Douglas L. Costa

The construction D + XDS[X]

If D is a commutative integral domain and S is a multiplicative system in D, then Tfs) = D + XD,[X] is the subring of the polynomial ring D,[X] con- sisting of those polynomials with constant term in D. In the special case where S = D* = D\(O), we omit the superscript and let T denote the ring D + XK[XJ, where K is the quotient field of D. Since Tfs) is the direct limit of the rings D[X/s], where s E S, we can conclude that many properties hold in T@) because these properties are preserved by taking polynomial ring extensions and direct limits. Moreover, the ring Tcs) is the symmetric algebra S,(D,) of D, considered as a D-module. In addition, Ds[Xj is a quotient ring of Tts) with respect to S; in fact, in the terminology of [lo], Tfs) is the composite of D and D,[iYj over the ideal XDJX]. (The most familiar of the composite constructions is the so-called D + M construction [l], where generally M is the maximal ideal of a valuation ring.) The ring T ts), therefore, provides a test case for many questions about direct limits, symmetric algebras, and composites. The state of our knowledge of T is considerably more advanced than that of VJ; generally speaking, we often show that a property holds in T if and only if it holds in D. In other cases we show that Tcs) does not have a given property if D, # K. For example, if T(S) is a Priifer domain, then D,[xJ is a Prtifer domain and D, is therefore equal to K. We show that T is Priifer (Bezout) if and only if D is Prtifer (Bezout). Yet Tts) is a GCD-domain if D is a GCD- domain and the greatest common divisor of d and X exists in

Parameterizing families of non-noetherian rings

(1994). Parameterizing families of non-noetherian rings. Communications in Algebra: Vol. 22, No. 10, pp. 3997-4011.

Seminormality and root closure in polynomial rings and algebraic curves

Let D be an integral domain with identity and let K be the quotient field of D. Then D is said to be root closed if whenever 01 E K with (Y~ E D for some positive integer n, then 01 E D. The domain D is called (2, 3)-closed if whenever 01 E K with a2, a3 E D, then OL E D and D is called F-closed if whenever OL E K with nar E D for some positive integer n and OLD, 01~ E D, then ol E D. Clearly, if D is root closed, then D is (2,3)-closed and if D is (2,3)-closed, then D is F-closed. The property of being root closed arose in Sheldon’s work [7] on how changing D changes the quotient field of D[[Xj]. As for (2, 3)-closure, its significance is due to the fact that an integral domain D is (2, 3)-closed if and only if D is seminormal if and only if Pit(D) = Pic(D[X, ,..., X,]), where Pit denotes the Picard group [4, Theorem 11. Concerning F-closure, it is shown in [l] that the domain D is F-closed if and only if D[X’j is D-invariant-i.e., if and only if whenever (D[Xj) [XI ,..., X,l ho 5’[Y, ,..., Yn], then S is D-isomorphic to D[x]. As noted in the preceding paragraph, these properties are related, although in their original non-arithmetic forms, they did not appear to be. It is a consequence, albeit a deeply hidden one, of the proof of Theorem 1 of [4] that if D is (2,3)-closed, so is D[Xj. In this paper, we give arithmetic proofs that each of these three properties respects polynomial extension. In fact, we do this in greater generality and our approach is a unifying one in that we show that if an integral domain D is “n-root closed”, then so is D[Xj. The notion of ‘%-root closed” is then utilized to delineate the arithmetic distinction between normality and seminormality for algebraic curves. Recall that Bombieri [2] has shown that the geometric distinction between a normal curve and a seminormal curve is that the seminormal curve may have “ordinary” singular points. We prove that the coordinate ring of an irreducible algebraic curve over an algebraically closed field K is integrally closed if and only if it is (2, 3)-closed plus n-root closed for some n prime to the characteristic of K. In the process of showing that our results cannot be extended to arbitrary reduced rings, we give a negative answer to a question implicit in [5, Ex.11,

