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The Integral Closure of Subrings Associated to Graphs
199, 281]289 Ž1998. JA977171
JOURNAL OF ALGEBRA ARTICLE NO.
The Integral Closure of Subrings Associated to Graphs Aron Simis* Departamento de Matematica, Uni¨ ersidade Federal da Bahia, 40210, ´ Sal¨ ador, Bahia, Brazil
Wolmer V. Vasconcelos† Department of Mathematics, Rutgers Uni¨ ersity, New Brunswick, New Jersey 08903
and Rafael H. Villarreal ‡ Departamento de Matematicas, Centro de In¨ estigacion ´ ´ y de Estudios A¨ anzados del IPN, Apartado Postal 14-740, 07000, Mexico City, D.F., Mexico ´ Communicated by Craig Huneke Received February 3, 1997
Let G be a graph on the vertex set V, and R s k w V x a polynomial ring over a field k. We give a very explicit description of the integral closure of the monomial subring k w G x ; R, generated by the squarefree monomials of degree two defining the edges of the graph G. Q 1998 Academic Press
1. INTRODUCTION Let R s k w x 1 , . . . , x n x be a polynomial ring over a field k, and let F s f 1 , . . . , f q 4 be a finite set of monomials in R. The monomial subring spanned by F is the k-subalgebra, k w F x s k f 1 , . . . , f q ; R. * Partially supported by CNPq, Brazil. E-mail address: [email protected]. † Partially supported by the NSF. E-mail address: [email protected]. ‡ Partially supported by CONACyT and SNI, Mexico. E-mail address: [email protected]. ´ 281 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
282
SIMIS, VASCONCELOS, AND VILLARREAL
In general, it is very difficult to certify whether k w F x has a given algebraic property}e.g., Cohen]Macaulay, normal}or to obtain a measure of its numerical invariants}e.g., Hilbert function. This arises because the number q of monomials is usually large. It is similarly difficult to carry out manipulations such as those required in the Noether normalization process or to find a set of generators for the integral closure k w F x of k w F x. The focus of our attention is the algebra k w F x . In general, it is difficult to predict which elements lie in the integral closure of an affine domain, but in the case of k w F x one has the following general description: k w F x is also a monomial subalgebra that is generated by monomials f s x 1b1 ??? x nb n s x b ,
b g Nn,
with the following two properties Žsee w7, Chap. 7x for an algorithm.: v v
f s Ł f ia i , a i g Z, f m s Ł f ib i , m, bi g Zq.
The first condition asserts that f lies in the field of fractions of k w F x, the other being the special form the integrality condition takes for such elements. Alternatively, kw F x s k
x a N a g Z A l Rq A 4 ,
where A denotes the exponent vectors of the monomials in F, and Rq A is the cone generated by A. In this note we determine k w F x , when F is generated by squarefree monomials of degree two. In this case the underlying graph theoretic aspects are very helpful toward a solution. Consider any graph G with vertex set V s x 1 , . . . , x n4 . Let F be the set of all monomials x i x j in R, such that x i x j 4 is an edge of G. For simplicity of notation we denote k w F x by k w G x. To explain our result we begin by exhibiting some monomials in the integral closure of k w G x. Given an induced subgraph w of G consisting of two edge disjoint odd cycles Z1 s
SUBRINGS ASSOCIATED TO GRAPHS
283
z 0 , z1 , . . . , z r s z 0 4 and Z2 s z s , z sq1 , . . . , z t s z s 4 , joined by a path Žthose subgraphs will be called bow ties ., we associate the monomial Mw s z1 ??? z r z sq1 ??? z t . We observe that Z1 and Z2 are allowed to intersect and that only the variables in the cycles occur in Mw , not those in the path itself. Our main result is THEOREM 1.1. Let G be a graph and let k be a field. Then the integral closure k w G x of k w G x is generated as a k-algebra by the set B s f 1 , . . . , f q 4 j Mw N w is a bow tie 4 , where f 1 , . . . , f q denote the monomials defining the edges of G. As a corollary we have now a full characterization of when k w G x is integrally closed. A related issue, was raised in w6x, where the authors considered the Rees algebra of the ideal I s I Ž G . of R generated by the f i Žintroduced in w8x.. This ring, Rw It x, is also a monomial subalgebra of Rw t x, R w It x s k x 1 , . . . , x n , f 1 t , . . . , f q t , but now generated by forms of degree 1 and 3. It was conjectured in w6x that the integral closure of Rw It x would be generated by elements similar to bow ties. This will follow from our main result. As a consequence, we show that if G is a connected graph, then k w G x is normal if and only if I Ž G . is a normal ideal.
