closures of I

J o u r n a l

w

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Real closures of commutative rings*). I

B y Manfred Knebusch at Regensburg Dedicated to Helmut Hasse on his jj^th birthday

Introduction

L e t A be a connected commutative r i n g w i t h 1, and let A denote the universal covering ( = separable closure) of A i n the sense of Galois theory (cL e. g. [8]). T h e main goal of the present paper is to give a contribution to the following p r o b l e m : Classify all coverings B e i of A ( = direct limits of finite etale connected extensions of A), such that 1 < [A : B] < oo. F o r A a field a complete answer to this problem has been given b y A r t i n and Schreier [2], [3]: F o r every such covering B we have [A : B] = 2, and B is a real closure of A w i t h respect to an ordering of A. In this way the isomorphy classes of coverings B of A w i t h 1 < [A : B] < oo correspond uniquely to the orderings of A.

T o generalize Artin-Schreier's theory to rings we have to find a suitable substitute for the orderings of a field. T o m y firm conviction this substitute are the signatures. A signature a of the r i n g A is defined as a homomorphism from the W i t t ring W(A) of symmetric inner product spaces over A [20] to the ring Z of integers. T h i s definition is m o t i v a t e d b y a result due to H a r r i s o n [10] and L e i c h t - L o r e n z [19], w h i c h says that for A a field the signatures of A correspond uniquely to the orderings of A. T h e value of the signature a corresponding to a given ordering on a inner product space E is Sylvester's index of inertia of E w i t h respect to the ordering, i . e. the number of positive coefficients minus the number of negative coefficients i n an a r b i t r a r i l y chosen diagonalization of E.

Thus we consider pairs (A, a) consisting of a connected commutative r i n g A and a signature a of A. There is an evident notion of morphism q>: (A, a)-> (B, r) between pairs (cf. § 2), w h i c h for A and B fields just means, that cp is a homomorphism from A to B compatible w i t h the orderings corresponding to a and T. W e call cp a covering, i f the r i n g homomorphism cp: A -> B is a covering. W e further call a pair (R, Q) real closed, if (R, Q) does not admit coverings except isomorphisms. F i n a l l y we call a covering oc : (A, G)-> (R, Q) w i t h (i?, Q) real closed a real closure of the pair (A, a). U s i n g Zorn's lemma, i t is easily seen t h a t every pair (A, a) has at least one real closure.

L e t oc: (A, tr)-> (R, Q) denote a fixed real closure of (A, a). W e shall prove i n § 3 and § 5 the following t w o general theorems:

(0. 1) Any other real closure of (A, a) is isomorphic to oc over A¹).

*) A part of the results of this paper has been announced in [13].

J) In the case of fields a proof of (0.1) by the methods of this paper is already contained in [14].

(0. 2) [A : R] <^ 2. If some prime number p is a unit in A, then [A : R] =z 2. Fur- thermore in the case that 2 is a unit, A — R[\/—1 ] .

In part II of this paper we shall see, that the real closures of a local r i n g A have nearly a l l the pleasant properties discovered b y A r t i n and Schreier i n the case of fields:

(0. 3) Q is the unique signature of R. Furthermore W(R) = Z if in addition 2 is a unit in R. This is generally false if 2 is not a unit. But the Witt ring W(A, J) of hermitian inner product spaces over A with respect to the involution J =j= i d of A/R always equals Z.

Thus our signature a may be identified with the canonical map from W(A) = W(A, id) to W(Ä, J).

(0. 4) R has no automorphisms over A except the identity.

(0. 5) The signatures a of A correspond uniquely to the conjugacy classes of involu- tions J #= i d in the Galois group of AI A, the fixed ring of such an involution J being a real closure of {A, a).

S l i g h t l y more generally the results (0.3) (0. 4) and probably also (0. 5) remain true for A semi-local. I further obtained m u c h evidence, that for A semi-local indeed all coverings B of A w i t h 1 < [A : B] < oo are real closures of A.

Basic tools to prove the results ( 0 . 1 )—( 0 . 5) are provided b y t w o papers [16], [17]

w r i t t e n j o i n t l y w i t h A . Rosenberg a n d R . W a r e , a n d b y an i m p o r t a n t theorem of A . Dress (see Theorem 2. 1 i n § 2). I n particular, the statement (0. 4) follows almost immediately from (0. 3) and the arguments i n the proof of P r o p o s i t i o n 4. 8 of [17].

The result (0. 3) strongly suggests to study more generally rings A equipped w i t h involutions J^A (which are allowed to be the i d e n t i t y ) . T h i s w i l l be done i n this paper.

In § 1 we develop a theory of coverings for such rings and more generally w i t h o u t addi- t i o n a l work for rings on w h i c h an a r b i t r a r y fixed finite group n is acting. Signatures and real closures can be defined for connected rings w i t h i n v o l u t i o n i n an analogous w a y as above, a n d results similar to (0. 1 )—( 0 . 5) w i l l be proved.

If A is a r i n g w i t h J^A = i d , and or is a signature of A, then we call, since now, a real closure of (A, a) i n the category of rings w i t h o u t i n v o l u t i o n , as defined above, a strict real closure of (^4, cr), a n d we reserve the notion ^ur e a l closure" to the m a x i m a l cov- erings of (A, a) i n the category of rings w i t h i n v o l u t i o n . These notions are closely relat- ed: L e t A denote as before the universal covering of A i n the category of rings w i t h o u t i n v o l u t i o n , a n d let (R, Q) be a strict real closure of (A, a). I n the case [A : R] = 1 the pair (R, Q) is also a real closure of (A, a) {J^R = id}. I n the case [A : R] = 2 & real closure of (^4, a) is given b y the r i n g A equipped w i t h the automorphism / 4= i d of A jR as i n - v o l u t i o n a n d a suitable signature of (A, J).

W e call the i n v o l u t i o n J^A non degenerate, i f A is finite etale of degree t w o over the r i n g A⁰ of fixed elements of J^A. W e shall prove i n § 6 the rather surprising fact, that for a connected r i n g A equipped w i t h an a r b i t r a r y i n v o l u t i o n a n d a n a r b i t r a r y signature a a real closure of (A, cr) has a non degenerate i n v o l u t i o n , i f at least one prime number p is a u n i t i n A.

If J^A = i d or J^A is n o n degenerate, then the theory of real closures of A can be reduced to the theory of strict real closures of A⁰. F o r A a field we always meet one of these cases. T h u s i t is reasonable from our point of view, t h a t A r t i n and Schreier never studied fields w i t h i n v o l u t i o n .

I wish to t h a n k A . Dress, A . Rosenberg, and R . W a r e for discussions and letters w h i c h have proved to be helpful for the theory presented here. The experienced reader w i l l perceive the close connections between the methods used i n this paper and Dress' theory of Mackey-functors, i n p a r t i c u l a r i n Section 3.

§ 1. E q u i v a r i a n t coverings

W e study c o m m u t a t i v e rings (with 1) on w h i c h a fixed finite group n acts from the left b y r i n g automorphisms. Such rings w i l l be called w r i n g s . F o r our applications i n this paper only the case n — Z / 2 Z is needed, but the purely formal study of this section does not present serious a d d i t i o n a l difficulties for a r b i t r a r y finite n, and could equally well be done for schemes. A l l propositions of this section are well k n o w n i n the case n — 1.

