2}. 2 2 0 — e - e — (aO-ao) + (2aoa1 a1)x + (a1 a2+2aoa2)x 3 2 4 3 4 Now applying the relations, x = x2 - px and x = (l—p)x2 - px, we get 0 = (a2-a ) + (2a a -a -2a a p—a2 )x o o o 1 1 1 2 2p 2 2 2 + (al-a2+2a0a2+2a1a2+a2(l-p))x . 22 [l,x,x2] being a free basis of A over R implies the following relations: 2 (-1-) a0 = a0 (-2-) Zaoa1 - a1 - 2a1a2p - agp = 0 (-3-) ai - a2 + 2aoa2 + Zala2 + a§(l-p) = 0. Since R is connected, equation (-l-) gives a0 = 0 or a0 = 1. The conclusion follows by examining equations (-2-) and (-3-) when a0 = 0 and when a0 = 1. Lemma 2.4. Let R be a connected ring. If A = 3R[:2cl (x -x +px> S such that S is not a connected ring, then S a.- Re e R(1-e) , where p e rad R, has an inertial subalgebra (as rings) for some idempotent e e.A. Proof: Since R is connected, um(A) is well defined, and um(A) = 2 for any maximal ideal m of It. Suppose A has an inertial subalgebra S. Then by Proposition 1.7, 2 = um(A) = rankR(S). If S is not connected then S = Sele S(l-e) for some idempotent e €,A, e # 0,1. If rankR(Se) = 0 then (Se)P = 0 for all prime ideals p of R and so Se = 0 and e = 0. Similarly rankR(S(l-e)) # 0. Since rank of direct sums is additive (Chapter 1, property IX, p13), rankR(Se) = rankR(S(l—e)) = 1. Since Se and S(l-e) are projective modules over a connected ring, they are faithful Remodules. 23 Se is thus a finitely generated, projective, faithful Re- module. By [6, Corollary 1.11, p.8 and Corollary 4.2, p.56] Re is an Re-direct summand of Se. Therefore there exists a finitely generated Re—module U such that Se 2 Re 9 U as Re-modules. Localizing at each prime ideal p of R aeRe gives UP = 0, and thus U = 0. Thus Se esRe as rings and similarly S(l-e) a R(l-e), and therefore S 9-. Re a R(l—e) (as rings). Lemma 2.5. Let A = __§l§%____. for p e rad R. Let (x -x +px> S be a subring of A, S = Re e>R(l-e) for e an idempotent of A of the form e = alx + a x2 where a1 6 rad R and a 2 is an idempotent of R/rad R. Then A = S + N implies zen-.21.. A 2 and a is a unit of R. rad(R)-A 2 a (R(radRzlxl A = S + N implies there exist r1,r2 e R and n 6 N such Proof: N = and 2|? that x = rle + r2(l-e) + n = r2 + (rl-r2)e + n. Since - _ - -2 . - _ - - _- - -2 - . e — a2x in A/(rad(R) A), x - r2 + (r1 r2)a2x + n In A/(rad(R)-A). Therefore in A/N, R = I2 + (I1-?2)§2§. Thus r2 = 5 and x = rlazx. Therefore 1 = rlaz. Since a2 18 both a unit and an idempotent of A/N, :2 = I. But then by -2 [7, Lemma 1.2, p.46], 52 = I in .A/(rad(R)-A). Thus 5 = x and a2 is a unit of R. 24 The following lemma is a generalization by Greco of a result for local rings due to Nakayama [2, Lemma 3, p.134]. Lemma 2.6. [7, see proof of Theorem 3.1, p.54] Let f(x) 6 R[x] be a monic polynomial. Suppose A = Efxx > = 91 O n for ideals 91,515 of A. Suppose further that A/rad (R) -A a: (go (x)> @ (ho (x)> for monic, c0prime polynomials 90(X).ho(x) 6 (R/radR)[x] such that QI/(rad R-QI) = (go(x)> and B/(rad R-B) = (ho (x)>. Then there exist monic, coprime polynomials g(x),h(x) 6 R[x] such that f(x) = g(x)-h(x). and g(x) =go(x), h(x) =ho(x) in FEET—R[XJ' We can now complete the proof of Theorem 2.2: Case 2. Assume S is not connected. By Lemmas 2.3, 2.4, and 2.5 S must be of the form S = Re 0 R(l-e), for e an idempotent in A of the form e = alx + a2x2 where a1 6 rad R, 52 = l in R/rad R. Thus A = Ae @ A(1-e) = e = e . Furthermore x2 + aglalx a x2 modulo rad(R) -A and x2 + aglalx - age-1 2 x2 - 1 modulo rad(R) -A. Finally = in A/(rad (R) ~A), since E and (x—l) = (1-x)(x2-l) in A/(rad(R).A) implies 5. Thus by Lemma 2.6, there exist monic polynomials g(x),h(x) in R[x] such that x3 - x2 + px = g(x) 'h(x) with 3T2?)