2. A COMBINATORIAL DESCRIPTION OF NORMALIZATIONS Let us fix some notation that will be used throughout. Our main references for graph theory and affine semigroups are w4; 2, Chap. 6x, respectively. Let G be a graph. A walk of length n in G is an alternating sequence of vertices and edges w s ¨ 0 , z1 , ¨ 1 , . . . , ¨ ny1 , z n , ¨ n4 , where z i s ¨ iy1 , ¨ i 4 is the edge joining ¨ iy1 and ¨ i . A walk is closed if ¨ 0 s ¨ n . A walk may also be denoted ¨ 0 , . . . , ¨ n4 , the edges being evident by context. A cycle of length n is a closed walk, in which the points ¨ 1 , . . . , ¨ n are distinct. A path is a walk with all the points distinct. A tree is a connected graph without cycles and a graph is bipartite if all its cycles are even. A vertex of degree one will be called an end point. If e is an edge of G, we denote by G _ e4 the spanning subgraph of G obtained by deleting e. The removal of a vertex ¨ from graph G results in
284
SIMIS, VASCONCELOS, AND VILLARREAL
that subgraph G _ ¨ 4 of G consisting of all the vertices in G except ¨ and all the lines not incident with ¨ . As usual, V Ž G . and EŽ G . stand for the vertex set and edge set of G, respectively. PROPOSITION 2.1. Let G be a connected graph and let k be a field. If G has at most one odd cycle, then k w G x is normal. Proof. We may assume that G has a unique odd cycle, for otherwise G is a bipartite graph and k w G x is normal by w6, Theorem 5.9x. Pick an edge e of G which is on the unique odd cycle and set H s G _ e4 . Notice that H is a connected bipartite graph because e is not a bridge Žit is on a cycle.. Hence k w H x is a normal domain of dimension dim k w G x y 1 Žsee w8, Lemma 3.1x.. Consider the epimorphism w
k w H xw t x ª k w G x ª 0,
induced by w Ž t . s f e ,
where t is a new indeterminate and f e is the monomial corresponding to e. Since k w H xw t x and k w G x are both Noetherian domains of the same dimension, w is an isomorphism. Therefore k w G x is normal. LEMMA 2.2. Let G be a graph without e¨ en cycles. Then the intersection of any two cycles is either a point or the empty set. Proof. It is a simple parity count. Let G be a graph on the vertex set V Ž G .. Given a subset U ; V Ž G ., the neighbor set of U is defined as N Ž U . s ¨ g V Ž G . N ¨ is adjacent to some vertex of U 4 . Notice that if Z is a cycle then Z : N Ž Z .. DEFINITION 2.3. Let G be a graph without even cycles. A cycle Z of G is said to be a terminal cycle if N Ž Z . s Z j ¨ 4 , for some ¨ f Z. LEMMA 2.4. Let G be a connected graph without e¨ en cycles. Assume that Z1 l N Ž Z2 . s B, for any two cycles Z1 , Z2 of G. If G is not a unicyclic graph, and degŽ ¨ . G 2 for all ¨ g V Ž G ., then G has at least two terminal cycles. Proof. Let Z1 , . . . , Zr be the cycles of the graph G. We now construct a new graph T having vertex set r V Ž T . s Z1 , . . . , Zr 4 j Ž V Ž G . _ D is1 Zi . .
Thus the vertices of T are the cycles of G, together with the vertices of G not lying on any cycle of G. Two points Y1 , Y2 of T are adjacent if and only if Y1 l N Ž Y2 . s B. Here we regard a vertex ¨ of G as the one point
SUBRINGS ASSOCIATED TO GRAPHS
285
set ¨ 4 . Notice that T is a tree and that the terminal cycles of G correspond to the end points of T. Hence G must have at least two terminal cycles. DEFINITION 2.5. A bow tie w of a graph G is a subgraph of G consisting of two edge disjoint odd cycles Z1 s z 0 , z 1 , . . . , z r s z 0 4
and
Z2 s z s , z sq1 , . . . , z t s z s 4
joined by a path z r , . . . , z s 4 . In this case we set Mw s z1 ??? z r z sq1 ??? z t . If w is a bow tie of a graph G, as above, then Mw is in the integral closure of k w G x. Indeed if f i s z iy1 z i , then Mw s
Ł fi Ł
i odd
i even r-iFs
fy1 i
and
Mw2 s f 1 ??? f r f sq1 ??? f t .
Hence Mw g k w G x . We fix a final element of notation by setting B s f 1 , . . . , f q 4 j Mw N w is a bow tie 4 , where f 1 , . . . , f q denote the monomials defining the edges of G. LEMMA 2.6. Let G be a connected graph without e¨ en cycles, and let V be the ¨ ertex set of G. Assume the following three conditions: Ža. Z1 l N Ž Z2 . s B, for any two cycles Z1 , Z2 of G. Žb. degŽ ¨ . G 2 for all ¨ g V. Žc. G has an e¨ en number of ¨ ertices. Then Ł ¨ g V ¨ is a monomial in k w B x. Proof. The proof is by induction on the number of cycles of the graph G. Set z s Ł ¨ g V ¨ . If G has exactly two cycles, then G is a bow tie and we can write z s f d , where f g k w G x and d s MG is the product of the vertices in the two cycles. We may now assume that G has at least three cycles. By Lemma 2.4 there are two terminal cycles Z1 s x 0 , x 1 , . . . , x r s x 0 4 ,
Z2 s y 0 , y 1 , . . . , ys s y 0 4 .
There is a path x r , x rq1 , . . . , x r 14 , so that r q 1 F r 1 , degŽ x j . s 2 for all r - j - r 1 , and degŽ x r 1 . G 3. Likewise there is a path ys , ysq1 , . . . , ys14 , so that s q 1 F s1 , degŽ y j . s 2 for all s - j - s1 , and degŽ ys1 . G 3. We may
286
SIMIS, VASCONCELOS, AND VILLARREAL
assume r 1 and s1 even, for otherwise, if r 1 is odd, we write z s Ž x 1 x 2 . Ž x 3 x 4 . ??? Ž x r x rq1 . ??? Ž x r 1y2 x r 1y1 .