A homomorphism cp\ A-> B from a rc-ring A to a jr-ring B is of course an o r d i n a r y r i n g h o m o m o r p h i s m , m a p p i n g 1 to 1, w h i c h is compatible w i t h the jr-actions. The homo- m o r p h i s m cp is called finite etale, if <p is finite etale as an ordinary r i n g homomorphism ([9]? § 18. 3). A 7r-ring A is called connected, if A does not contain any idempotent different from 0 and 1 w h i c h is i n v a r i a n t under n. Assume A is connected. T h e n clearly A has only finitely m a n y p r i m i t i v e idempotents e^x, . . ., e^r, on w h i c h n acts t r a n s i t i v e l y . Assume i n a d d i t i o n t h a t cp : A -> B is a finite etale homomorphism into a Ti-ring B. T h e n the projective module Bcpie^ over Aeⁱ has for every e^t constant rank, since Aeⁱ is a con- nected r i n g i n the ordinary sense. Since n acts t r a n s i t i v e l y on the e^t, these ranks are a l l equal. T h u s the r i n g B w i t h o u t ji-action, w h i c h is denoted b y | B\, is a projective module of constant rank over \A |. T h i s rank w i l l be denoted b y [B : A] and w i l l be called the degree of the finite etale h o m o m o r p h i s m cp. Notice t h a t i n the case [B: A] > 0, i . e . B 4= 0, the map <p must be injective. Unless the contrary is e x p l i c i t l y stated, we assume since now i n this paper, t h a t a l l occurring rings are 4= 0.

A n idempotent e of a 7i-ring A w i l l be called a n-idempotent, if e is i n v a r i a n t under n, and a jr-idempotent e 4= 0 w i l l be called n-primitive, if e is not the sum of two orthogonal

^-idempotents e^x and e² w h i c h are b o t h 4= 0. I n this paper only jr-rings A w i l l occur w i t h \A I containing only finitely m a n y idempotents. L e t {e¹, . . ., e^t} be the set of n- p r i m i t i v e jz-idempotents of A. W e call the rc-rings Aⁱ: = Ae^t the components of A (and

t

regard them as subsets of A). Clearly A is the direct product J J A^t of the A^t i n the cate- gory of Ti-rings, the projections pⁱ:A->Aⁱ being defined b y p^a) = ae^t.

F o r two 7r-homomorphisms cp: A-> B and a: A -> C the tensor product B ®^AC w i t h respect to <p and oc is defined as the usual tensor product \B \ ®\^A\\C\, equipped w i t h the jr-action g(b ® c) = (gb) ® (gc) for g i n n. W e denote the jr-homomorphism B -> B ®^AC, 6h- > 6 ® 1 , b y 1 ® <% and the jr-homomorphism C - > B ®⁴C , m 1 ® c, by cp ® 1. The c o m m u t a t i v e diagram

B ® 6

(1. 0) q> <p®l

is a pushout i n the category of ,-r-rings.

L e t A be a connected Tr-ring. W e call a 7i-homomorphism cp: A -> B a finite cov- ering, if (p is finite etale, the 7r-ring B is connected, and S + 0 .

Lemma 1. 1. Let cp\ A -> B and ß: B -> C be homomorphisms between connected st- rings A, B, C. Assume cp and ß o cp are finite coverings. Then also ß is a finite covering.

Proof. Consider the diagram ( 1 . 0 ) w i t h oc: = ß o (p. Here B ® C is a finite product

r

JjEⁱ of connected <p-rings E^t. L e t oc^x,...,oc^r denote the components of 1 ® oc and i = l

<p^x, . . ., <p^r denote the components of cp ® 1. Since 1 ® oc and cp ® 1 are finite etale, a l l ocⁱ and cpj are coverings. B y the pushout property of our diagram there exists a unique homomorphism pi from B ® C to C w i t h pi o (99 ® 1) = i d^c and ° (1 ® oc) = ß. Since C is connected and 4= 0, the homomorphism ^ maps a l l ^ - p r i m i t i v e jr-idempotents of B ® C to 0 except one. Thus ^ factors through a unique canonical projectionpⁱ: B ® C->Eⁱ, ju = y op.. F r o m pi o (9? ® 1) = i d^c we obtain y o 99. = i d^c. T h i s implies i n particular, that the kernel of y : E^t-> C is generated b y an idempotent ([8], p . 96), w h i c h must be i n v a r i a n t under n. Since Eⁱ is connected this idempotent must be 0, i . e. y is injective.

Thus we see that y is an isomorphism. Since ß = y o p. o (1 ® #) = y © «^f and aⁱ is a finite covering, also ß is a finite covering.

W e call a h o m o m o r p h i s m cp: A -> B from a connected Ti-ring A to a Ti-ring B a covering, if cp is the direct l i m i t of a direct system ((p^f: A -> J5^{i 7} i , j € / ) of finite cov- erings. B y L e m m a 1. 1 then also a l l tp^: B^t-> B^j are coverings, and i n particular i n - jective. T h u s the canonical maps ipⁱ: B^t^ B from the B^{ into the direct l i m i t B are injective, B is connected, a n d cp: A -> B is injective. Regarding B as an overring of A we c a n say more s i m p l y t h a t a n injection A B is a covering, if every finite subset of B is contained i n a r i n g B' w i t h A < B' < B and yl ^ B' a finite covering.

Proposition 1. 2. TAe composite xp o cp of two coverings cp: A -> B and xp: B -> C is again a covering.

Proof. W e regard cp and xp as inclusion maps. I t suffices to show t h a t for every finite subcovering B <-> C of B ^ C the composite A C is a covering. T h u s we m a y assume t h a t C is finite over B. I t is not difficult to show, t h a t there exists a finite sub- covering A ^ B' of A ^ B a n d a finite covering % : B' -> D , such t h a t 1 ® %: B -> Z? ®^B,D is a covering of B isomorphic to B C. B u t (1 ® %) °<p can also be w r i t t e n as the composite

w i t h i the inclusion map from B' to B. N o w i \ B' ^ B is the direct l i m i t of inclusion maps j: 5 ' ^ B" w i t h ^4 J9" finite coverings. B y L e m m a 1. 1 each j: B' 5 " is a finite covering. (1 ® %) o <p is the direct l i m i t of the finite coverings

B' - j + D -r^- B" ®^B.D.

T h u s (1 ® x) °<P i ^{s a} covering of A, hence also xp <xp.

Proposition 1. 3. Assume cp. A-> B and oc: A -> C are coverings of a connected Ti- ring A. Then every homomorphism ß: 2?-> C with ß o cp — oc is also a covering.

Proof. T h i s h a d been stated for oc a n d cp finite already i n L e m m a 1 . 1 . W e regard C as an overring of A and oc as the inclusion map from A to C.

i) T h e assertion is true if (p is a finite covering. Indeed, C is the union of a directed system ( C⁰ i € I) of subrings containing cp(B) such t h a t a l l A are finite coverings.