Retracts of polynomial rings

Let R be a commutatrve rmg and A a commutative R-algebra. A is an extension of R if rt contains R as a subrmg. A 1s a projective R-algebra If rt is a projectrve object in the category of commutative R-algebras. As in the category of Rmodules, projective R-algebras are retracts of free R-algebras, that is, retracts of polynomial rings over R The purpose of this paper IS to discuss the question: Is every projective extension A of R the symmetric algebra of an R-module ? This questron is raised implicitly in [3] by an incomplete diagram of logical implications. In Section 1 we study retracts of arbitrary rings. In Section 2 we digress briefly to present some examples which distmguish between retracts and inert subrmgs. In Section 3 we attack the mam questron, obtaining some positive results on retracts of R[X,I, retracts of R[X, , X,J, where R 1s artiman, and retracts of kI3 7 . , X,], where k is a field. We shall see in Section 4, however, that the answer to the general question is negative. In fact an example due to Hamann [4] yields a retract of RIXl , X;] whrch 1s an extension of R but not a symmetrrc algebra, where R may be taken to be a one dimensional local domain. We wash to thank M. Hochster for kindly allowing us to include Theorem 1.10 in our paper.

On the normal subgroups of SL(2, A)

Let A be a commutative ring having 2 in the stable range. Let N be a subgroup of SL(2, A) having level ideal J. It is shown that if either A is von Neumann regular or 2 is invertible in A, then N is normal in SL(2, A) if and only if N contains the commutator group H(J) = [E(2, A), L(2, A, J)]. Structure theorems for normal subgroups of SL(2, A) are deduced from this result.

On the normal subgroups ofGL(2, A)

In this paper we determine the normal subgroups of GL(2, A) for A a commutative local ring. The papers of Klingenberg [a], Lacroix [3], and our paper [ 1 ] constitute the previous work on this problem. As a consequence of our results, we are able to describe the normal subgroups of GL(2, A) for all Artinian rings A; hence, in particular, for all finite rings A. If N is a subset of GL(2, A), we denote by I(N) the level ideal of N, that is, the smallest ideal of A such that N consists of scalar matrices modulo I(N). For J an ideal of A, GC(2, A; J) = { TE GL(2, A) 1 I(T) E J> is the normal subgroup consisting of all matrices which are scalars modulo J, while SL(2, A; J) = { TE GL(2, A) 1 det T= 1 and Tz Imod J> is the principal congruence subgroup for J. Klingenberg proved that if i E A and the residue field of A is not G8’(3), then a subgroup N of GL(2, A) is normal if and only if SL(2, A; Z(N)) EN. Equivalently, N is normal if and only if SL(2, A; J) E Nc GC(2, A; J) for some ideal J, in which case J= Z(N). His paper also shows that this criterion for normality holds in GL(n, A) for n > 3 without any hypotheses on the local ring A. Our paper [ 1 ] characterizes normal subgroups of GL(2, A) for any %,-ring A with

Sequences of linear type

6 A. Since every local ring is an SR,-ring, this leaves only the case of local rings in which 2 is not a unit, i.e., in which the residue field has characteristic 2. Lacroix, in [3], deals with local rings in which the residue field has characteristic 2 but is not GI;(2), obtaining the same normality criterion as Klingenberg. This leaves open the case of local rings with residue field GE(2). We shall use the “method of reduction” employed in Cl] to give a normality criterion for subgroups of GL(2, A) for A a local ring with GJ’(2) as residue field. Moreover, the method will allow us to recapture the results of Klingenberg and Lacroix with great efficiency. Thus, we will be able to 395 0021~8693/90

Zero-dimensionality and the GE2 of polynomial rings

3.00

ON THE SPECTRUM OF THE GROUP RING

Let A be a commutative ring, 91 an ideal of A, S,(2l) the symmetric algebra of

Seminormality and projective modules over polynomial rings

8, and R,,,(Vl) the Recs algebra of VI. There is a canonical surjection x S,(%) --+ K,(%) which is homogeneous of degree zro and therefore gives canonical surjcctions r,: F,(%) + ‘91’ for every integer t 3 0. Many recent papers [ 1, 2, 4: 5, 7, Cc] have investigated the problem of determining which ideals 91 have the property that “A is an isomorphism. Such ideals have been dubbed ideals of‘ lineur type’ [S]. For example, Huneke showed that every ideal generated by a d-sequence is of linear type [71. If (II is generated by n elements a, ,..., a,, then for t 3 I there is an exact scquencc