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Hunt.-. 0 m “3mm“ 1 "William”!{Tllfi!fl?(fl’fi'/HT!TW!WWW“ ' Michigan State 3 1293 00629 9287 University COHEN-MACAULAY UNIONS 0F LINES IN PE AND THE COHEN-MACAULAY TYPE presented by Frank Judson Curtis III has been accepted towards fulfillment of the requirements for Ph.D. degreein Mathematics ' Major professor Date February 12, 1990 MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before due due. T J -._.J r—‘J JI . JL L—l—____ MSU Is An Affirmative Action/Equal Opportunity Institution COHEN—MACAULAY UNIONS OF LINES IN P}: AND THE COHEN—MACAULAY TYPE by Frank Judson Curtis III A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1990 L05 4349 ABSTRACT COHEN-MACAULAY UNIONS OF LINES IN Pfi AND THE COHEN—MACAULAY TYPE by Frank Judson Curtis III Let V be a union of projective lines in projective n—space. We consdier the questions of when k[V]m, the coordinate ring localized at the irrelevant maximal ideal, is Cohen—Macaulay, and what the Cohen—Macaulay type is when k[V]In is Cohen—Macaulay. Chapter 1 reviews the definitions and some of the history of the problem. Chapter 2 shows how it can be interpreted as a problem involving the graded ring k[V] and modules over k[V]. This approach yields a linear algebra algorithm, similar to an algorithm by M. Baruch and W. C. Brown, which can answer both questions (Chapter 3). The method also provides a graph theoretical solution in the case where V is a graph on n+1 linearly independent points (Chapter 4). Chapters 5 and 6 are applications of a method, deve10ped by A. V. Geramita and C. Weibel, which uses pullback rings. In Chapter 5, the class of simply connected unions V is defined and both questions are answered for this class. A Hilbert function condition on k[V], which is necessary for Cohen—Macaulay V. (due to A. V. Geramita, P. Maroscia, and L. Roberts), is reviewed in chapter 6, and an example constructed to Show the condition is not sufficient. The last chapter expands the class of known Cohen—Macaulay unions of linear varieties by Showing that any unmixed union of linear varieties W is contained in a union of linear varieties V of the same dimension which is Cohen—Macaulay. The type of V is also established. ACKNOWLEDGMENTS I would like to thank my advisor, Professor William C. Grown, for his guidance and encouragement, and particularly for the careful criticism and many suggestions which he provided at every stage in the preparation of this thesis. I would also like to thank my typist, Loretta Ferguson, for a job well done. iv Chapter 1. Chapter 2. Chapter 3. Chapter 4. Chapter 5. Chapter 6. Chapter 7. TABLE OF CONTENTS Introduction Graded Rings Matrix Methods Graphs Simply Connected Unions of Lines Hilbert Function Existence Theorems 15 3O 44 52 Figure 4.1 Figure 4.2 Figure 5.1 Figure 5.2 Figure 6.1 Figure 6.2 Figure 6.3 LIST OF FIGURES vi 18 19 32 36 46 46 49 CHAPTER 1 INTRODUCTION The study of commutative algebra arose historically as an adjunct to the study of algebraic geometry, but it has not been limited to this role in the development of its methods and concepts. In fact, while commutative algebra has been able to provide a more rigorous foundation for algebraic geometry, it has at the same time produced new concepts whose significance in algebraic geometry remains largely unexplored. In this paper I will be concerned with two such properties, the Cohen-Macaulay pr0perty for rings and the Cohen—Macaulay type of a Cohen—Macaulay ring. The geometric objects I will consider are the localized coordinate rings of reducible projective varieties consisting of unions of line in P“, where k is an arbitrary algebraically closed field. The two basic questions are: (1) When are such rings Cohen—Macaulay? (2) What is the Cohen—Macaulay type in those cases where the coordinate ring is Cohen—Macaulay? Given the motivation for considering these questions, it is to be understood that any answers to them will be of greater interest to the extent to which the criteria invoked are clearly geometric, and that algebraic computational solutions are not as informative. Unfortunately, it appears that a satisfactory general solution to even the first question is very difficult to obtain. In fact, one of the basic problems in studying these questions is the scarcity of examples of unions of lines whose coordinate rings are known to be Cohen—Macaulay. For