B y L e m m a 1. 1 the maps B->Cⁱ induced b y ß are also finite coverings. T h u s ß is a covering.

ii) I n the general case the m a p ß is certainly injective. F o r we c a n write B as a union of a directed system (B^jtJ) of subrings containing <p{A) such that a l l maps A -> B^j induced b y cp are finite coverings. B y part i) of our proof a l l restrictions ß \ B^

are coverings a n d hence injections. Thus ß is injective.

iii) W e now prove the proposition i n the general case. W e choose a directed system (C^{, i€ I) of subrings of C containing A such that a l l inclusions A <^> Cⁱ are finite cov- erings. L e t Dⁱ denote the r i n g generated b y ß(B) and Cⁱ i n C. W e shall show that for every i € I the homomorphism B-+ D^{ induced b y ß is a finite covering. T h e n i t w i l l be clear that ß is a covering. W e fix some Cⁱ a n d call i t C, and we denote the corresponding D^t b y D'. T h e tensor product B ®^AC' is finite etale over B, and thus is a finite product

t

II of connected rc-rings L e t cp^x, ..., cp^t denote the components of cp ® 1 : C -> B ® C",

7 = 1

and let y^x, . . ., y^t denote the components of the canonical m a p from B to B ® C. T h e Yi are finite coverings. B y the pushout property of B ®^AC we obtain from ß: B->C and from the inclusion m a p C ' ^ C a map B ®^AC'-+ C, whose image i n C is clearly D'.

The corresponding m a p d: B ®^AC-> D' must factor through a canonical projection B ®^AC E^j for a unique j w i t h 1 rgj j ^ t. W e denote the m a p from E^j to D' corre- sponding to 6 b y (5, a n d obtain a commutative diagram

cp is a covering and y⁷ is a finite covering. Since also A ^ C is a finite covering we obtain from P r o p o s i t i o n 1. 2 and p a r t i ) of the proof that cp^ is a covering. Since C C is a covering we further obtain from part ii) of our proof t h a t d is injective. T h u s d is a n iso- m o r p h i s m . T h e m a p J5-> D' induced b y ß is clearly ö o y.. It is a finite covering.

W e call a rc-ring C simply connected, i f C is connected a n d every covering of C is an isomorphism. W e further call a n y covering cp: A -> C of a connected rc-ring A w i t h C s i m p l y connected a universal covering of A.

Lemma 1. 4. Assume C is a connected n-ring which neglecting the n-action has a de- composition \C\ = C¹ X • • • X Cn with n the order of n and thus all Cⁱ connected. Assume all C^% are simply connected in the usual sense (= "separably closed" in [8]). Then C is simply connected.

Proof. L e t cp: C -> D be a finite covering, a n d let e^x, . . eⁿ be the p r i m i t i v e idem- potents of C. The homomorphism cp induces finite etale homomorphisms q>^{: Ce^{-+ Dq>(e^t) of rings w i t h o u t ^-action. A l l cp (e^t) are #= 0 a n d thus a l l Dq>(e^ must be connected, since otherwise D w o u l d contain more t h a n n p r i m i t i v e idempotents. Since Ce^{ ^ C^{ is s i m p l y connected, every cpⁱ is bijective. T h u s cp is bijective.

Proposition 1. 5. Every connected n-ring A has universal coverings.

W e prove this now only i n the special case t h a t n is a group {1, / } w i t h 2 elements, sufficient for our applications. T h e general case w i l l be settled i n a n appendix of this paper²). Consider first the case t h a t | A \ is connected, a n d let | A \ -> D be a universal

2) See end of this paper.

Journal für Mathematik. Band 274/275 9

covering of |J 4 | , regarded as an inclusion m a p . W e introduce on D x D the jr-action J(x, y) = (y, x), a n d denote this n-ring b y C. T h e m a p cp : A -> C, z (z, / z ) is a TT- h o m o m o r p h i s m a n d i n fact a covering. F u r t h e r m o r e C is s i m p l y connected b y L e m m a 1 . 4 . W e n o w consider the case t h a t \A | is n o t connected. T h e n A ^ B x B w i t h a connected r i n g B a n d the jr-action on B X B given b y 2/) = (y, x). L e t y. B^ D be a uni- versal covering of B i n the usual sense. W e define a ^ - a c t i o n on D x D again b y

y) = T h e n xpx tp: B x B -> D X D is a ^-covering, and D x D is a simply connected jr-ring b y L e m m a 1. 4 .

Proposition 1. 6. Let A be a connected²) 71-ring and oc: A -+ C be a homomorphism into a simply connected n-ring C (e. g. a universal covering of A). Furthermore let cp: A -> B be a finite etale n-homomorphism. Then there exist exactly [B: A] n-homomorphisms ß: B^C with ß o cp ^ oc.

Proof. W e regard the tensor product B 0^A C w i t h respect to cp a n d oc, see diagram ( 1 . 0). T h e homomorphism <p ® I : C-> B ®^AC is again finite etale. T h u s B 0 C is a

t

finite product HE^{ of connected jr-rings E^x. L e t oc^x, . . ., oc^t a n d cp^x, . . . , <p^t denote the

i = l

components of 1 0 oc a n d <p 0 1 respectively. T h e cp^t are (finite) coverings a n d thus iso- morphisms, since C is s i m p l y connected. I n p a r t i c u l a r t = [B 0 C: C] = [B: A]. T h e homomorphisms ß^t: = cp^¹ o * . from B to C clearly a l l satisfy oc = ß^to cp. O n the other hand an a r b i t r a r y jr-homomorphism ß : B-+ C w i t h ß o cp = * corresponds b y the push- out property of the tensor product to a unique h o m o m o r p h i s m y : 5 0 C -> C w i t h 7 ⁰ (<p 0 1 ) = i d^c and y o ( 1 0 oc) = /?. Since C is connected, 7 factors through a unique canonical projection p^t: B 0 C -> 2?,.. F r o m y o (9? 0 1 ) = i d^c we obtain y = 9 p "¹ o p .^? and from y o ( 1 0 oc) = ß we obtain /? = cpT¹ o ocⁱ —

F r o m this P r o p o s i t i o n 1 . 6 we i m m e d i a t e l y obtain b y use of Zorn's l e m m a the following

Corollary 1. 7. Let cp: A -> B be a covering of a connected n-ring A and let oc: A-^ C be a homomorphism into a simply connected n-ring C. Then there exists at least one homo- morphism ß : B-> C with ß o cp = oc.

A p p l y i n g this corollary a n d the previous P r o p o s i t i o n 1. 3 to the case that cp a n d oc are b o t h universal coverings of A, we obtain

Theorem 1. 8. Any two universal coverings of a given connected n-ring A are iso- morphic over A.

In the sequel we choose a fixed universal covering of our connected ji-ring A a n d regard this as an inclusion m a p . W e denote this universal covering b y A «-> A. W e call a subring B cz A a covering of A if B contains A a n d the inclusion m a p A B is a covering.

Proposition 1. 9. Assume (B^f, id I) is a family of coverings of A with B^t< A. Then the ring B generated by the B^t in A is also a covering of A.

