Knebusch, Manfred

English

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2 o Separatum PUBLICATIONS HATHEMATICAE T O M U S 24. F A S C . 1 — 2 . D E B R E C E N 1977 F U N D A V E R U N T A . R ^ N Y I , T . S Z E L E E T O . V A R G A A D j U V A N T I B U S Z . DARÖCZY, B. G Y I R E S , A . RAPCSÄK, L. TAMÄSSY R E D I G I T : B. B A R N A M . Knebusch Remarks on the paper "Equivalent topological properties of the space of signatures of a semilocal ring9' by A. Rosenberg and R. Ware I N S T I T U T U M M A T H E M A T I C U M U N I V E R S ITATI S D E B R E C E N I E N S IS H U N G A R I A PUBLICATIONS MATHEMATICAE 24. KÖTET F A S C . 1—2. DEBRECEN 1977 ALAPfTOTTÄK: R E l N Y l A L F R E D S Z E L E T I B O R E S V A R G A O T T Ö DARÖCZI ZOLTÄN G Y I R E S B £ L A , RAPCSÄK A N D R Ä S , TAMÄSSY L A j O S KÖZREMÜKÖDESEVEL S Z E R K E S Z T I : B A R N A B£LA A D E B R E C E N I T U D O M Ä N Y E G Y E T E M M A T E M A T I K A I I NT£ Z E T E Remarks on the paper "Equivalent topological properties of the space of signatures of a semilocal ring9' by A. Rosenberg and R. Ware By Manfred Knebusch (Princeton, N. J.)* Dedicated to the memory of Andor Kertesz ROSENBERG and WARE deal in their paper [8] with semilocal rings in which 2 is a unit. I observed that the assumption that 2 is a unit can be removed in their results by adding further ideas to their proofs. Theorem 2.1 in [8] can even be gen-eralized to semilocal rings with involutions. The interested reader should understand in detail the paper [8] of Rosenberg—Ware before studying the present paper, since the main ideas are developed there. We essentially use the same notations as in [8]. Let A be an arbitrary connected semilocal ring (without involution) and let E be a symmetric bilinear space over A . For every signature a of A we have |a(2s)|^dim E and ex (is) = dim E mod 2. As in [5] we call E positive definite at the signature a (resp. negative definite, resp. definite), if <7(£*) = dim E (resp. a(E) = - d i m E9 resp. |<x(is)| = dim is), and we call E indefinite at a if |cr(is)|<dim is. Notice that if a: A^R is a homomor-phism from A into a real closed field R inducing a then E is positive definite (negative definite, ...) at a if and only if the bilinear space i s® A R is positive definite (negative definite, ...) over R in the classical sense. Such homomorphisms a always exist [4, §4]. If E is a quadratic space over A, i.e. a projective module equipped with a quadratic form q: E^A, such that the associated bilinear form is nondegenerate [cf. 7, p. 110], then we denote the associated bilinear space by £. For any signature a of A we shortly write a(E) instead of a(E\ and we call E positive definite (negative definite, ...) if E has this property. Lemma 1. Let E be a quadratic space which is positive definite at a. Then cr(c) = l for every unit c of A represented by E. PROOF. Suppose that a(c) = — l. Then the bilinear space F:=£±( — c) is positive definite at a. Now F represents the element 2c—c=c. Thus (r(c) = + l [6, Lemma 2.3], a contradiction. *) Supported in part by an NSF grant and by a Fulbright travel grant. We choose a natural number A ^ l such that both 2A —1 and 4A—1 are units in A, which is always possible [5, p. 49], and introduce the binary quadratic space G with basis u, v over A and quadratic form q(Au+fiv) = k2+Xn+ii*h. As in [5] this space G will play a useful role in our present paper. Lemma 2. a(G)=2 for every signature a of A. PROOF. A S observed in [5, p. 55f] ÖJL<~ 1> - <1, 2h - 1 , (2* -1)(1 -4fc>. Apply a t a this relation. We now turn to Rosenberg—Ware's first theorem [8, Th. 2.2]. We do not need to discuss the statements (3)—(5) in this theorem, since this is already done there [7, Remark 2.3]. We stick to Rosenberg—Ware's notations. In particular we denote by X the space of signatures of A and for a a unit of A by W{a) the clopen set of all a in X with <x(a) = —1. Theorem 1. If the sets W(a) from a basis of X(WAP)9 then every clopen set of X has the form W(a) with some a in U(A) (SAP). To prove this we look at the step (1)=>(2) in Rosenberg—Ware's proof of their theorem 2.2. We arrive at the relation 2mE±2mF~2m+n(\) with the bilinear spaces E, F