- = x2 and h(x) = x - 1 in (R/radR)[x]. This contradicts the choice of p. 25 Theorem 2.2 states that if R is a connected, inertial coefficient ring then R/rad R is connected and so (R,rad R) trivially is an L.I. pair. we next use the decomposition space to extend Theorem 2.2 to an arbitrary inertial coefficient ring. Proposition 2.7. Let A be a finitely generated R- algebra. (A,N) is an L.I. pair if (Ax,N*) is an L.I. pair for all x e X(R). Proof: Let u E A be such that u2 - u E N. We must find an idempotent e 6 A such that u - e €,N. Since Rx is a flat R-module [17, Proposition II.18, p.33] without ambiguity we can let Nk denote the image of N under the canonical homomorphism. A 4 A/I(x)-A = Ax. Now ux = u + I(x)-A is an element of A.x such that fix is an idempotent element of Ax/Nk. Since (Ax,N*) is an L.I. pair and since an idempotent in Ax can be lifted to an idempotent in A [17, PrOposition II.20, p.34], there exists an idempotent f(x) €.A such that ux = [f(x)]x + [n(x)]x for some n(x) e_N. By Preposition 1.16 for each x e.X(R) there exists an idem, potent e(x) e R such that u-e(x) = f(x)-e(x) + n(x)-e(x) and uy = [f(x)]y + [n(x)]y for all y e N(e(x)). (N(e(x))}x€x(R) Is an Open cover of X(R) and thus by the partition property, PrOposition 1.14, there exists a finite refinement of disjoint Open and closed sets 26 U1,...,Um. By Pr0position 1.15 (iv), Ui = N(ei) for some idempotent ei €,R. Since {N(ei)}T;1 covers X(R) and since N(ei) n N(ej) = ¢, by Proposition 1.15 (i), (ii), and m (iii) 1 = Z) ei and ei-ej = 0 for each i # j. Further- i=1 more N(ei) EN(e(xi)) for some xi EX, and thus N(ei) n N(e(xi)) = N(ei) implies ei-e(xi) = ei: hence u-ei = fi°ei + ni-ei where fi = f(xi) and n1 = n(xi). Let m e = Z)(fi-ei). Since the ei are pairwise orthogonal and i=1 each fi is an idempotent, e is an idempotent. New m m m u = .Z uei = .23 (fiei+niei) = e + .23 niei, and therefore i=1 i=1 i=1 u - e E N. Corollary 2.8. If R is an inertial coefficient ring, (R,rad R) is an L.I. pair. gggog: Since a homomorphic image of an inertial coefficient ring is an inertial coefficient ring (Chapter I, property VIII, p.8) Rx = R/I(x) is a connected inertial coefficient ring for every x e X(R). By Theorem 2.2 (Rx,rad(Rx)) is an L.I. pair for every x E X(R). Since (rad R)x _C._'. rad(Rx), by Lemma 1.10 (ii) (Rx, (rad R)x) is an L.I. pair for every x e X(R): hence by Proposition 2.7 (R,rad R) is an L.I. pair. we are now able to prove the general case: 27 Theorem 2.9. Let R be an inertial coefficient ring and A be a finitely generated R-algebra. Then (A,N) is an L.I. pair. Proof: R/annihR A, being a homomorphic image of R, is an inertial coefficient ring and A is a faithful R/annihRA- algebra. Thus replacing R/annihRA by R 'we may assume A is a faithful R-algebra. Let c €.A be such that c2 - c 6 N. we must find an