Ł ¨,
¨ gV1
where V1 s V _ x 1 , . . . , x r 1y14 , and then use induction. It is not hard to show that x 1 , . . . , x r 14 l y 1 , . . . , ys1y14 s x 1 , . . . , x r 1y14 l y 1 , . . . , ys14 s B. Set
s 1 s Ž x rq1 x rq2 . ??? Ž x r 1y2 x r 1y1 . and s 2 s Ž ysq1 ysq2 . ??? Ž ys1y2 ys1y1 . . If x r 1 / ys1 then we write s
r
z s s 1 s 2 Ł yi Ł x i is1
is1
Ł ¨,
¨ gV1
where V1 s V _ x 1 , . . . , x r 1y1 , y 1 , . . . , ys1y1 4 , hence using induction we get z g k w B x. We may now assume that r 1 , s1 are even and x r 1 s ys1. Consider the subgraph H s G _ x 1 , . . . , x r 1y1 , y 1 , . . . , ys1y1 4 . If degŽ x r 1 . G 4, then applying the induction hypothesis to H we obtain z g k w B x. Assume degŽ x r 1 . s 3. Note that H has a terminal cycle Z3 s z 0 , z1 , . . . , z t s z 0 4 and there is a path z t , z tq1 , . . . , z t 14 , so that t q 1 F t 1 , degŽ z j . s 2 for all t - j - t 1 , and degŽ z t 1 . G 3; by the previous arguments we are reduced to the case r 1 , s1 , t 1 even and x r 1 s ys1 s z t 1. Observe that this forces the equality V s x 1 , . . . , x r 1, y 1 , . . . , ys1y1 , z1 , . . . , z t 1y14 . To complete the proof we use the identity r
s
z s Ž z1 z 2 . Ž z 3 z 4 . ??? Ž z t 1y1 z t1 . s 1 s 2 Ł x i Ł yi , is1
is1
to derive z g k w B x. Given a graph G we set AG s logŽ f 1 ., . . . , logŽ f q .4 , where f 1 , . . . , f q are the monomials corresponding to the edges of G and logŽ f i . is the exponent vector of f i . The affine subsemigroup of N n generated by AG will be denoted by CG . Thus CG s
Ý
Na .
ag AG
The cone generated by CG will be denoted by Rq CG ; it is equal to Rq CG s
½
r
Ý ai a i N ai g Rq , a i g AG , r g N is1
Note that k w G x is equal to the affine semigroup ring k w CG x.
5
.
SUBRINGS ASSOCIATED TO GRAPHS
287
LEMMA 2.7. Let G1 and G 2 be finite sets of monomials in disjoint sets of indeterminates. Then k w G1 j G 2 x , k w G1 x mk k w G 2 x and k w G1 j G 2 x , k w G1 x mk k w G 2 x , k w G1 x k w G 2 x , is the subring generated by the generators of k w G1 x and k w G 2 x . Proof. Set G s G1 j G 2 . By w2, Proposition 6.1.2x we have kwGx s k
x a N a g ZCG l Rq CG 4 .
On the other hand there is a decomposition ZCG l Rq CG s Ž ZCG 1 l Rq CG 1 . [ Ž ZCG 2 l Rq CG 2 . . Using this equality the assertion follows readily. Proof of the Main Result. Let V s x 1 , . . . , x n4 and E s e1 , . . . , e q 4 be the vertex set and edge set of G, respectively. The proof is by induction on q, the number of edges of G. The case q s 1 is clear. Assume q G 2 and that the result is true for graphs with less than q edges. If e i s x j , x k 4 , we set f i s x j x k . First we remove all the isolated vertices of G, without affecting the subring k w G x. By Lemma 2.7 and the induction hypothesis we may assume that G is a connected graph. We may also assume that degŽ ¨ . G 2 for all ¨ g V; because if degŽ ¨ . s 1 and f i s ¨ x j , then setting G9 s G _ e i 4 we have k w G x s k w G9xw f i x, the latter being a polynomial ring over k w G9x, hence k w G x s k w G9 x w f i x and f i is not part of a bow tie, which by induction yields the required equality. Let z s x a g k w G x be a minimal generator of the integral closure of w k G x. There is a positive integer m so that z m s f 1m l1 ??? f qm l q s f 1b 1 ??? f qb q ,
where l i g Z and bi g Zq , for all i.
Ž 1. Observe that if bi s l i s 0 for some i, then each bow tie of G _ e i 4 is a bow tie of G and the induction hypothesis yields z g k w B x. Hence we may further assume that all the monomials f 1 , . . . , f q must be present in Eq. Ž1.. There are two cases to consider. Case ŽI.. Assume that G contains an even cycle Z. We set ¨ i s logŽ f i .. For convenience we use the edges of G to represent the cycle Z by the sequence e i1, . . . , e i k 4 , where two consecutive edges are adjacent. If bi r s 0 for some 1 F r F k, then using the relation k
Ý js1
j Ž y1. ¨ i j s
k
Ý Ž y1. j log Ž f i . s 0, j
js1
Ž 2.