this reason, much of this paper will be devoted to the task of identifying classes of examples whose coordinate rings are Cohen-Macaulay. Before reviewing the history of the subject, I will pause to review some definitions and introduce notation to be used throughout the paper. Let k be an algebraically closed field. Let .9 be a homogeneous prime ideal of height n—l in k[XO,...,Xn] with 9=(f1,...,fn_l), where deg fi=1 for i=1,...,n—1. .9 is a prime ideal and L=V(.9’) 9 P11: is a li_ne in P111. Thus L is a linear variety of projective dimension 1. Linear varieties of projective dimension 0 are mints in PE. V will denote a union of s lines in Pn s , V: U L. k i=1 1. 8 Let I=J(V)=. n .95. A will denote the coordinate ring, 1 1 A=k[V]=k[XO,...,Xn]/I. We wish to consider the local ring Am, where m=(xo,...,xn) is the homogeneous maximal ideal of A. A 2—generated mAm—primary ideal q is a parameter ideal, and any set of 2 generators for such an ideal is a system of parameters. The multiplicity of a parameter ideal q is the leading coefficient of its Hilbert polynomial. Definition/Theorem 1.1. [ZS, p. 400; HK] The following are equivalent for a Noetherian local ring (R,m) with dim R=d: 1) R is a Cohen—Macaulay ring 2) 111 contains a regular sequence of length d. 3) e(q)=l(R/q) for some parameter ideal q, where e(q) is the multiplicity and 1(R/q) is the length of R/q. 4) e(q)=l(R/q) for every parameter ideal q. 5) One system of parameters is a regular sequence. 6) Every system of parameters is a regular sequence. 7) Ext §(R/m,R)=0 for i=0,...,d~—l. Any reduced Noetherian local ring of dimension 1 contains regular elements, and is thus Cohen—Macaulay, by (2). In particular, the localized coordinate ring of a union of points in P: is always Cohen-Macaulay. Definition. The Cohen-Macaulay type r of a Cohen-Macaulay ring (R,m) is a r=dimR/m(d’(R/q)) for any parameter ideal q [K, p. 189]. J(R/q) is the $219 97(R/q)={yER/q|ym=0} g [qzm]/q. R is Gorenstein if r=1. Theorem 1.2 r is also [K, p.202, HK, p.4] 1) The number of irreducible components in an irredundant decomposition of q into irreducible ideals. 2) dimR/m(Ext §(R/m, R)). For the rings Am which are Cohen-Macaulay, r is also the last nonzero betti number in a minimal free resolution of A m over k[Xo, ..... ’Xn](X0,---,Xn)' It will sometimes be convenient to call V Cohen-Macaulay whenever A!n is and refer to the Cohen—Macaulay type of A m as the Cohen—Macaulay type of V. The first partial answer to the first question (when is A m Cohen—Macaulay) is due to Hartshorne. Theorem 1.3 (by [H1, pr0p. 2.1]). Let A be the homogeneous coordinate ring of a union of irreducible curves in Pfl. If A is Cohen-Macaulay, then the union of curves is connected. This says that V must be connected in order to be Cohen—Macaulay. 3 So, for example, if V consists of two skew lines in Pk’ then V is not Cohen—Macaul ay. Unfortunately the converse is false, as the following theorem illustrates. We recall that the nonsingular quadratic surface, 7’ (X0X2—X1X3), is isomorphic to PfixPfi; i.e., it is a ruled surface with two rulings [M, p. 27]. Theorem 1.4. Geramita and Weibel ([GW, 5.1]). Let V be a union of lines on a quadric surface, with m lines from one ruling and n lines from the other ruling. Then Am is Cohen—Macaulay if and only if lm-IIISI. In the same paper it is shown unions of lines through a common point (pencils) are always Cohen-Macaulay. Geramita and Weibel have also proven the following theorem. Here, a _v_ert_ex of V is a point of intersection of two or more lines. For a reduced Noetherian ring R, with total quotient ring Q(R), R is said to be seminormal if the following pr0perty holds: whenever aEQ(R); a2, a3ER, then aER [S]. Theorem 1.5 [GW, 5.9]. If V is connected, and the lines through each vertex are linearly independent, then V is Cohen—Macaulay if and only if V is seminormal. As a consequence, the examples of seminormal unions V given in [DR] are all Cohen—Macaulay. The Cohen—Macaulay type of V has, so far a I know, been previously studied only in some special cases which will be discussed in Chapter 4. The Cohen-Macaulay type of unions of points has been studied more extensively (see bibliography). CHAPTER 2 GRADED RINGS The ring A, as the coordinate ring of a projective variety, is naturally a graded ring. In this chapter, we will relate the local properties of Am (being Cohen-Macaulay of a