idempotent e €.A such that c - e €.N. Let B = R[c] denote the R-subalgebra of A generated by c. B is a finitely generated, commutative R-algebra. By Pr0position 1.3 N n B _c; rad B. Let (R/radR)[5], where E = c + N, denote the R/rad Resubalgebra of A/N’ generated by 5. Define a n . n . homomorphism ¢:B +(R/radR)[E] by H 2 ricl) = Z Ei(<-:)1. i=0 i=0 I) is surjective and ker W E N n B c_: rad B. Now B/ker w e- (R/radR)[E] is a homomorphic image of (R/gadlfil’Lx] a R/rad R e (x -x> R/rad R and therefore is a separable R-algebra. Since B/rad B a rgékgikzrw , B/rad B is a separable R-algebra. By Chapter I, prOperty VII, p.8, B is an inertial coefficient ring, and thus by Corollary 2.8 (B,rad B) is an L.I. pair. By Lemma 1.10 (ii), (B,N’rlB) is an L.I. pair. Then c2 - c e N n B implies that there exists an idempotent e 6 B such that c - e €,N n B. But then e e.A and c - e 6 N. 28 Notice that in the preceding proof we showed that if R is an inertial coefficient ring and if E €.A/N is idem- potent, then there exists an idempotent e e B = R[c] lifting 5. Thus e is a polynomial over R in c. Furthermore, the following result is a consequence of Corollary 1.12 and the proof of Theorem 2.9. Corollary 2.10. The following are equivalent prOperties of a commutative ring R: (i) All finitely generated R-algebras have the idempotent lifting property. (ii) All finitely generated, commutative R— algebras are Hensel rings. Any algebra A having the idempotent lifting property must satisfy the two prOperties below. A consequence of Theorem 2.9 is that these results hold for any finitely gen- erated algebra over an inertial coefficient ring. 1) Any countable sequence of pairwise orthogonal idempotents in A/N. can be lifted to a sequence of pairwise orthogonal idempotents in A [16, Proposition 2, p.73]. 2) If .A/N a ann' the full nxn matrix ring over a finitely generated R/rad R-algebra B, then there exists a finitely generated R- C: 2 B and A esc . rad C nxn the full nxn matrix ring over C [14, see algebra C such that proof of Theorem 1, p.55]. 29 Corollaryo2.ll. If R is an inertial coefficient ring then R/fl is a Hensel ring for any ideal u of R. Proof: By Chapter I, PrOperty VIII, p.8, R/fl is an inertial coefficient ring and by Theorem 2.9 all finitely generated R/fl-algebras have the idempotent lifting property. Thus by Corollary 2.10 R/u is a Hensel ring. A consequence of Corollary 2.11 is that all homomorphic images of an inertial coefficient ring have the following properties of a Hensel ring R: R .- rad R - module P of rank n there exists a unique (up to 1) For every projective isomorphism) projective R-module P of P [7’ rank n such that P = m Corollary 5.4, p.58]. 2) The homomorphism $(R) 4 8(R/rad R) is an isomorphism, where 8(R) denotes the Brauer group of R [22]. Corollary 2.12. The following are equivalent properties of a commutative ring R: (i) For all finitely generated, commutative R-algebras A such that A/N is R-separable, there exists a separable Resubalgebra S of A such that A=S+N. (ii) All finitely generated, commutative R-algebras have the idempotent lifting property. 