288
SIMIS, VASCONCELOS, AND VILLARREAL
that comes from the even cycle, we may rewrite Eq. Ž1. as z m s f 1m m 1 ??? f qm m q s f 1b 1 ??? f qb q , where m i g Z for all i , and m i r s bi r s 0, which by induction yields z g k w B x. We may now assume bi r ) 0 for all 1 F r F k. We may harmlessly assume that bi1 F ??? F bi k. Using Eqs. Ž1. and Ž2. we obtain q
ma s
k
Ý bi¨ i s bi Ý ¨ i 1
is1
j
q
js1
Ý c i¨ i s 2 bi Ý 1
i/i 1
¨ij q
j even
Ý ci¨ i , i/i 1
where the c i ’s are non-negative integers. The net effect is that now we may rewrite Eq. Ž1. as z m s f 1m d 1 ??? f qm d q s f 1d1 ??? f qd q ,
where d i g Z, d i g Zq , for all i ,
and d i1 s d i1 s 0. Hence by induction z g k w B x. Case ŽII.. Assume that all the cycles of G are odd. Let Z1 and Z2 be two distinct cycles of G, represented by the sequences of edges e i1, . . . , e i k 4 and e l 1, . . . , e l s 4 , respectively. If Z1 l Z2 / B, then according to Lemma 2.2 Z1 l Z2 s x 4 , for some x g V, say x g suppŽ f i1 . l suppŽ f l 1 .. Using the identity
Ý ¨i
j
q
j odd
Ý j even
¨ lj s
Ý
¨ij q
j even
Ý ¨l
j
j odd
we may proceed as in Case ŽI.. If Z1 l Z2 s B and N Ž Z1 . l Z2 / B, say f t s xy, with x g suppŽ f i k . and y g suppŽ f l 1 .. Again using the equality
Ý ¨i j odd
j
q
Ý ¨l j odd
j
s 2¨t q
Ý
¨ij q
j even
Ý
¨ lj ,
j even
we may proceed as in Case ŽI. to derive z g k w B x. Therefore we are reduced to the case N Ž Z1 . l Z2 s B, for any two cycles Z1 , Z2 . If x j is not in the support of f 1b 1 ??? f qb q , for some j, consider the set D s i N x j g supp Ž f i" l i . 4 . Then Ý i g D l i s 0 and bi s 0, for all i g D. Note that, for any i g D, we may eliminate the monomial f i from Eq. Ž1., hence by induction z g k w B x. As a consequence we are reduced to the case suppŽ z m . s V, that is, z s x 1a1 ??? x na n , with a i G 1 for all i. By w3, Corollary 3.6x we obtain that k w G x is generated as a k-algebra by monomials of degree at most 2Ž n y
SUBRINGS ASSOCIATED TO GRAPHS
289
u nr2v.. Altogether we get n F degŽ z . F 2Ž n y u nr2v., hence n must be even and z s x 1 ??? x n . Using Lemma 2.6 we conclude that z g k w B x, as required. COROLLARY 2.8. Let G be a connected graph and let RŽ I Ž G .. be the Rees algebra of the edge ideal I Ž G .. Then k w G x is normal if and only if RŽ I Ž G .. is normal. Proof. Assume that k w G x is normal. Let C Ž G . be the cone over the graph G; it is obtained by adding a new vertex x to G and joining every vertex of G to x. According to w9, Remark 2.5x there is a isomorphism RŽ I Ž G .. , k w C Ž G .x. Let Mw be a bow tie of C Ž G . with edge disjoint cycles Z1 and Z2 joined by the path P; it is enough to verify that Mw g k w C Ž G .x. If x f Z1 j Z2 j P, then w is a bow tie of G and Mw g k w G x. Assume x g Z1 j Z2 , say x g Z1. If Z1 l Z2 / B, then Mw g k w C Ž G .x. On the other hand if Z1 l Z2 s B, then Mw g k w C Ž G .x because in this case Z1 and Z2 are joined by the edge x, z 4 , where z is any vertex in Z2 . It remains to consider the case x f Z1 j Z2 and x g P, since G is connected there is a path in G joining Z1 with Z2 . Therefore Mw s Mw 1 for some bow tie w 1 of G and Mw g k w G x. Conversely assume that RŽ I Ž G .. is normal; then by w6, Theorem 7.1x we obtain that k w G x is normal. Note added in proof: After this paper had been submitted the authors learned that T. Hibi and H. Ohsugi have independently given descriptions of the integral closure of these toric rings.
REFERENCES 1. W. Bruns, Computing the integral closure of an affine semigroup, an implemented algorithm to compute normalizations, 1996. 2. W. Bruns and J. Herzog, ‘‘Cohen]Macaulay Rings,’’ Cambridge Univ. Press, Cambridge, 1993. 3. W. Bruns, W. V. Vasconcelos, and R. Villarreal, Degree bounds in monomial subrings, Illinois J. Math. 41 Ž1997., 341]353. 4. F. Harary, ‘‘Graph Theory,’’ Addison]Wesley, Reading, Massachusetts, 1972. 5. H. Matsumura, ‘‘Commutative Algebra,’’ Benjamin]Cummings, Reading, Massachusetts, 1980. 6. A. Simis, W. V. Vasconcelos, and R. Villarreal, On the ideal theory of graphs, J. Algebra 167 Ž1994., 389]416. 7. W. V. Vasconcelos, ‘‘Computational Methods in Commutative Algebra and Algebraic Geometry,’’ Springer-Verlag, New YorkrBerlin, in press. 8. R. Villarreal, Rees algebras of edge ideals, Comm. Algebra 23 Ž1995., 3513]3524. 9. R. Villarreal, Normality of subrings generated by square free monomials, J. Pure Appl. Algebra 113 Ž1996., 91]106.
