Rings of weak dimension 1 and syzygetic ideals

Rings of Weak Dimension One and Syzygetic Ideals Francesc Planas-Vilanova Abstract We prove that rings of weak dimension one are the rings with all (three-generated) ideals syzygetic. This leads to a characterization of these rings in terms of the AndreQuillen homology. Let I b e an ideal of a commutative ring A . There is a canonical morphism of graded A - algebras  : S ( I ) ! R ( I ) from the symmetric algebra of I onto its Rees algebra. The ideal I is said to b e of linear typ e if  is an isomorphism. If  2 : S 2 ( I ) ! I 2 is an isomorphism, I is said to b e syzygetic. In  C ] (Theorem 4), Costa showed that a domain A is Pr ufer if and only if I is of linear typ e for every two-generated ideal I of A and I is syzygetic for every three-generated ideal I of A .In this note we show that the preliminary hyp othesis that A is a domain can be removed bychanging the Pr ufer condition to the condition wd ( A )  1, weak dimension of A one or less. Moreover, the condition that every two-generated ideal of A b e of linear typ e is not necessary. Concretely, Theorem 1 Let A be a commutative ring. The fol lowing conditions are equivalent: i ) wd ( A )  1 . ii ) Every ideal of A is of linear type. iii ) Every ideal of A is syzygetic. iv ) Every three-generated ideal of A is syzygetic. Recall that wd ( A ) is the supremum of the at dimensions of all A -mo dules. Von Neumann regular rings are those of weak dimension zero. Semihereditary rings (i.e. rings with all its nitely generated ideals pro jective) have weak dimension one or less. In fact, A is a semihereditary ring if and only if wd ( A )  1and A is coherent. A semihereditary domain is called a Pr ufer ring. For a domain A , to be Pr ufer is equivalentto wd ( A )  1 (see  B ] 1995 Mathematics Subject Classication . Primary 13F05, 13A30. Secondary 13D03. Key words and phrases . Ideal of linear typ e, syzygetic ideal, weak dimension, homology of commutative rings. or  R ]). In particular, if A is a domain, Theorem 1 characterizes Pr ufer rings as domains with every three-generated ideal b eing syzygetic. Before proving Theorem 1 we need the following two lemmas. Lemma 2 Let ( A m ) bea local ring. Supose that there exist two nonzero elements a b 2 A with ab =0 .If ( a ) is syzygetic, then ( a b ) it is not. Proof . Let I =( a b )  m b e the ideal of A generated by the zero divisors a , b .If x 2 ( a ) \ ( b ), x = ac = bd , and multiplying by a , c 2 (0 : a 2 ). Since ( a ) is syzygetic, c 2 (0 : a ) (see  V ], page 31) and x = ca =0. Therefore, ( a ) \ ( b )=0. Let 0 ! Z 1 ! A 2 f ! I ! 0be the free presentation of I dened by f ((1  0)) = a , f ((0  1)) = b . Consider 0 ! N ! A  X Y ] s ! R ( I ) ! 0, the induced free presentation of R ( I )=  q  0 I q t q , dened by s ( X )= at , s ( Y )= bt . If I =( a b )were syzygetic, the quadratic relation XY on a , b could b e written in terms of linear relations on a , b ( V ], page 29), i.e. XY =( a 1 X + b 1 Y )( c 1 X + d 1 Y )+  +( a r X + b r Y )( c r X + d r Y ) (1) with c i X + d i Y 2 N 1 = Z 1 . In particular, c i a = ; d i b 2 ( a ) \ ( b ) = 0. Therefore, c i a = d i b = 0 and as a b 6 =0, then c i d i would b e zero divisors, in particular, elements of m . Comparing the co ecients of XY in b oth members of (1) we would get the contradiction 1= P r i =1 a i d i + b i c i 2 m . Lemma 3 Let ( A m k ) be a local ring. Let I beanon principal nitely generated ideal of A . If I is syzygetic, then I 2 it is not. Proof .As I is not principal, dim k ( I= m I ) = n > 1. By hyp othesis,  2 is an isomorphism and hence,  2  1 k is