introduced there {~ denotes Witt equivalence}. From this we obtain a Witt-equivalence 2m(E®G)L2m(F®G) ~ 2m+nG of quadratic spaces (cf. e.g. [7, p. I l l ] for the tensor product of a bilinear and a quadratic space). Since the cancellation law holds true for quadratic spaces over A [2], this relation implies an isometry 2m(E®G)±2m(F®G) ^ 2m+nG±_2m+nH, with H the quadratic hyperbolic plane over A. By [1, Satz 2.7] there exists a unit c of A such that — c is represented by 2m(E®G) and c is represented by 2 m (F®G). Now using Lemma 1 and Lemma 2 we obtain as in Rosenberg—Ware's proof that W(aj)C) ...riW(a„) = W(c). With more effort we obtain a proof of Theorem 1 more generally for A a con-nected semilocal ring equipped with an involution. Let X denote the space of signatures of (A, / ) , cf. [6], and let X0 denote the space of signatures of the fixed ring A0 of Then X is a closed subspace of X0, cf. [6, p. 211]. Finally we denote for c in A the norm c«/(c) by Nc. Lemma 3. Assume that all the fields A0/m with m running through the finitely many maximal ideals of A0 contain at least four elements. Let E be a bilinear space over A0 such that the hermitian space E®AoA is Witt-equivalent to zero. Then there exists units cl9 . . . , c r of A such that (*) <1, Nc^®...®(l, Ncr)®E^ 0 over A0. PROOF. The kernel of the natural map from W(A0) onto W(A9 J) is the ideal generated by the classes of the spaces (1, — Nc) with c running through the units of A [6, Prop. 2.5]. Thus E~(aJ(l9 -Nc1)±...±(ar)(\9 -ATcr> with units a, of A0 and units ct of A. Clearly the relation (*) above holds true with these units cl9 cr. From the description of the kernel of the map from W(A0) onto W(A9 J) just given it is further clear, that under the assumptions about A0 in Lemma 3 a sig-nature a of A0 lies in X if and only if a(Nc) = \ for every c in U(A) [6, Cor. 2.6]. We now assume that the space A" of signatures of (A9 J) has the property WAP9 saying that the clopen sets W(a):= {<reX\a(a) = -l} with a in U(A0) form a basis of X. We want to prove that these sets W(a) are already all clopen sets of X (SAP). It suffices to find for given units a l 9 a n and b l 9 b „ of A0 with W(aJ 0 ... 0 W(an) = W(bJ U . . . U W(bn) a unit c of A0 such that this set coincides with W(c) [8]. As in [8] we introduce the bilinear spaces £:=<g)<l , -a , .> , F:=<g)<l,-&,> over A09 and as in [8] we see that (2mE±2mF)®AoA - 2W+M<1> over (A9 J) with m some natural number. Assume now that all fields A0/m contain at least 4 elements. By Lemma 3 there exists a bilinear space L:=(l9NCl)®...®(l9Ncr) with units cl9...9cr of A suchthat 2mE®L±2mF®L - 2 w + n L over A0. We multiply this relation with the binary quadratic space G from above and learn that the quadratic space 2 m E ® L® G _L 2 m F ® I ® G is isotropic. Thus there exists a unit c of A0 suchthat 2mE®L®G represents — c and 2mF®L®G represents c. Now observing that a(L)=2r and a(G)=2 for any a in X we see as in [8] that W{a^C[...C\ W(an) coincides with W(c). This finishes our proof that WAP implies SAP in the case that all fields A0/m contain at least 4 elements. We shall obtain the proof in general immediately from this and the following lemma. Lemma 4. Let {A, J) be an arbitrary semilocal *) ring with involution. Let C denote the extension A[T]/(f(T)) with2) f(T) = T3+6T2+29T+l, and let t denote the image of T in C. Let J' denote the involution of C which extends J and maps t onto itself Then every residue class field of the fixed ring C 0 = A0[t] = A0[T]/(f(TJ) of J' contains at least 1 elements. Every signature a of A0 extends to a unique signature of C 0 , denoted by a\ and a is a signature of (A, J) if and only if a' is a signature of (C, / ' ) . For the regular norm NCo/Ao(c) of a unit3) c of C0 we have •(c) = <r(AWc)). The proof of this lemma will be given below. Let X' denote the space of sig-natures of (C, J') and for b a unit of C 0 let W'{b) denote the set of all T in X' with T(6) = — 1. By our lemma we have a canonical homeomorphism o-'—a from X' onto X and this homeomorphism maps W'(b) to W(NCo/Ao(b)). Further-more the inverse image of a subset W(a) of X, a in U(A0)9 is the set W'(a). Thus it clearly suffices to prove WAP=>SAP for ( C , / ' ) instead for (A, J). This had been done above since all residue class fields of C 0 contain at least 4, in fact 7, elements. It remains to prove Lemma 4. Our polynomial f(T)=T3+6T2+29T+l is irreducible over the prime fields F 2 , F 3 , F 5 . Thus indeed all residue class fields of A0 contain at least seven elements. The other statements of Lemma 4 will now be proved by a transfer method similar to [4, § 3]. We introduce the y4-linear form s: C-+A with $(1) = 1, s(t)=s(t2)=0 and the restriction s0: C0^A0 of s to an y40-linear form on C 0 . The hermitian form s(xJ/(y)) on the ^4-module C is non degenerate and thus s induces a transfer map s*:W(C,J') + W(A,J) sending the class of an hermitian space (E, $) over (C, J') to the class of the hermitian space (E, so$) over (A, J). In the same way we obtain a transfer map st:W(C0)-»W(A0). Since [C0:A0] is odd every signature of A0 extends to at least one signature of C 0 [6, p. 236]. The argument in [6] also shows, that every signature of A extends to at least one signature of C. *) The proof below shows that Lemma 4 remains true for A an arbitrary commutative ring with 1, if we use the more general concept of signatures developed in [4]. 2) cf. the cubic polynomial in [3, p. 26]. 3) J V c o M o ( c ) = ( t e t e r m i n a n t ° * t h e -linear map cx on C 0 . We now consider a fixed signature a of A0 and a fixed extension T of a to C 0 . We choose a homomorphism ß from C 0 into a real closed field Ä inducing ry i.e. such that the diagram W(o £ . wen) commutes, with Q the unique signature of R, (cf. [4, § 4] for the existence of ß). Let a: A-+ R denote the restriction of ß to A0. Our polynomial f(T) has pre-cisely one root in the field R of real numbers {Observe that f'(T) has no root in R}. Thus f(T) has also a unique root in R9 and we learn that ß is the only homomorphism from C 0 to R extending a. There exist two more homomorphisms from C 0 to i ? ( V - l ) over a. Let y be one of them. The commutative diagram A0 - R with <p the inclusion map and q>' the diagonal embedding is a tensor product dia-gram for C 0 and R over A09 and we identify RXR(Y^l) with C 0 ® A o Ä in this way. The /Minear map ^ 0 ®id from this tensof product to R decomposes into a pair of Ä-linear forms sx:R + R9 s2:R{^T) + R9 and we have sx(x)=bx with some b^O in R. Since the diagram W(C0) ^ >W(C0®R) *of J(50®1)* W(A0) W(R) «* commutes we obtain for z in W(C0) a+s0*(z) = <b)^(z)+5 2* n(z). But the additive map from ^ ( Ä ( ^ T ) ) = Z / 2 Z to * H R ) = Z must vanish,, and we have «*s0*(z) = (b)ß*{z). Applying Q we obtain, since e ° ß * = T > e°«*=ff: ÖTSJ(Z) = Q(b)T(z). Now the /l0-bilinear form s0(xy) on C 0 has with respect to the basis 1, —t2 the matrix I 0 0 0 0 1 0 1 6, and thus s£ maps the unit element of W(C0) to the unit element of W(A0). Inserting z=l in the formula above we see that Q (b) = 1, hence ( * ) <"J(z) = T(Z) for every z in W(C0). In particular T must be the only signature of C 0 extending a, since any other extension of a has to fulfill the same equation (*). Now Lemma 4 is proved up to the last statement. For c a unit of C 0 we have a A W < 0 = i8(c)^(/n)/Ä(y(c)). Applying Q we obtain indeed <rMCofAo(c) = T(C). Remark. It is now easy to deduce from (*) the analogous formula as*(w) = T(W) for w in W(C, / ' ) and c a signature of (C, 7). Apparently Lemma 4 is just a special case of a theory of signatures of "Frobe-nius extensions" which I hope to make explicit in the near future. We now turn to Rosenberg—Ware's Theorem 3.1. The involution of A is assumed to be trivial, since we want to avoid the complications of "quadratic forms over a ring with involution". Theorem 2. The following statements are equivalent for A an arbitrary semi-Jocal ring. SAP: Every clopen set of X has the form W(a). HMP: If a quadratic space E over A is indefinite at all signatures of A then mE is isotropic for some natural number m. We first prove SAP=>HMP. We need the following lemma which will be proved below. Lemma 5. If A is an arbitrary commutative ring and z is an element of the quadratic Witt group WQ(A) [7, p. 112] which has image zero in W(A), then 8z=0. Let £ be a quadratic space of rank n over our semilocal ring A and