JOURNAL OF ALGEBRA ARTICLE NO.
The Integral Closure of Subrings Associated to Graphs Aron Simis* Departamento de Matematica, Uni¨ ersidade Federal da Bahia, 40210, ´ Sal¨ ador, Bahia, Brazil
Wolmer V. Vasconcelos† Department of Mathematics, Rutgers Uni¨ ersity, New Brunswick, New Jersey 08903
and Rafael H. Villarreal ‡ Departamento de Matematicas, Centro de In¨ estigacion ´ ´ y de Estudios A¨ anzados del IPN, Apartado Postal 14-740, 07000, Mexico City, D.F., Mexico ´ Communicated by Craig Huneke Received February 3, 1997
Let G be a graph on the vertex set V, and R s k w V x a polynomial ring over a field k. We give a very explicit description of the integral closure of the monomial subring k w G x ; R, generated by the squarefree monomials of degree two defining the edges of the graph G. Q 1998 Academic Press
1. INTRODUCTION Let R s k w x 1 , . . . , x n x be a polynomial ring over a field k, and let F s f 1 , . . . , f q 4 be a finite set of monomials in R. The monomial subring spanned by F is the k-subalgebra, k w F x s k f 1 , . . . , f q ; R. * Partially supported by CNPq, Brazil. E-mail address: [email protected]. † Partially supported by the NSF. E-mail address: [email protected]. ‡ Partially supported by CONACyT and SNI, Mexico. E-mail address: [email protected]. ´ 281 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
282
SIMIS, VASCONCELOS, AND VILLARREAL
In general, it is very difficult to certify whether k w F x has a given algebraic property}e.g., Cohen]Macaulay, normal}or to obtain a measure of its numerical invariants}e.g., Hilbert function. This arises because the number q of monomials is usually large. It is similarly difficult to carry out manipulations such as those required in the Noether normalization process or to find a set of generators for the integral closure k w F x of k w F x. The focus of our attention is the algebra k w F x . In general, it is difficult to predict which elements lie in the integral closure of an affine domain, but in the case of k w F x one has the following general description: k w F x is also a monomial subalgebra that is generated by monomials f s x 1b1 ??? x nb n s x b ,
b g Nn,
with the following two properties Žsee w7, Chap. 7x for an algorithm.: v v
f s Ł f ia i , a i g Z, f m s Ł f ib i , m, bi g Zq.
The first condition asserts that f lies in the field of fractions of k w F x, the other being the special form the integrality condition takes for such elements. Alternatively, kw F x s k
x a N a g Z A l Rq A 4 ,
where A denotes the exponent vectors of the monomials in F, and Rq A is the cone generated by A. In this note we determine k w F x , when F is generated by squarefree monomials of degree two. In this case the underlying graph theoretic aspects are very helpful toward a solution. Consider any graph G with vertex set V s x 1 , . . . , x n4 . Let F be the set of all monomials x i x j in R, such that x i x j 4 is an edge of G. For simplicity of notation we denote k w F x by k w G x. To explain our result we begin by exhibiting some monomials in the integral closure of k w G x. Given an induced subgraph w of G consisting of two edge disjoint odd cycles Z1 s
SUBRINGS ASSOCIATED TO GRAPHS
283
z 0 , z1 , . . . , z r s z 0 4 and Z2 s z s , z sq1 , . . . , z t s z s 4 , joined by a path Žthose subgraphs will be called bow ties ., we associate the monomial Mw s z1 ??? z r z sq1 ??? z t . We observe that Z1 and Z2 are allowed to intersect and that only the variables in the cycles occur in Mw , not those in the path itself. Our main result is THEOREM 1.1. Let G be a graph and let k be a field. Then the integral closure k w G x of k w G x is generated as a k-algebra by the set B s f 1 , . . . , f q 4 j Mw N w is a bow tie 4 , where f 1 , . . . , f q denote the monomials defining the edges of G. As a corollary we have now a full characterization of when k w G x is integrally closed. A related issue, was raised in w6x, where the authors considered the Rees algebra of the ideal I s I Ž G . of R generated by the f i Žintroduced in w8x.. This ring, Rw It x, is also a monomial subalgebra of Rw t x, R w It x s k x 1 , . . . , x n , f 1 t , . . . , f q t , but now generated by forms of degree 1 and 3. It was conjectured in w6x that the integral closure of Rw It x would be generated by elements similar to bow ties. This will follow from our main result. As a consequence, we show that if G is a connected graph, then k w G x is normal if and only if I Ž G . is a normal ideal.