Proof. Since the Bⁱ themselves are generated b y families of finite coverings of A, we m a y assume t h a t a l l B^t are finite over A. F u r t h e r m o r e B is the u n i o n of the rings generated b y the finite subfamilies of (B^t, i € I). T h u s we m a y assume i n a d d i t i o n t h a t I is finite, a n d then even t h a t our family consists of two rings B^x, B². T h e map b^x 0 b²*-*b¹b² from B^x ®^AB² t o A factors t h r o u g h a component E of B^x 0 B². Since £ is a finite cov-

3) It suffices to assume that the ring B is an ^-module of constant rank.

ering of A> E is mapped injectively into A b y P r o p o s i t i o n 1. 3 (already L e m m a 1. 1 suffices). E has the image B i n A w h i c h hence is also a covering of A.

W e denote b y G(A) the Galois-group of A, i . e. the automorphism group of A/A.

F r o m the Propositions 1. 2 a n d 1. 3 and from Theorem 1. 8 we immediately obtain

Proposition 1.10. If B < A is a covering of A and X: B-> A is a homomorphism from B to A over A, then X can he extended to some a in G(A).

W e call the covering B <: A of A galois over A, i f every such A maps B into B; i n other terms, B is galois i f and o n l y i f every a i n G(A) keeps B stable. T h e automorphism group of a galois covering B\A w i l l be denoted b y G(BjA). If B is finite over A this group has order [B: A] b y Proposition 1. 6.

F o r an arbitrary covering B < A oi A the subring C of A generated b y the images A ( 5 ) of all ^4-homomorphisms X from 2? to A is a covering of A b y Proposition 1. 9, w h i c h clearly is galois. W e call C the galois hull of BjA. If 5 is finite over A, then B admits only finitely m a n y yl-homomorphisms into A b y Proposition 1. 6, a n d hence C is also finite over A.

In particular the finite galois coverings C c i of A constitute a directed family of subrings of A, whose union is A. T h e restriction maps G(A)-* G(CjA) induce an isomorphism

G ( i l ) ^ > l i m G(C/A)

w i t h C r u n n i n g through a l l finite galois coverings of A i n A. W e use this isomorphism to make G(A) a profinite topological group.

F o r any subgroup H of G(A) we denote as usual b y A^H the r i n g of a l l elements i n A fixed under H. Clearly A^H = A^H w i t h H the closure of H i n G(A). W e now state the fundamental theorem of equivariant Galois theory.

Theorem 1.11. The coverings B < A of A correspond uniquely to the closed sub- groups H of G(A) by the relations

B = A^H, H = G(B).

The covering B is finite over A if and only if G(B) has finite index in G(A), and then (G(A):G(B)) = [B:A].

F o r the proof we need t w o lemmas.

Lemma 1.12. Assume B <. A is a covering of the connected n-ring A, and G is a group of automorphisms of B over A. Then the ring A' = B° is a covering of A and B is a galois covering of A'. If G is finite then G = G{B\A').

Proof. I n B the subrings B' > A w h i c h are finite coverings of A a n d stable under all automorphisms of B/A constitute a filtered family whose u n i o n is B. T h i s remark allows to reduce the proof to the case t h a t B is finite over A. T h e n also G is finite b y Proposition 1. 6. W e n o w verify that \B\ is w i t h respect to G a galois extension of \A'\

i n the sense of A u s l a n d e r - G o l d m a n a n d Chase-Harrison-Rosenberg [5]. F o r this i t suf- fices to show the following (cf. [5], p . 18):

i) \B\ is separable over \A'\.

ii) F o r every idempotent e #= 0 of B a n d different elements cr^{1 ?} a² of G there exists some b i n B w i t h <r¹(fc)e #= a²{b)e.

N o w i) is clear, since | B | is separable over the smaller r i n g \A\. T o prove ii) we m a y assume t h a t e is p r i m i t i v e . Suppose o^x and a² are elements of G w i t h a¹(6)e = a²(b)e for a l l b i n B. A p p l y i n g some g i n n to this equation we obtain a^x(b)g(e) = a²(b)g(e) for all b i n B. Since n permutes the p r i m i t i v e idempotents of B transitively we obtain a^x(b) = o²(b) for a l l b a n d thus a^x — o².

Since \ B\ is a galois extension of \A'\ w i t h respect to G , the r i n g \ B\ is finite etale over \A'\ a n d [JS : 4 ' ] = | G | . W e m a y conclude that also \A'\ is finite etale over \A\

(e. g. [8], p . 95). W e obtain t h a t A' is a covering of A and B is a covering of A'. F r o m [Z?:^4'J = | G | a n d Proposition 1. 6 i t follows that B is a galois covering of A' a n d G = G(BIA').

Lemma 1.13, Assume B <. A is finite and galois over A with group G. Then B^G = A.

If H is a subgroup of G with B^H = A, then H = G.

Proof. B y L e m m a 1. 12 the r i n g B° is a covering of A a n d [B:B^G]= \G\ = [B:A].

T h u s B^G = A. F u r t h e r again from L e m m a 1. 12 we obtain \H \ = [B: A] = \G\. T h u s H = G.

F r o m the L e m m a s 1. 12 a n d 1. 13 the proof of Theorem 1. 11 is immediate.

The Theorem 1. 11 clearly has the following

Corollary 1.14. Assume (B^f, id I) is a family of subrings of A which are coverings of A. Then also the intersection of the B^t is a covering of A.

Assume B is a finite galois covering of A w i t h group G. F o r every g i n G we con- sider the 7r-homomorphism

f⁰: B ®^AB-+ 5 , f^g(b^x ® b²) = g{b,)b².

L e t f:B®^AB^JJB denote the homomorphism into the product of \G\ copies of B, indexed b y G , whose components are the f^g. Since \B\ is a galois extension of | A | (cf.

proof of L e m m a 1. 12), we obtain from [5], Theorem 1. 3 the following

Proposition 1.15. For any finite galois covering B of A the 7r-homomorphism f:B®^AB->IIBis an isomorphism.

G

W e shall also need the following corollary of this proposition.

Corollary 1.16. Assume cp: A^ B is a galois covering and oc: A -» C is a homo- morphism into a connected n-ring C, such that there exists at least one homomorphism ß: B-+C with ß © cp = oc. Then there exist exactly [B: A] such homomorphisms. For any two of them, ß^x, ß², there exists a unique a in G(B/A) with ß² = ßi° ^a-

Proof. I t clearly suffices to consider the case [B: A] < oo. T h e homomorphisms ß: B-+C w i t h ß o <p = oc correspond uniquely t o the ^ - p r i m i t i v e 7r-idempotents e of D: = B ®^AC w i t h [De: C] = 1 (cf. proof of P r o p o s i t i o n 1. 6). N o w we can write

D = (B ®^AB) ®^ßC

w i t h some fixed homomorphism ß⁰:B->C over A. It follows from P r o p o s i t i o n 1. 15, that D has e x a c t l y \G\ idempotents of the type described above, on w h i c h G acts freely and t r a n s i t i v e l y . T h u s G also acts freely a n d t r a n s i t i v e l y o n the corresponding set

H o m⁴( ß , C). T h i s is our assertion.