30 Proof: The fact that (ii) implies (i) fellows from Theorem 1.5. The proof of Theorem 2.9 shows that (i) implies (ii). If Ingraham's conjecture is true then it is un- necessary to restrict the algebras A in Corollary 2.12 to commutative R-algebras, for then the lifting of idempotents from A/N to A in all finitely generated R-algebras A is equivalent to the lifting of the separability of A/N to a separable R-subalgebra S of A in all finitely generated R—algebras. CHAPTER III NEW INERTIAL COEFFICIENT RINGS In this chapter we shall show that a ring R is an inertial coefficient ring if and only if for every x e_X(R) each connected ring Rx is an inertial coeffi- cient ring. we shall use this criterion to produce new inertial coefficient rings. WQC. Brown [4, Theorem 1, p.370] used the decomposition space X(R) to show von Neumann regular rings are inertial coefficient rings. Our result is a generalization of his result, for when R is a von Neumann regular ring each Rx is a field and therefore each Rx is an inertial coefficient ring. Our proof is closely patterned after Brown's proof. The technique is to show that an inertial subalgebra exists if and only if a particular finite collection of equations holds. To find an inertial subalgebra of an R-algebra A ‘we use the fact that certain equations hold in each Rx-algebra A.x and then using the topology on X(R) 'we patch together elements of A to obtain equations in A. which hold in every A.X and therefore hold in A. we shall use the following criterion for the separability of a finitely generated R—algebra S. 31 32 Lemma 3.1. Let S be an Realgebra generated as an R-module by 81.....8 Then S is separable if and only n. if there exists biibi e S i = l,...,m such that m, (1) .23 bibi = l. and i=1 m m (11) 1:31 sjbi sR bi = 1:31 bi 0R bisj holds m S 0 So for j = 1,...,n. R Proof: Proof follows from [6, Preposition 1.1 (iii), p.40]. The main result of this chapter is the following theorem. Theorem 3.2. R is an inertial coefficient ring if and only if R.x is an inertial coefficient ring for all x e X(R). Proof: If R is an inertial coefficient ring then Rx = R/I(x) is an inertial coefficient ring by Chapter I, property VIII, p.8. Conversely, suppose Rx is an inertial coefficient ring for all x E X(R), and let A be a finitely generated R—algebra such that A/N is R-separable. Then Ax = A/(I(x)-A) is a finitely generated Rx—algebra. Since Rx is a flat R- module [17, PrOposition II.18, p.33] without ambiguity we can let Nk denote the image of N under the canonical homomorphism A 4 A/(I(x)-A) = Ax. Furthermore, since 0 4 N 4 A 4 A/N 4 0 33 is an exact sequence of R—modules and Rx is R—flat, a (A/N)x is a separable Rx—algebra. Since each Rx '5' lexn’ an inertial coefficient ring and NK 5 rad (Ax), for every x 6 X(R) there exists a separable Rx-algebra S“ such that Sx + Nx = Ax [12, Corollary, p.3]. By Chapter 1, property V, p.8, each 3x is a finitely generated Rx-algebra: for each x 6 X(R) let sl(x), . . .,sn (x) (x) E A be such that (sl(x) )x' . . . , (sn (x) (x) )x are Rx—module Ax x W" generators of S . Let S = 2') R-s:.L (x) be the R-submodule i=1 of A generated by {si(x)}n(x). Then (Sx)x=SS‘. i=1 Let a1, ...,ap be R-module generators of A. Since for each x 6 X(R) ’3‘): = (Sx)x is a separable Rx-algebra such that Nx + (Sx)x = Ax, there exist elements rijk (x), ri (x),t£'j (x),rhj (x),r1;j (x) E R, elements z‘(x) 6 N, and elements bh(x),b1;(x) e 8" for i,j,k = l,...,n(x), 1. = l,...,p, and h = l,...,m(x) such that: n(x (—1-) (si(x))x(sj(x))x = kEi (r (x))x(sk(X))x ijk for i,j = l,...,n(x). n(x) (-2-) 1x = i2; (ri(X))x(si(x))x. n(X) (-3-) (31.)): = (z‘(x))x + 3.51 (tfl'j(x))x(sj(x))x for 1. = l,...,p. m(x) <-4-> 1x = I31 (bhmnxcngnx. 