2. A COMBINATORIAL DESCRIPTION OF NORMALIZATIONS Let us fix some notation that will be used throughout. Our main references for graph theory and affine semigroups are w4; 2, Chap. 6x, respectively. Let G be a graph. A walk of length n in G is an alternating sequence of vertices and edges w s ¨ 0 , z1 , ¨ 1 , . . . , ¨ ny1 , z n , ¨ n4 , where z i s ¨ iy1 , ¨ i 4 is the edge joining ¨ iy1 and ¨ i . A walk is closed if ¨ 0 s ¨ n . A walk may also be denoted ¨ 0 , . . . , ¨ n4 , the edges being evident by context. A cycle of length n is a closed walk, in which the points ¨ 1 , . . . , ¨ n are distinct. A path is a walk with all the points distinct. A tree is a connected graph without cycles and a graph is bipartite if all its cycles are even. A vertex of degree one will be called an end point. If e is an edge of G, we denote by G _ e4 the spanning subgraph of G obtained by deleting e. The removal of a vertex ¨ from graph G results in
284
SIMIS, VASCONCELOS, AND VILLARREAL
that subgraph G _ ¨ 4 of G consisting of all the vertices in G except ¨ and all the lines not incident with ¨ . As usual, V Ž G . and EŽ G . stand for the vertex set and edge set of G, respectively. PROPOSITION 2.1. Let G be a connected graph and let k be a field. If G has at most one odd cycle, then k w G x is normal. Proof. We may assume that G has a unique odd cycle, for otherwise G is a bipartite graph and k w G x is normal by w6, Theorem 5.9x. Pick an edge e of G which is on the unique odd cycle and set H s G _ e4 . Notice that H is a connected bipartite graph because e is not a bridge Žit is on a cycle.. Hence k w H x is a normal domain of dimension dim k w G x y 1 Žsee w8, Lemma 3.1x.. Consider the epimorphism w
k w H xw t x ª k w G x ª 0,
induced by w Ž t . s f e ,
where t is a new indeterminate and f e is the monomial corresponding to e. Since k w H xw t x and k w G x are both Noetherian domains of the same dimension, w is an isomorphism. Therefore k w G x is normal. LEMMA 2.2. Let G be a graph without e¨ en cycles. Then the intersection of any two cycles is either a point or the empty set. Proof. It is a simple parity count. Let G be a graph on the vertex set V Ž G .. Given a subset U ; V Ž G ., the neighbor set of U is defined as N Ž U . s ¨ g V Ž G . N ¨ is adjacent to some vertex of U 4 . Notice that if Z is a cycle then Z : N Ž Z .. DEFINITION 2.3. Let G be a graph without even cycles. A cycle Z of G is said to be a terminal cycle if N Ž Z . s Z j ¨ 4 , for some ¨ f Z. LEMMA 2.4. Let G be a connected graph without e¨ en cycles. Assume that Z1 l N Ž Z2 . s B, for any two cycles Z1 , Z2 of G. If G is not a unicyclic graph, and degŽ ¨ . G 2 for all ¨ g V Ž G ., then G has at least two terminal cycles. Proof. Let Z1 , . . . , Zr be the cycles of the graph G. We now construct a new graph T having vertex set r V Ž T . s Z1 , . . . , Zr 4 j Ž V Ž G . _ D is1 Zi . .
Thus the vertices of T are the cycles of G, together with the vertices of G not lying on any cycle of G. Two points Y1 , Y2 of T are adjacent if and only if Y1 l N Ž Y2 . s B. Here we regard a vertex ¨ of G as the one point
SUBRINGS ASSOCIATED TO GRAPHS
285
set ¨ 4 . Notice that T is a tree and that the terminal cycles of G correspond to the end points of T. Hence G must have at least two terminal cycles. DEFINITION 2.5. A bow tie w of a graph G is a subgraph of G consisting of two edge disjoint odd cycles Z1 s z 0 , z 1 , . . . , z r s z 0 4
and
Z2 s z s , z sq1 , . . . , z t s z s 4
joined by a path z r , . . . , z s 4 . In this case we set Mw s z1 ??? z r z sq1 ??? z t . If w is a bow tie of a graph G, as above, then Mw is in the integral closure of k w G x. Indeed if f i s z iy1 z i , then Mw s
Ł fi Ł
i odd
i even r-iFs
fy1 i
and
Mw2 s f 1 ??? f r f sq1 ??? f t .
Hence Mw g k w G x . We fix a final element of notation by setting B s f 1 , . . . , f q 4 j Mw N w is a bow tie 4 , where f 1 , . . . , f q denote the monomials defining the edges of G. LEMMA 2.6. Let G be a connected graph without e¨ en cycles, and let V be the ¨ ertex set of G. Assume the following three conditions: Ža. Z1 l N Ž Z2 . s B, for any two cycles Z1 , Z2 of G. Žb. degŽ ¨ . G 2 for all ¨ g V. Žc. G has an e¨ en number of ¨ ertices. Then Ł ¨ g V ¨ is a monomial in k w B x. Proof. The proof is by induction on the number of cycles of the graph G. Set z s Ł ¨ g V ¨ . If G has exactly two cycles, then G is a bow tie and we can write z s f d , where f g k w G x and d s MG is the product of the vertices in the two cycles. We may now assume that G has at least three cycles. By Lemma 2.4 there are two terminal cycles Z1 s x 0 , x 1 , . . . , x r s x 0 4 ,
Z2 s y 0 , y 1 , . . . , ys s y 0 4 .
There is a path x r , x rq1 , . . . , x r 14 , so that r q 1 F r 1 , degŽ x j . s 2 for all r - j - r 1 , and degŽ x r 1 . G 3. Likewise there is a path ys , ysq1 , . . . , ys14 , so that s q 1 F s1 , degŽ y j . s 2 for all s - j - s1 , and degŽ ys1 . G 3. We may
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SIMIS, VASCONCELOS, AND VILLARREAL
assume r 1 and s1 even, for otherwise, if r 1 is odd, we write z s Ž x 1 x 2 . Ž x 3 x 4 . ??? Ž x r x rq1 . ??? Ž x r 1y2 x r 1y1 .