§ 2. Definition of signatures and real closures

Since now n is always a group consisting of t w o elements 1, / . F o r a n y 7i-ring A we denote b y J^A the i n v o l u t i o n on A induced b y J. F o r a i n A we often write ä instead of J^A (a). W e say t h a t the r i n g A is local, resp. semilocal, resp. D e d e k i n d , etc. i f the r i n g IA I w i t h o u t 7r-operation has this property. T h e r i n g of elements fixed under J^A w i l l be denoted b y A⁰ and w i l l usually be regarded as a jz-ring w i t h t r i v i a l operation.

L e t W(A) denote the W i t t r i n g of hermitian inner product spaces over A. T h e elements of W(A) are suitable equivalence classes of pairs (E, 0) w i t h E a finitely gen- erated projective v4-module and 0 a n o n singular h e r m i t i a n form on E, linear i n the first argument and antilinear w i t h respect to J^A i n the second. T h e case J^A = i d is allowed.

W e refer the reader to [16], § 1 and to [20] for the basic definitions. (In [16] the t e r m

un o n degenerate" is used instead of ^un o n singular".)

The equivalence relation for h e r m i t i a n inner product spaces used i n the definition of W ( A ) w i l l be denoted b y ~ , and the equivalence class of an inner product space (E, 0) w i l l be denoted b y [E, 0]. W e often write E instead of (E, 0) and [E] instead of

\E, 0]. F o r a n y 7r-homomorphism <p: A -> B we denote the induced r i n g homomorphism [E] H > [E ®^AB] from W(A) to W(B) b y ^ .

If A is connected b u t \A | is not connected then W(A) = 0 (e. g. [16], p . 125). If IA I is connected, then every inner product space E over A has a constant rank, denoted b y d i m £ , and the m a p [E] H> d i m £ m o d 2 from W(A) to Z / 2 Z is well defined. W e call this h o m o m o r p h i s m the dimension index v a n d its kernel the fundamental ideal 1(A) of W{A).

A n inner product space (E, 0) w i t h E a free A-module w i l l often be denoted b y an h e r m i t i a n m a t r i x (a^fj) w i t h a^i} = 0(e^t, c>) for some basis e^x, . . ., eⁿ of E. I n partic- ular every u n i t a of A⁰ yields a free space (a) of rank one. A n orthogonal s u m

(a^x) ± • • • J . (aⁿ) w i l l also be denoted by (a^x, . . ., aⁿ).

Definition 2 .1 . A signature cr of a c o m m u t a t i v e r i n g A is a homomorphism from the r i n g W(A) to the r i n g Z of integers, cf. I n t r o d u c t i o n . T h e r i n g A is called real, i f the set Sign (A) of signatures of A is not empty. Otherwise A is called non real.

Remark 2. 2. If A is semi-local a n d \A/^0l\ > 4 for a l l m a x i m a l ideals 9R of A, then A is n o n real i f a n d only i f there exists an equation

— 1 = a^x\ + h a^ra^r

w i t h finitely m a n y a^{ i n A. T h i s has been proved i n [17], § 4 under the additional as- s u m p t i o n t h a t A contains an element fi w i t h [i + ~jx = 1. A proof not using this assump- t i o n w i l l be published i n the near future [15]. (The assumption about the residue class fields Ajyjl above is only needed i n the case J^A 4= i d , and perhaps can also be eliminated i n this case.)

F o r E a n inner product over A a n d a a signature of A we usually write a(E) instead of a ([E]). T h e role played b y the signatures i n the theory of W i t t rings is indicated b y the following

Theorem 2. 3. Assume \ A \ is connected. If A is non real then 1(A) is the only prime ideal of W ( 4 ) , and 2ⁿ • W(A) = 0 for some n J> 1. If A is real then the minimal prime ideals PofW(A) correspond uniquely to the signatures a of A, the prime ideal P correspond- ing to a being the kernel of a,

T h i s has been shown for A semi-local i n [16]. F r o m the semi-local case one easily obtains a proof of Theorem 2. 3 i n general b y use of the following theorem, due to A . Dress.

Theorem 2. 4. Let A be an arbitrary commutative n-ring. For every minimal prime ideal PofW(A) there exists a maximal ideal m of A⁰ and a minimal prime ideal Q ofW(A^m), such that P is (he inverse image of Q with respect to the canonical map from W(A) to W(A^m).

T h e proof of this i m p o r t a n t theorem, whose details have been thoroughly checked b y the present author, w i l l appear i n the near future (Dress, oral communication*)). T h e m a i n tool used i n this proof is L e m m a 10. 1 i n [6] (with the group G there being 1).

Assume cp : A -> B is a jr-homomorphism, a is a signature of A a n d r a signature of B. W e say that r extends a (with respect to cp), or t h a t o* is the restriction of r to A, if the diagram

W(A) ^ ,W(B)

Z

commutes. W e often denote the restriction o* b y r\A, i f there is no doubt w h i c h m a p cp is considered.

A c c o r d i n g to the Theorems 2. 3 a n d 2. 4 every signature of A extends to at least one localization ^ 4^m. I n § 4 we shall prove the sharper statement t h a t every signature of A extends to a residue class field A (p) = A^lpA^p w i t h p a suitable prime ideal of A stable under J^A.

L e t a be a signature of A. F o r a n y unit a of A⁰ we have [(a)]² = 1 i n W(A) a n d hence a{a) = ± 1. Thus a yields a character of A$ w i t h values ± 1. U A is semi-local, then a is u n i q u e l y determined b y this character. T h i s is evident i f \A \ is connected, since then W(A) is generated b y the elements [(a)]. B u t i t is also true i f \A | is not con- nected, cf. [17], end of § 2. I n the semi-local case we usually identify a w i t h the corre- sponding character of A$. T h e reader is advised to consult § 2 of the paper [17] for a more detailed description of these characters, a n d t o consult § 4 of the same paper, if he wants to see how to deal w i t h signatures of semi-local rings i n m u c h the same w a y as w i t h orderings of fields.

L a t e r on we shall need the following

Proposition 2. 5. Assume A is a commutative n-ring with \ A \ connected. Then for every signature a of A and z in W(A)

v(z) = a(z) m o d 2.

Proof. L e t m be a m a x i m a l ideal of A⁰ such t h a t a extends to a signature T of A^m, and let z' denote the image of z i n W(A^m). T h e n a(z) = r(z') a n d v(z) = v(z'). T h u s we have t o show v(z') = r(z') mod 2, i . e. we have reduced the proof to the case that A⁰ is a local r i n g . I n this case P r o p o s i t i o n 2. 5 is clear from the fact t h a t W(A) is generated b y the classes of free spaces (a) of rank one (or cf. [16], E x a m p l e 3. 11 last line).

W e n o w consider pairs (A, a) consisting of a Ti-ring A a n d a signature a of A.

A morphism cp: (A, a)-> (B, r) between such pairs is a jr-homomorphism cp from A to B such t h a t (r = T O ^ , i . e. T extends a w i t h respect to cp. T h e pair (^4, a) is called con-

*) Added in proof. A . Dress, The weak local-global principle in algebraic Z-theory, to appear in Com- munications in Algebra.

nected if A is connected. I n a similar w a y we use terms like " l o c a l " , " s e m i l o c a l " , "Dede- k i n d " , etc. for pairs. If (A, a) is connected then \A\ is connected, since otherwise W(A) = 0 and A would not possess signatures.