34 m(x) (-5-) 131 [(ssjlx))x(bh(X))x 8h): (b1;(X))x] m(x) =h231 [(bh(x))x cRx (1013M)x (8 (x)) x] j x x 0x . _ in (S )x th (S )x for j — 1, ...,n(x). n(x) n(x) (-6-) b (X) = 2'} rh jj(x)s (x) and bh(x) = 231 rh .(x)sj (x) 1‘ i=1 i=1 for h = l,...,m(x). Using Preposition 1.16 and.by intersecting the apprOpriate neighborhoods of x if necessary, for each x 6 X(R) there exists an idempotent e(x) 6 R such that [e(x)]x = 1x and equations 1-5 hold for all y E N(e(x)) when we replace the subscript x 'with the subscript y (for m(x) example, 131 ([sj(x)]y[bi(x)]y shy [bi(x)]y) = m(X) o . X SK 0 1:: ([bi(x)]y Ghy [bi(X)]y[8j(X)]y) holds in (S )y Chy( )Y)- The neighborhoods [N(e(x))} where each e(x) x€X(R) ' is chosen as above, form an Open cover of X(R). By PrOposition 1.15 and the partition property (PrOposition 1.14), there exist pairwise orthogonal idempotents {ei]i=l contained in R such that [N(eiH:=1 is a disjoint Open cover of X(R) refining [N(e(x))} Let xi denote a point of X(R) x€X(R)' such that N(ei)sN(e(xi)), i=1,...,t. Let n= maximum {n(xi )] and m = maximum [m(xi )}. For each k i=1, ...,t =l,...,t define sj(xk) = 0, rj(xk) = 0, rij£(xk) = 0 for all i,j,n, n(xk) < i gn, n(xk) < j gn, or n(xk) < ign, define 35 ti,j(xk) =0 for all i,j. 0313p and n(xk) (jgn, and define bj(xk) = b5(xk) = 0 for all j, m(xk) < j g;m. We patch together the data from the stalks Rx by defining the following: t 33 = E; sj(xk)ek j = 1. . .n t r3:111:31 rj(xk)ek 3:1" 'n t rijz=k=1rij£(xk)ek i,j,l.= l,...,n t 23 = kEa zj(xk)ek j = l,...,p _ E: t ( ) = l,...,p 1] k=1 ij xk ek j = 1,...,n 1: b3 = RE: bj(xk)ek j = l,...,m t bj=k§1bj(xk)ek j=l,...,m. Any x e X(R) is contained in N(ei) for some i and is not contained in N(ej) fer j # i. Thus for all prime ideals p e x, 1 - e1 6 p and therefore (ei)x = 1 . x For all p €,x and j # i, 1 - ej £ p and ej(l-ej) = 0 e,p imply that e3. 6 p, and therefore (ej)x = 0%. Thus when x ems.) (sj)x = (sj> over a von Neumann regular ring R, we first compute the stalks of these rings. Lemma 3.5. Let f(y) = i a.yj be an idempotent i=0 3 in R[[y]]. Then f(y) = a0 where a0 is an idempotent of R. 39 Proof: Equating constant terms and coefficients of O . O . y in the equation ( Z] ajyj)2 = Z) ajyJ gives the relations: j=o j=0 (—1-) a = a and (—2-) 2a a1 = a1. 2 O 0 By equation (-1-) a is an idempotent of R. we shall show 0 by induction that aj = O for j.2 1. .Multiplying equation (-2-) by a0 and uSing equation (-l-) gives 2aoa1 = aoa1 or aoa1 = 0: thus Zaoa1 = a1 = 0. Now suppose aj = 0 C 0 2 a j for j S i. As before (a0 + Z‘, a.yj) = a0 + 2'} a.y j=i+1 j j=i+1 3 implies that 2ai+1ao = ai+1 and so 2ai+1aO = ai+1ao giv1ng ai+1ao = o and hence ai+1 = 2ai+1aO = 0. Proposition 3.6. If R is a ring such that Rx[y1,...,ym] (respectively Rx[[y1,...,ym]], Rx<>) is an inertial coefficient ring for all x €.X(R) then