Ł ¨,
¨ gV1
where V1 s V _ x 1 , . . . , x r 1y14 , and then use induction. It is not hard to show that x 1 , . . . , x r 14 l y 1 , . . . , ys1y14 s x 1 , . . . , x r 1y14 l y 1 , . . . , ys14 s B. Set
s 1 s Ž x rq1 x rq2 . ??? Ž x r 1y2 x r 1y1 . and s 2 s Ž ysq1 ysq2 . ??? Ž ys1y2 ys1y1 . . If x r 1 / ys1 then we write s
r
z s s 1 s 2 Ł yi Ł x i is1
is1
Ł ¨,
¨ gV1
where V1 s V _ x 1 , . . . , x r 1y1 , y 1 , . . . , ys1y1 4 , hence using induction we get z g k w B x. We may now assume that r 1 , s1 are even and x r 1 s ys1. Consider the subgraph H s G _ x 1 , . . . , x r 1y1 , y 1 , . . . , ys1y1 4 . If degŽ x r 1 . G 4, then applying the induction hypothesis to H we obtain z g k w B x. Assume degŽ x r 1 . s 3. Note that H has a terminal cycle Z3 s z 0 , z1 , . . . , z t s z 0 4 and there is a path z t , z tq1 , . . . , z t 14 , so that t q 1 F t 1 , degŽ z j . s 2 for all t - j - t 1 , and degŽ z t 1 . G 3; by the previous arguments we are reduced to the case r 1 , s1 , t 1 even and x r 1 s ys1 s z t 1. Observe that this forces the equality V s x 1 , . . . , x r 1, y 1 , . . . , ys1y1 , z1 , . . . , z t 1y14 . To complete the proof we use the identity r
s
z s Ž z1 z 2 . Ž z 3 z 4 . ??? Ž z t 1y1 z t1 . s 1 s 2 Ł x i Ł yi , is1
is1
to derive z g k w B x. Given a graph G we set AG s logŽ f 1 ., . . . , logŽ f q .4 , where f 1 , . . . , f q are the monomials corresponding to the edges of G and logŽ f i . is the exponent vector of f i . The affine subsemigroup of N n generated by AG will be denoted by CG . Thus CG s
Ý
Na .
ag AG
The cone generated by CG will be denoted by Rq CG ; it is equal to Rq CG s
½
r
Ý ai a i N ai g Rq , a i g AG , r g N is1
Note that k w G x is equal to the affine semigroup ring k w CG x.
5
.
SUBRINGS ASSOCIATED TO GRAPHS
287
LEMMA 2.7. Let G1 and G 2 be finite sets of monomials in disjoint sets of indeterminates. Then k w G1 j G 2 x , k w G1 x mk k w G 2 x and k w G1 j G 2 x , k w G1 x mk k w G 2 x , k w G1 x k w G 2 x , is the subring generated by the generators of k w G1 x and k w G 2 x . Proof. Set G s G1 j G 2 . By w2, Proposition 6.1.2x we have kwGx s k
x a N a g ZCG l Rq CG 4 .
On the other hand there is a decomposition ZCG l Rq CG s Ž ZCG 1 l Rq CG 1 . [ Ž ZCG 2 l Rq CG 2 . . Using this equality the assertion follows readily. Proof of the Main Result. Let V s x 1 , . . . , x n4 and E s e1 , . . . , e q 4 be the vertex set and edge set of G, respectively. The proof is by induction on q, the number of edges of G. The case q s 1 is clear. Assume q G 2 and that the result is true for graphs with less than q edges. If e i s x j , x k 4 , we set f i s x j x k . First we remove all the isolated vertices of G, without affecting the subring k w G x. By Lemma 2.7 and the induction hypothesis we may assume that G is a connected graph. We may also assume that degŽ ¨ . G 2 for all ¨ g V; because if degŽ ¨ . s 1 and f i s ¨ x j , then setting G9 s G _ e i 4 we have k w G x s k w G9xw f i x, the latter being a polynomial ring over k w G9x, hence k w G x s k w G9 x w f i x and f i is not part of a bow tie, which by induction yields the required equality. Let z s x a g k w G x be a minimal generator of the integral closure of w k G x. There is a positive integer m so that z m s f 1m l1 ??? f qm l q s f 1b 1 ??? f qb q ,
where l i g Z and bi g Zq , for all i.
Ž 1. Observe that if bi s l i s 0 for some i, then each bow tie of G _ e i 4 is a bow tie of G and the induction hypothesis yields z g k w B x. Hence we may further assume that all the monomials f 1 , . . . , f q must be present in Eq. Ž1.. There are two cases to consider. Case ŽI.. Assume that G contains an even cycle Z. We set ¨ i s logŽ f i .. For convenience we use the edges of G to represent the cycle Z by the sequence e i1, . . . , e i k 4 , where two consecutive edges are adjacent. If bi r s 0 for some 1 F r F k, then using the relation k
Ý js1
j Ž y1. ¨ i j s
k
Ý Ž y1. j log Ž f i . s 0, j
js1
Ž 2.