A covering (resp. finite covering) of a connected pair (A, a) is a morphism q>: [A, (r)-> (B, T)

into a connected pair (B, T) such that the 7i-homomorphism cp: A -> B is a covering (resp.

finite covering) as explained i n § 1. W e call a pair (B, Q) real closed, if (B, Q) is connected and does not admit coverings except isomorphisms. A n y covering <p : (A, a)-> (B, Q) w i t h (B, Q) real closed is called a real closure of (A, a).

B y § 1 every covering (p : (A, a) -> (B, r) of (.4,0*) is isomorphic to a covering

\p: (A, a) -> (B', T' ) w i t h A <. B' <. A and y\A->B' the inclusion map. F r o m this remark one easily obtains b y use of Zorn's lemma

Proposition 2. 6. Every connected pair (A, a) has at least one real closure cp:(A,a)->(B,Q).

Remark 2. 7. If cp: (A, a)-> (B, Q) is a real closure of (A, a), then certainly (p: A -> B is not a universal covering, since b y the proof of Proposition 1. 5 the r i n g \A \ is not connected.

In the next section we shall see (Theorem 3. 9) that a n y t w o real closures of (A, or) are isomorphic over (A, a).

A pair (7¹, T) is called strictly real closed, if T has t r i v i a l i n v o l u t i o n J^T = id^T, a n d (T, r) is connected and does not admit coverings b y pairs w i t h t r i v i a l i n v o l u t i o n except isomorphisms. A strict real closure of a connected pair (^4, a) w i t h t r i v i a l i n v o l u t i o n is a covering <p : (A, a) -> (T, r) w i t h (T, r) strictly real closed.

Proposition 2. 8. (i) Every connected pair (A, a) with trivial involution has at least one strict real closure, (ii) If (T, r) is strictly real closed and rp: (T, r)-» (B, Q) is a real closure of (T, r), then rp(T) = B⁰ and [B : B^Q] <; 2.

Proof. T h e first assertion is again clear b y Zorn's l e m m a . T o prove the second we m a y assume w i t h o u t loss of generality T <c B a n d that tp is the inclusion map. Clearly T cz B⁰. N o w the r i n g B⁰ is a covering of T b y L e m m a 1. 12. Since the signature Q\B⁰ extends r we must have B⁰ = T. Furthermore b y the same l e m m a [B : B⁰] = 2 i f J^R #= i d , otherwise B = B⁰.

Since n o w we also use the following terminology: L e t (A, a) be a connected pair and remember t h a t A denotes an arbitrarily chosen fixed universal covering of A. W e say t h a t a connected pair (B, T) is a covering (resp. real closure, etc.) of (^4, a), if A c: B c: A a n d the inclusion m a p i: A ^ B is a morphism from (^4, a) to (B, r) w h i c h is a covering (resp. real closure, etc.).

§ 3. The trace formula

L e t A be a 7r-ring and cp: A -> B a finite etale Ti-homomorphism. T h e n the trace Tr^,: B-+A of this finite etale extension ([8], p. 91 ff.) is an A-linear map, w h i c h is compat- ible w i t h the Ti-actions. W e have a well k n o w n transfer homomorphism i n

T r * : W{B)-+W(A)

of a d d i t i v e groups m a p p i n g the class of a space ( £ , 0) over B to the class [ E ^ , T r ^ o 0 ] , w i t h E^V denoting E considered as an A-module b y y, cf. [7], § 2. W e shall use the fol- l o w i n g criterion for extending signatures.

Lemma 3. 1. Let o be a signature of A and assume there exists an element y in W(B) with a (Tr*(?/)) 4= 0. Then a can be extended to B.

T h i s can be proved b y the same argument as used i n [17] i n the semi-local case, cf. the proof of L e m m a 5. 3 i n t h a t paper.

L e t now a be a fixed signature of A and let cp: A -> B be a fixed finite etale :rc-homo- m o r p h i s m . W e denote b y S(<p, a) the set of a l l signatures r of B w h i c h extend a w i t h respect to cp.

Definition. A trace formula w i t h respect to cp and a is a map n: S(y, a)->Z such t h a t n(r) = 0 except for finitely m a n y T i n S(<p, o*), and

(3.2) o(Tv*(z)) = Hn(r)r{z)

* r\a

for a l l z i n W(B). Here the sum is taken over a l l r i n S(<p, or), w i t h the convention t h a t this sum is zero if £ ( 9 ? , or) is e m p t y .

Remark. W e shall see below t h a t a c t u a l l y S (<p, a) is always a finite set.

Lemma 3. 3. For given cp and a there exists at most one trace formula.

Proof. Assume n and n' are two different trace formulas for a. W e choose some r⁰ i n S((p, o) w i t h n(r⁰) =f= n'(r⁰). L e t M denote the finite set of a l l r i n S(cp, a) such t h a t 71 ( T ) and n'(r) are not b o t h zero. F o r every T i n M let P(r) denote the kernel of

T : W(B)->Z.

Since a l l these P(r) are m i n i m a l prime ideals of W(B), the intersection of a l l P(r) w i t h T i n M and T 4= T⁰ is not contained i n P(r⁰). T h u s we can find some z i n W(B) w i t h r^Q(z) 4= 0 b u t r(z) = 0 for a l l other T i n M. N o w e v a l u a t i n g o-(Tr*(z)) using b o t h trace formulas n and n' we o b t a i n the c o n t r a d i c t i o n

n(*o)*o(^z) = n^f(r⁰)r^Q(z).

W e now state the m a i n result of this section.

Theorem 3. 4. (i) For given <p and a there exists a trace formula n. (ii) In this formula n(r) > 0 for every r in S(<p, a). In particular S((p, a) is finite, (iii) If oc: {A, a) -> (Z?, Q) is a morphism into a real closed pair (Z?, Q) then for any r in S((p, a) the number n(r) equals the cardinality of the set of all morphisms from {B, r) to (i?, Q) over (A, a).

Remark 3. 5. F o r every pair (^4, a) there exists some m o r p h i s m <x into a real closed pair Q). Indeed, if A is connected y o u can take a real closure of (^4, a). If A is arbi- t r a r y there exists b y Theorem 2. 4 some m o r p h i s m (A, a)-> (A^M1 y) w i t h m a m a x i m a l ideal of A⁰. T h i s m o r p h i s m can be composed w i t h a real closure of (A^M, y).

W e postpone the proof of Theorem 3. 4, and first draw some consequences from this theorem. F o r the coefficients n(r) of the unique trace formula belonging to (p and 0*

we now write more precisely cp) or 71 (r, ^4), and we call w(r, cp) the multiplicity of r w i t h respect to <p or A.

Inserting the u n i t element of W(B) i n t o our trace formula we o b t a i n a ( T r ; ( l ) ) = 2 7 n( T , <p).