R[y1,...,ym] (respectively R[[y1,...,ym]], R<>) is an inertial coefficient ring. m: Let S=R[y1,...,ym], T=R[[y1,...,ym]]. and U = R<>. Using Lemma 3.5 and induction on m one can show that any idempotent in S,T, or U is an idemr potent of R. Since the decomposition space of a commutative ring A is the collection of maximal Boolean ideals of A, it follows immediately that X(S) = X(T) = X(U) = X(R). Since for each x 6 X(S), I(x) is the ideal of S generated by the idempotents of x, and since idempotents of S are contained 40 in R, then I(x) = 1-8 where I the ideal of R generated by the idempotents of x, and 8x = S/I(x) = R/I[y1,...,ym] = Rx[y1,...,ym]. Similarly Tx = Rx[[y1,...,ym]] and U* = Rx<>. The result now follows from Theorem 3.2. Corollary 3.7. If R is a von Reumann regular ring, R[y1,...,ym], R[[yl,...,ym]], and R<> are inertial coefficient rings. Proof: For each x e X(R), Rx is a field and hence Rx[y1,...,ym] (13, Corollary 2, p.553), Rx[[y1""'ym]] (18, Theorem 30.3, p.104), and Rx<> (18, Theorem 45.5, p.193) are known to be inertial coefficient rings. n If S = e 23 R1 is a finite direct sum of rings Ri i=1 then for every x 6 X(S) S = (Ri)x for some xi 6 X(Ri). i x This fact suggests that Theorem 3.2 mdght.be of value in studying infinite direct sums (with 1 adjoined) and direct products. As our final example of a new inertial coefficient ring we shall see that infinite direct sums (with 1 adjoined) and a few very special direct products can be shown to be inertial coefficient rings using the decomposition space. Let R ‘be a "ring" perhaps‘without an identity. R can be embedded in a ring which has an identity element in the usual manner: Let R"Ir = R e Z where Z denotes the integers. Define addition in R* coordinatewise and multiplication by (a,i)-(b.j) = (ab+ib+ja,ij) for a,b 6 R and i,j E z. The 41 element (0,1) is the identity element of R*. The following lemma follows easily from the definition of multiplication . * in R. Lemma 3.8. All idempotents of R* are of the form (e,0) or (-e,1) where e is an idempotent of R. Let R = e E Ra be the direct sum of a collection of commutative rings {Ra}. we next compute the points of X(R*) and the stalks (R*)x by finding the maximal Boolean ideals of R* for this particular R. Any idempotent e of R has only finitely many nonzero coordinates each of which must be th an idempotent. Let ea denote the a coordinate of e. Lemma 3.9. All x e:x(R*) are of the form x0 = {(e,o):e an idempotent of R} or B a x [(e,0):e an idempotent of R and ea 6 xB e X(RaH U {(-e,1) :e an idempotent of R and ea ,é x6}. (R*) a z, the integers and (Rf)xfi 22(Ru) x0 x5' and x2 are a Proof: One can easily show that xo maximal Boolean ideals of R*. we shall show that any maxi- mal Boolean ideal x of R* is one of these ideals. Let x(o) = {eaI-(e,0) e x}. If for every 0., 1a e x(a). then x0 5 x implies x0 = x. 42 Suppose there exists an a such that lo. A x(a) . It is easily checked that x(a.) is now a maximal Boolean ideal of Rd and so x(o.) e X(Ra)' say x(a) = xB. If (e,0) e x then by definition of x , ea 6 xfi, whence (e,0) 6 x2. To show x 5 x2 we shall show that (-f,1) e x implies fa. ,é xB, or equivalently that 1a - fa 6 xB. Let e = (0, . . .,0,1a,0, .. .,0) e R. Since (e,0) - (-f,l) = (e-ef,0) E x and e - ef = (0,...,0,1a-fa,0,...,0), by the definition of x£3 we have 1a - fa 6 x5. Thus x2 5x and so xB=x. o. It is clear that R; = Z. One can check that the O ring homomorphism cp:R @ Z -0 (Ra)x given by cp(r,j) = T(ra+3a). where = 3'10 and -r:Ra 4 (Ra) is the J' a canonical homomorphism, induces an isomorphism between * R = (R 9 Z) and (R ) . x5 x2 0. XS Proposition 3.10. If {Ra}oa is a collection of * inertial coefficient rings, then (9 23 Rd) is an inertial C161 coefficient ring. Proof: The result follows from Theorem 3.2, Lemma 3.9, and the fact that the integers form an inertial coefficient ring. D Maple 3.11. (e Z Z/an)* where Z denotes the n==l integers is an inertial coefficient ring which has radical which is nil but not nilpotent. 43 PrOposition 3.12. Let n,R denote the direct product of (Ro}aeI' where eachI Rd is isomorphic to a fixed finite, connected ring R. n R is an inertial coefficient ring. I ‘grgggg If I and R are given the discrete tOpology, n R e:C(I,R). Let B(I) denote the Stone-gech compactificition of I. It is not hard to show that 9(1) is totally disconnected, and hence a profinite space. By Pro- position 3.3 C(B(I),R) is an inertial coefficient ring, since any finite ring is an inertial coefficient ring. The natural ring homomorphism ¢:C(B(I),R) 4 C(I,R) given by restriction is surjective since R is compact. Then C(I,R), being a homomorphic image of an inertial coefficient ring, is itself an inertial coefficient ring. Corollary 3.13. Let n Ra denote the direct product I of {RaIaEI' where each Rd is isomorphic to a finite ring of cardinality less than some fixed integer n. n Ra is an I inertial coefficient ring. Proof: Since there are only a finite number of distinct isomorphism classes of connected rings of cardinality less than .M n and since [I Ra a a Z (11 R1), a finite direct sum of rings I i=1 Ii n Ri' where each n Ri is a direct product of a collection of I. I. i 1 rings each isomorphic to a fixed finite, connected ring Ri' then H Rd is an inertial coefficient ring. I 44 we have thus far been unable to use the decomposition space to determine whether more general direct products, e.g. IIZ. are inertial coefficient rings. The main result of this chapter, that R is an inertial coefficient ring if and only if each R.x is an inertial coefficient ring, is parallel to a result which follows from Proposition 2.7 in Chapter II, that all finitely generated R-algebras A have (A,N) an L.I. pair if and only if all finitely generated Rx-algebras B have (B,rad B) an L.I. pair for all x e X(R). This result further suggests the equivalence suggested by Ingraham of inertial coefficient rings and rings R all of whose finitely generated R—algebras A have the idempotent lifting prOperty, since both these pro- perties can be determined from the connected stalks R of x the ring R. BIBLIOGRAPHY 10. 11. 12. BIBLIOGRAPHY Auslander, M., and Goldman, 0., "The Brauer group of a commutative ring", Trans. Amer. Math. Soc. 97 (1960), 367-409. Azumaya, G., "On maximally central algebras”, Nagoya Math. J. 2 (1951). 119-150. Bourbaki, N;, Elements of Mathematics - Commutative Algebra, Addison wesley (1972). 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