also an isomorphism. Therefore, dim k  I 2 = m I 2  =dim k  S k 2 ( I= m I )  = n ( n +1) 2 = p Thus, dim k  S k 2 ( I 2 = m I 2 )  = p ( p +1) 2 = n ( n +1)( n 2 + n +2) 8 . Since  4 is an epimorphism,  4  1 k is also an epimorphism. So, one has dim k  I 4 = m I 4   dim k  S k 4 ( I= m I )  = n ( n +1)( n 2 +5 n +6) 24 Finally, one observes that if n 6 = 0  1, then 3( n 2 + n +2) > ( n 2 +5 n +6). In particular, S k 2 ( I 2 = m I 2 ) 6' I 4 = m I 4 and S A 2 ( I 2 ) 6' I 4 . Proof of the theorem. Recall wd ( A )  1isequivalenttoevery ideal of A b eing at ( R ]). If I is a at ideal of A , then I is an ideal of linear typ e (see Prop osition 3  MR ]or P ]). This proves i ) ) ii ). Let us see iv ) ) i ). 2 Let I b e an ideal of A . To show I is at, one can supp ose I is nitely generated since any ideal is the direct limit of nitely generated ideals and the direct limit of at mo dules is again a at mo dule. Write I =( x 1 ::: x n ) and let us see that I m is a free A m -mo dule for every maximal ideal m of A .If I 6 m , I m = A m . If I  m , consider J =( x 1 x 2 )  I . By hyp othesis iv ), J and J 2 are syzygetic ideals of A . Lo calizing at m , lemma 3 provides an element z 2 J with J m =( z 1 ). Therefore I m =( z 1  x 3 1 ::: x n 1 ). Now, take J 0 =( z x 3 )  I . Rep eating the pro cess we deduce that I m is a principal ideal of A m . By hyp othesis iv ) and lemma 2, A m is a domain, in particular, I m is an ideal generated byanonzero divisor, i.e. afree A m -mo dule. Remark 4 From Lemma 2 and Costa's Theorem 3 of  C ] one deduces that for a commutativering A to b e lo cally an integrally closed domain is equivalent to b eing of linear typ e for every two-generated ideal of A . Moreover, the same example given by Costa in his pap er shows that this last condition is strictly stronger than every two-generated ideal of A b eing syzygetic. In terms of the Andre-Quillen homology (see  A ]) and as a corollary of Theorem 1, rings of weak dimension one are characterized as follows: Corollary 5 Let A be a commutative ring. The fol lowing conditions are equivalent: i ) wd ( A )  1 . ii ) H 2 ( A B   )=0 for every quotient ring B = A=I of A . iii ) H 2 ( A B  B )=0 for every quotient ring B = A=I of A by a three-generated ideal I of A . Proof . It follows from the fact that if I is an ideal of A , B = A=I and  2 : S 2 ( I ) ! I 2 is the canonical morphism, then H 2 ( A B  B )=Ker  2 .Moreover, if I is syzygetic, H 2 ( A B  W )= Tor B 1 ( I=I 2 W )forany B -mo dule W (see, for instance,  BR ]). Acknowledgements :Iamvery grateful to Jose M. Giral for the very helpful discussions concerning this pap er. References  A] M. Andre: Homologie des algebres commutatives. Grundlehren 206 . Heidelb erg: Springer 1974.  BR] J. Barja, A. G. Ro dicio: Syzygetic Ideals, Regular sequences and a Question of Simis . J. Algebra 121 (1989), 310-314.  B] N. Bourbaki: Algebre. Chapitre 10. Algebre homologique. Masson. Paris, 1980.  C] D. Costa: On the Torsion-Freeness of the Symmetric Powers of an Ideal . J. Algebra 80 (1983), 152-158. 3  MR] A. Micali, N. Roby: Algebres symetriques et syzygies . J. Algebra 17 (1971), 460-469.  P] F. Planas Vilanova: Ideals de tipus lineal i homologia d'Andre-Quillen. Ph.D. Thesis. Universitat de Barcelona, 1994.  R] J. Rotman: An Introduction to Homological Algebra. Academic Press, 1979.  V] W. V. Vasconcelos: Arithmetic of Blowup Algebras. LMS Lecture Note Series 195 .Cambridge University Press, 1994. Departament de Matematica Aplicada I. ETSEIB. Universitat Politecnica de Catalunya. Diagonal 647, E-08028 Barcelona . E-mail address: [email protected] c.es 4