T\0

In the case that A is connected there exist b y Proposition 1. 6 at most [B: A] jr-homo- morphisms from B to B over A i n the situation described i n Theorem 3. 4. (iii). Thus our Theorem 3. 4 has the following

Corollary 3. 7. Let <p: A -> B be a finite etale n-homomorphism. Then o*(Tr*(l)) J> 0 for every signature a of A, and a is extendable to B if and only if <r(Tr*(l)) > 0. If A is connected, then a has at most [B: A] extensions to B.

W e mention a rather t r i v i a l application of this handy criterion for extendability of signatures.

Proposition 3. 8. Assume cp: A -> B is finite etale and [B^m : A^m] is odd for all maximal ideals m of A⁰. Then every signature a of A can be extended to B.

Proof. There exists some m a x i m a l ideal m of A⁰ such t h a t a extends to a signa- ture y of A^m. L e t z denote the image of T r * ( l ) i n W(A^m). W e have v(z) = 1 a n d thus b y Proposition 2. 5

T h i s implies a ( T r * ( l ) ) 4= 0.

F o r another proof of Proposition 3. 8 cf. [17], Proposition 5. 4.

Theorem 3. 9. Assume (A, a) is connected and oc : (A, a) -> (B, Q) is a morphism with (B, Q) real closed. Let (p: (A, a) (B, r) be a covering.

(i) There exists at least one morphism ß: (B, r)-> (B, Q) with oc = ßo<p.

(ii) If oc and <p are real closures then every such ß is an isomorphism.

Proof, (i) W e m a y assume A < B a n d t h a t cp is the inclusion m a p from A t o B.

B y Zorn's l e m m a there exists a m a x i m a l r i n g C <: B w i t h A < C and A ^ C a covering, such that there exists a morphism ß^x from (C, r\C) to (B, Q) extending oc. If C 4= B then we can find a r i n g D w i t h C %D <.B and C ^ B a finite covering. B y Theorem 3. 4 the morphism ß^x can be extended to a morphism from (Z>, r\D) to (/?, Q). T h i s contradicts the m a x i m a l i t y of C. Thus C = B.

(ii) If oc is a covering, then b y Proposition 1. 3 also ß is a covering. Thus i f i n ad- dition (p is a real closure then ß must be an isomorphism.

Corollary 3.10. Assume A and B have trivial involutions. Then Theorem 3. 9 re-

mains true with the words "real closed" and "real closure" replaced by "strictly real closed"

and "strict real closure".

Proof. Statement (ii) is clear b y the same argument as above. T o prove (i) we choose a real closure y: (B, Q) -> (S, rj) of the strictly real closed r i n g (B, Q). W e regard y a n d <p as inclusion maps. B y Theorem 3. 9. we can extend y o oc t o some morphism d: (B, T ) - > (S, YJ). Clearly d(B) <c S⁰. B y Proposition 2. 8 the r i n g S⁰ coincides w i t h B.

T h e morphism ß: (B, r) -> (B, Q) induced b y d fullfills ß o <p = a.

W e now enter the proof of Theorem 3. 4. W e consider the situation described i n part (iii) of this theorem. S t a r t i n g from the diagram (1. 0) w i t h the letter C there re- placed b y B we obtain a diagram

a ( T p * ( l ) ) = y(z) s i mod 2.

W(B) W(B ®^A B)

W(A) * W(B).

Journal für Mathematik. Band 274/275 10

It is easily checked t h a t this diagram is c o m m u t a t i v e , cf. [7], L e m m a 2. 1. T h u s for z in W(B)

a o T r * (z) = g o <x+ o T r * (z) = g o T r *^{@ 1} o (1 ® * ) * (z).

t

N o w i ? ® R is a direct product TLE^I of finitely m a n y connected jr-rings E^T. L e t t = i '

p^t: B 0 R->E^T denote the corresponding projections, 1 ^ i <^£, further let

*i^{: =} Pi ° (1 ® *) ^{a n c}* <Pi^: = Pi ° (95 ® 1)

be the components of 1 ® # and 99 ® 1 respectively. The 99^ : /? -> 2?, are finite coverings.

W e have

t t

e^oTr* 0 i° ( l ® 2 ^ T r * o ^ o ( l ®*)+(z)= HQOTT oa^(z).

L e t i be a fixed index i n [1, t]. If [2?^f: /?] > 1, then £ cannot be extended to E^T, since

(R,Q) is real closed. T h u s b y L e m m a 3. 1 the corresponding summand g o T r * o ocIS|E (2)

is zero. If [2?,: Ä ] = 1 then T r * = (<p^t*)^{_ 1} as is easily verified, a n d we obtain T r * o*„{z) = ßⁱ+{z)

w i t h ß^t = (p^^{1 0}'a^t. N o w these ß^t are precisely a l l homomorphisms from B to R over A , cf. the proof of P r o p o s i t i o n 1. 6. W e thus obtain

( r o T p ; ( z ) = J S^eo / } , (²)

w i t h /? r u n n i n g through the finitely m a n y homomorphisms from B to R over A. F o r every such /? the signature g o /?^# clearly extends o\ W e now for every signature T i n 5 ( 9 9 , ex) the n a t u r a l number n(r) as the number of a l l ß w i t h @ ⁰ ß% — T. Clearly ra(r) = 0 except for finitely m a n y T, a n d as we have just seen

o-oTr*(z) = ^ W ( T ) T ( Z ) T|nr

for 2 i n W(B). K e e p i n g L e m m a 3. 3 i n m i n d the assertions (i) a n d (Iii) of Theorem 3. 4 are proved.

L e t now r⁰ denote a fixed signature i n S (99, a) and choose some m o r p h i s m ß⁰ from (2?, T⁰) into a real closed pair (2?, g) (cf. R e m a r k 3. 5). A p p l y i n g assertion (iii) of Theorem 3. 4 to the m o r p h i s m oc: = ß⁰ o 99 we see n ( r⁰) > 0. T h u s also assertion (ii) is proved.

In the case t h a t A a n d B are semi-local a n d have t r i v i a l i n v o l u t i o n s the trace formula (3. 2) h a d been conjectured i n [17], 5. 16 w i t h m u l t i p l i c i t i e s n(r) = 1. W e shall see i n part I I of the paper, t h a t indeed a l l n(r) = 1 i n this case.

W e now discuss a case where m u l t i p l i c i t i e s n(r) = 2 occur i n a t r i v i a l w a y . L e t A be a jr-ring. T h e i n v o l u t i o n J^A is a jr-automorphism of A. W e denote for z i n W(A) the image (J^A)*(z) b y z.

Lemma 3. 11. For every signature a of A and every element z of W(A) we have a(z) = a(z). In other words, J^A is an automorphism of (A, a).

Proof, a extends to a signature r of B : = A^m w i t h m a suitable m a x i m a l ideal of , 4⁰. L e t x denote the image of z i n W(B). T h e n x is the image of z, a n d i t suffices to prove r(x) = T(X). N O W W (B) is generated b y elements [(a)] w h i c h are fixed under

T h u s y = y for a l l y i n W(B).

W e call the i n v o l u t i o n J^A non degenerate if A is finite etale over A⁰ and [A^m : A^0m] = 2 for a l l m a x i m a l ideals m of A⁰. If A is connected a n d is non degenerate then the i n - clusion m a p A⁰^A is a covering of w r i n g s .