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SIMIS, VASCONCELOS, AND VILLARREAL
that comes from the even cycle, we may rewrite Eq. Ž1. as z m s f 1m m 1 ??? f qm m q s f 1b 1 ??? f qb q , where m i g Z for all i , and m i r s bi r s 0, which by induction yields z g k w B x. We may now assume bi r ) 0 for all 1 F r F k. We may harmlessly assume that bi1 F ??? F bi k. Using Eqs. Ž1. and Ž2. we obtain q
ma s
k
Ý bi¨ i s bi Ý ¨ i 1
is1
j
q
js1
Ý c i¨ i s 2 bi Ý 1
i/i 1
¨ij q
j even
Ý ci¨ i , i/i 1
where the c i ’s are non-negative integers. The net effect is that now we may rewrite Eq. Ž1. as z m s f 1m d 1 ??? f qm d q s f 1d1 ??? f qd q ,
where d i g Z, d i g Zq , for all i ,
and d i1 s d i1 s 0. Hence by induction z g k w B x. Case ŽII.. Assume that all the cycles of G are odd. Let Z1 and Z2 be two distinct cycles of G, represented by the sequences of edges e i1, . . . , e i k 4 and e l 1, . . . , e l s 4 , respectively. If Z1 l Z2 / B, then according to Lemma 2.2 Z1 l Z2 s x 4 , for some x g V, say x g suppŽ f i1 . l suppŽ f l 1 .. Using the identity
Ý ¨i
j
q
j odd
Ý j even
¨ lj s
Ý
¨ij q
j even
Ý ¨l
j
j odd
we may proceed as in Case ŽI.. If Z1 l Z2 s B and N Ž Z1 . l Z2 / B, say f t s xy, with x g suppŽ f i k . and y g suppŽ f l 1 .. Again using the equality
Ý ¨i j odd
j
q
Ý ¨l j odd
j
s 2¨t q
Ý
¨ij q
j even
Ý
¨ lj ,
j even
we may proceed as in Case ŽI. to derive z g k w B x. Therefore we are reduced to the case N Ž Z1 . l Z2 s B, for any two cycles Z1 , Z2 . If x j is not in the support of f 1b 1 ??? f qb q , for some j, consider the set D s i N x j g supp Ž f i" l i . 4 . Then Ý i g D l i s 0 and bi s 0, for all i g D. Note that, for any i g D, we may eliminate the monomial f i from Eq. Ž1., hence by induction z g k w B x. As a consequence we are reduced to the case suppŽ z m . s V, that is, z s x 1a1 ??? x na n , with a i G 1 for all i. By w3, Corollary 3.6x we obtain that k w G x is generated as a k-algebra by monomials of degree at most 2Ž n y
SUBRINGS ASSOCIATED TO GRAPHS
289
u nr2v.. Altogether we get n F degŽ z . F 2Ž n y u nr2v., hence n must be even and z s x 1 ??? x n . Using Lemma 2.6 we conclude that z g k w B x, as required. COROLLARY 2.8. Let G be a connected graph and let RŽ I Ž G .. be the Rees algebra of the edge ideal I Ž G .. Then k w G x is normal if and only if RŽ I Ž G .. is normal. Proof. Assume that k w G x is normal. Let C Ž G . be the cone over the graph G; it is obtained by adding a new vertex x to G and joining every vertex of G to x. According to w9, Remark 2.5x there is a isomorphism RŽ I Ž G .. , k w C Ž G .x. Let Mw be a bow tie of C Ž G . with edge disjoint cycles Z1 and Z2 joined by the path P; it is enough to verify that Mw g k w C Ž G .x. If x f Z1 j Z2 j P, then w is a bow tie of G and Mw g k w G x. Assume x g Z1 j Z2 , say x g Z1. If Z1 l Z2 / B, then Mw g k w C Ž G .x. On the other hand if Z1 l Z2 s B, then Mw g k w C Ž G .x because in this case Z1 and Z2 are joined by the edge x, z 4 , where z is any vertex in Z2 . It remains to consider the case x f Z1 j Z2 and x g P, since G is connected there is a path in G joining Z1 with Z2 . Therefore Mw s Mw 1 for some bow tie w 1 of G and Mw g k w G x. Conversely assume that RŽ I Ž G .. is normal; then by w6, Theorem 7.1x we obtain that k w G x is normal. Note added in proof: After this paper had been submitted the authors learned that T. Hibi and H. Ohsugi have independently given descriptions of the integral closure of these toric rings.
REFERENCES 1. W. Bruns, Computing the integral closure of an affine semigroup, an implemented algorithm to compute normalizations, 1996. 2. W. Bruns and J. Herzog, ‘‘Cohen]Macaulay Rings,’’ Cambridge Univ. Press, Cambridge, 1993. 3. W. Bruns, W. V. Vasconcelos, and R. Villarreal, Degree bounds in monomial subrings, Illinois J. Math. 41 Ž1997., 341]353. 4. F. Harary, ‘‘Graph Theory,’’ Addison]Wesley, Reading, Massachusetts, 1972. 5. H. Matsumura, ‘‘Commutative Algebra,’’ Benjamin]Cummings, Reading, Massachusetts, 1980. 6. A. Simis, W. V. Vasconcelos, and R. Villarreal, On the ideal theory of graphs, J. Algebra 167 Ž1994., 389]416. 7. W. V. Vasconcelos, ‘‘Computational Methods in Commutative Algebra and Algebraic Geometry,’’ Springer-Verlag, New YorkrBerlin, in press. 8. R. Villarreal, Rees algebras of edge ideals, Comm. Algebra 23 Ž1995., 3513]3524. 9. R. Villarreal, Normality of subrings generated by square free monomials, J. Pure Appl. Algebra 113 Ž1996., 91]106.