assume \a(E)\^n—2 for every a in X. Using SAP we produce as in [8, § 3] a bilinear space F=(bl9 . . . ,6 n - 2 ) o v e r ^ suchthat <r(F)=a(E) for all a in X. By Lemma 2 the quadratic spaces 2E and F® G, with G as above, have again the same value for all G in X. Thus for some m^O the bilinear spaces 2m+1£ and 2mF®G are Witt-equivalent. By Lemma 5 the quadratic spaces 2m+*E and 2 m + 8 i r ® G are Witt-equivalent. Since 2m+zF<g>G has smaller rank than 2m+*E and cancellation holds true for quadratic spaces we see that 2 m + 4 £ is isotropic. The proof of Lemma 5 is easy. Let L denote the unique positive definite quad-ratic space of rank 8 over Z . As is well known £~8(1 ) . Indeed, the natural map from W(Z) to W(R) is an isomorphism [7, p. 90]. If now E is a quadratic space over A with Z ~ 0 , then4) 0 ~ £®L ^ Z®E ~ SE. Remark. The cokernel of the canonical map from WQ(A) to W(A) is also killed by 8. Indeed for every bilinear space F over A we have %F ~ (F®L)~. We finally prove HMP=>SAP along the lines of [8]. For arbitrary units a, b of A we have to find a unit c of A with W(a)C\ W(b)= W(c). We introduce the quadratic space E:= < - l , a , b, ab)®G and observe that a(E) = ± 4 for every <j in X. By (HMP) 2mE is isotropic for some natural number m. Assume now that all residue class fields of A have at least 3 elements. Then we obtain as in [8] by use of [1, Satz 2.7] an equation t = 5 + with units /, s, c of A such that — t is represented by 2 m G, s is represented by 2m(a)G and c is represented by ( l ,a)G. Using Lemma 1 we deduce from this equation as in [8] that W(a)H W(b)= W(c). If A has residue class fields with 2 elements, then we switch over to the cubic extension C of Lemma 4, the involution J there now being trivial. We have units /, s, c of C with t=s+bc and — /, s, c represented by the spaces listed above over C. Thus we have in the space X' of signatures of C WXa)C\W'(b) = W\c). Applying the canonical homeomorphism from X' to X we obtain by Lemma 4 W(a)C)W(b) = W(Nc/A(c)). This finishes the proof of Theorem 2. In Theorem 2 we considered a Hasse—Minkowski principle for quadratic spaces. Actually this Hasse—Minkowski principle is equivalent to the analogous principle for bilinear spaces. This follows immediately from Lemma 2 above and the following two observations: Lemma 6. Let E be a bilinear space over A such that the quadratic space E®G is isotropic. Then 6E is isotropic. This had been proved in [5, § 5]. 4) We write £ ® £ instead of £®(L®ZA), etc. Lemma 7. Let E be a quadratic space over A such that the associated bilinear space E is isotropic. Then 2E is isotropic. Indeed, E contains a primitive vector x with 2q(x)=0. Thus the primitive vector x@x of the quadratic space E±E is isotropic. References [1] R. BAEZA, M . KNEBUSCH, Annullatoren von Pfisterformen über semilokalen Ringen, Math. Z. 140 (1974), 41—62. [2] M. KNEBUSCH, Isometrien über semilokalen Ringen, Math. Z. 108 (1969), 255—268. [3] M . KNEBUSCH, Runde Formen über semilokalen Ringen, Math. Ann. 193 (1971), 21—34. [4] M. KNEBUSCH, Real closures of commutative rings I, / . reine angew. Math. 274/275 (1975), 61—89. [5] M . KNEBUSCH, Generalization of a theorem of Artin—Pfister to arbitrary semilocal rings, and related topics, / . of Algebra 36 (1975), 46—67. [6] M. KNEBUSCH, A. ROSENBERG, R. WARE, Signatures on semilocal rings, J. of Algebra 26 (1973), 208—250. [7] J. MILNOR, D. HUSEMOLLER, Symmetric bilinear forms, Ergebnisse Math. u. Grenzgeb. Bd 73, Springer, New York—Heidelberg—Berlin 1973. [8] A. ROSENBERG, R. WARE, Equivalent topological properties of the space of signatures of a semi-local ring, Publ. Math. (Debrecen) 23 (1976), 283—289. INSTITUTE FOR A D V A N C E D STUDY, PRINCETON N . J., USA A N D FACHBEREICH M A T H E M A T I K DER UNIVERSITÄT REGENSBURG, G E R M A N Y . (Received May 17.1976.) 

Remarks on the paper "Equivalent topological properties of the space of signatures of a semilocal ring" by A. Rosenberg and R. Ware

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Remarks on the paper "Equivalent topological properties of the space of signatures of a semilocal ring" by A. Rosenberg and R. Ware

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