Proposition 3.12. Assume J^A is non degenerate. Then n(r, A⁰) = 2 for every signa- ture x of A. A signature a of A⁰ has at most one extension to A.

Proof. L e t a denote the restriction r\A⁰ of a given signature r oi A. T h e n J^A is an automorphism of (A, r) over (A⁰,a⁰), and we see from Theorem 3. 4 (iii), that n(r, A⁰) ^> 2. A g a i n b y this theorem and b y Proposition 1. 6 we have

2n(r',A⁰)^[A:A⁰] = 2.

r'\a

T h i s implies b o t h assertions.

W e n o w present some applications of Theorem 3. 9.

Proposition 3.13. Let (A, o) be a connected pair with J^A non degenerate, let a⁰ denote the restriction a\A⁰, and let (R, Q) be a real closure of (A⁰, a⁰) with A⁰ <=: R <c A⁰ = A.

Then A < R and {R, Q) is a real closure of (A, a).

Proof. (A, a) is a covering of (A⁰, a⁰). B y Theorem 3. 9 there exists a morphism % from (A, a) to (R, Q) over (A⁰, a⁰). Since A is a galois covering of A⁰, we have

A = j [ ( A ) c / l .

N o w Q\A extends a⁰ a n d Proposition 3. 12 implies Q\A = a. F i n a l l y b y Proposition 1. 3 (R, Q) is a covering of (A, a). Thus (/?, Q) is a real closure of (^4, a).

Proposition 3.14. Assume (A, a) is connected and has trivial involution. Let (R, Q) be a real closure of (A, a). Then (R⁰, Q\R⁰) is a strict real closure of (A, a).

T h i s follows from Proposition 2. 8, since b y Theorem 3. 9 a n y t w o real closures of (^4, a) are isomorphic over (A, a).

Theorem 3.15. Let cp: A-> B be a finite etale n-homomorphism and oc: A -+ C be an arbitrary n-homomorphism. Let r^x be a signature of B and r² be a signature of C whose restrictions r^x o q>^ and r² o oc# are equal. Then there exists at least one signature rj of the tensor product B ®^AC with respect to cp and oc such that rj\B = r^x and rj\C = r², the re-

strictions being taken with respect to the canonical homomorphisms 1 ®oc:B->B®C and

<p ® 1 : C^B ® C.

Proof. L e t a denote the signature r^x\A = T²\A. W e choose some morphism y ^: (C, T²) - > (if, Q) into a real closed pair (R, Q) (cf. 3. 5). A p p l y i n g Theorem 3. 9 to the morphisms <p : (A, a) -> (B, T^X) a n d

y ooc: (A,a)^ (C, r²) -> (B, Q)

we k n o w t h a t there exists a morphism ß : (B, r^x) -> (B, Q) w i t h ß o <p = y o oc. B y the pushout property of the tensor product we have a homomorphism d: B ® ^A C R w i t h ö o (1 ® oc) = ß a n d d o (cp 0 1) = y. T h e signature r\: = Q O has the restrictions rj o (1 ® oc)+ = x^x and rj o (cp ® 1)* = ^T2.

W e investigate the situation described i n Theorem 3. 15 i n a special case.

Proposition 3.16. Let (A, a) be a connected pair with J^A non degenerate, let a^Q denote the restriction a\A⁰ and let oc: (A⁰, a⁰) -> (T, r) be a morphism into a strictly real closed pair (T, r). Let R denote the tensor product A ®^A%T with respect to oc. (i) T extends to a unique

10*

signature Q of R, and (R, Q) is real closed, (ii) This signature Q extends a with respect to 1 ® oc : A -> R. (iii) If oc: (^4⁰, a⁰) -+ (7¹, T ) is a strict real closure of (A⁰¹ <r⁰), then

1 ® * : ( 4 , c r ) - » ( / ? , Q) is a real closure of (A^y a).

Proof. W e regard T as a subring of R, which is possible since T -> R is finite etale.

Clearly R⁰ = T and J^R is n o n degenerate. B y Theorem 3. 15 there exists a signature Q on R extending b o t h a and r , and b y Proposition 3. 12 there exists no other signature of R extending T. If (R\ q') is a real closure of (7¹, T ) , then [R': T] 2 b y Proposi- t i o n 2. 8. W e now see from Theorem 3. 9 t h a t (R, Q) is isomorphic to (R', Q') over ( 7 \ T ) , and hence (JR, e) is real closed. T h u s the assertions (i) and (ii) are proved. If oc is a cov- ering then also 1 ® oc is a covering, w h i c h proves (iii).

Proposition 3.17. Let (/?, e) a real closed pair. Then the pair (R⁰, Q⁰) with Q^Q = Q\R⁰ is strictly real closed.

Proof. L e t <p: ( Ä⁰, Q⁰) -+ (T,r) be a strict real closure of (R^0lg⁰). W e consider the diagram

w i t h oc the inclusion m a p . W e m a y also regard 1 ® oc as an inclusion m a p , since cp is a covering. One easily verifies T = (T ® R)⁰. T h u s T ® R is certainly connected and (p ® 1 is a covering. B y Theorem 3. 15 there exists a signature rj of T ® R extending b o t h r and Q. Since (Ä, p) is real closed this implies [T: R⁰] = [T ® R: R] = 1. T h u s (#o> Qo)^is s t r i c t l y real closed.

W e close this section w i t h a description of the real closures of pairs ( 4 , a) w i t h A a field i n classical terms. T e m p o r a r i l y we write a ^ - r i n g A as a pair ( B , / ) w i t h B = \ A \ and J = J^A.

Example 3.18. L e t K be a field, J be an i n v o l u t i o n of K, a n d cr be a signature of (K, J). T h i s means that on the fixed field K⁰ of / an ordering < is given w i t h xx > 0 for a l l x i n K*^y cf. P r o p o s i t i o n 3. 12 and [17], 1. 6. L e t K denote the algebraic closure of K.

Since every covering of K (in the category of rings w i t h o u t involution) is a field, K is the universal covering of K. B y the proof of P r o p o s i t i o n 1. 5 the universal c o v e r i n g (K, J)~ is the pair (K x K, ß) w i t h ß(x, y) = (y, x). Regarding (K, J) as a rc-subring of (K, 7 ) ~ we have to identify an element x of K w i t h the element J(x)) of K x K.

L e t T < K be a real closure of K^Q w i t h respect to the ordering < i n the classical sense ([2], p . 89). B y the fundamental theorem of algebra K = Tfy—l] and [K : T] 2, cf. [2], p. 89. L e t oc denote the generator of the Galois group of KjT. W e have

W(T, id) = W(K, oc) = Z ,

and we denote b y r and Q the unique signatures of ( J , id) and (K, oc) respectively. We further denote b y a^Q the signature of K^Q corresponding to the ordering < , i . e. the re- striction of a to (Ü L⁰, id). Clearly Q extends r and T extends a⁰. If / 4= i d , then J is n o n degenerate, a n d we see from P r o p o s i t i o n 3. 12 — or b y a direct argument — t h a t Q also extends a. W e embed (K, oc) into (K, J)~ i d e n t i f y i n g an element x of K w i t h the element (x,oc(x)) of (K, jr.