certain type) to the global prOperties of the graded ring A. Let f1, f26k[xo,....,Xn], with deg f1: deg f2=1. Denote by fi the image of fi in Am and in A. As dim Am=2, a = (fl, f2)_C_A m is a parameter ideal (i.e., {f1, f2} is a system of parameters) 4:) (f1, T2) is mAm—primary in A m :1 (f1, f2) is m—primary in A 4:» 1’ 2 + = (X0,....,Xn) 4:» 7(f1) n 7(12) n V=0 in PE Because k is infinite, we can choose elements f1 and f2 of degree 1 so that fl ¢ 53, for i=1,...,s, and f2 9! (f1, .95), for i=1,...,s. Then Jul, f25+l = (X0,....,Xn) and {f1, f2} will be a system of parameters. In the following pr0position, e(A) denotes the multiplicity of the graded ring A, defined as follows. If we let P A(n)=dimkAn be the Hilbert function of A, and PA(n) the corresponding Hilbert polynomial, then e(A) is the leading coefficient of PA(n). By [K, VI, pr0p. 2], grmA (Am) k[X0,...-.Xn]/({L(F)IF61}) m = k[XO,....,Xn]/I =A. _ n n+1 . . . As Pmm(n)——l(mm/mIn ), It follows that the Hilbert functions of m m and the graded ring A are the same. So e(A)=e(mm). Proposition 2.1. If l(A/q)=e(A), then AIn is Cohen—Macaulay. Proof. As q is m-primary, A/q is local with maximal ideal m/q. So A/qg(A/q) m / q gAm/qm, so the lengths are the same. l(Am/qm)2e(qm), as this is true for any parameter ideal, by [Z8 11, p. 296]. Let qu(n) and qu(n) denote the Hilbert function and polynomial respectively for qm, then qu(n)2Pmm(n) for all n. Thus qu(n)2Pmm(n) for all n>>0. As these polynomials both have degree 2=dim Am, we can compare leading coefficients to obtain e(qm)_>_e(mm). As noted in the discussion preceding the pr0position, e(mm)=e(A), so e(qm).>.e(A)- We have now shown that the following chain of inequalities holds: l(A/Q)=l(Am/am).>.e(qm)2e(A)- So if l(A/q)=e(A), then l(Am/qm)=e(qm), and AIn is Cohen-Macaulay(1.1(3)). u The converse of 2.1 is also true. Pr0position 2.2. If AIn is Cohen—Macaulay, then l(A/q)=e(A). Proof. If A m is Cohen—Macaulay, then {f1, f2} is a regular sequence in Am, so {f1, f2} is a regular sequence in A. Consider the Poincaré series: m . QA(Z) =i§0(dimkAi)zl m . CIA/(1(2)=i§0(dimk(A/Q)i)zl, where Ai denotes the ith graded part of A. We then have an exact sequence ”f 0 -+A(—1) _1.. A -» A/flA-vo is exact. So QA/f1A(z)=QA(z)—QA(_1)(Z) = QA(Z)'ZQA(Z)- a) . Repeating the argument with f we obtain Q A(z)=( E zl)2Q A (2). So if 2 i=0 /‘1 . m . a) . . . m“1 C q, 2 (dim A.)zl = 2 (i+1)zl£ dim (A/q).zl. Equating coefficients _ i=0 R l 0 i=0 k l i of z1 for i213 dimk Ai=k§0 (i+1—k)dimk(A/Q)k=k£0 (i+1)dimk(A/q)k - :0 k dimk(A/q)k=(i+1)l(A/q)-k§0 k dimk(A/q)k. So e(A)=l(A/q). a e(A) is the leading coefficient of the Hilbert polynomial of A. As projdim (A)=1, the degree of V is e(A)~l!=e(A). We now apply [H, prOp.I.7.6]. Each line has degree 1; all lines have projective dimension 1; and any two intersect in a variety of projective dimension 0 or —1. So the degree of V is the sum of the degrees of the Li’ i.e. deg (V)=s. Thus e(A)=s. We have now proven: Proposition 2.3. Am is Cohen—Macaulay if and only if l(A/q)=s. As l(A/q) can be determined by choosing as a composition series of A/q a refinement of: Ala 2 mm 2 m2q/>0 (where the subscript i denotes the ith graded piece of A/q), and dimk(q:m/q)= £1 dimk(q:m/q)i. i=0 In the next two chapters we will use these formulas to reduce determination of the Cohen-Macaulay pr0perty and computation of type to elementary problems in linear algebra, and to obtain a formula for the type for one class of unions V which are Cohen—Macaulay. CHAPTER 3 MATRIX METHODS In this chapter, we will use the results of Chapter 2 to show that we can tell whether AIn is Cohen-Macaulay, and compute the type if it is, by doing elementary matrix computations. These results are more of theoretical than practical interest, as the matrices whose ranks must be computed will be very large, given V with a large number s of lines. The reader should note that the method used for computing the type is generalization to lines of Baruch and Brown's method for computing the type in the case of points [BB, B]. Each line in V can be described as the linear span of two projective points, Li=3pan((b. "’ib n)’(cio’”" "Cin))' Let S=k[X0,....,Xn]. Let u be the map defined by10 S 31—. k[Tgl), 1151)] g... .9 1411(3) T(S)] V(Xi)=(b1iT]1) + CliTng "bs 1T(S)+c3iT(S)) for i=0,....,n. Let IIj be projection into the jth coordinate, II].- :k[T(1), T(1 )1 e 4» k[T]S). T9], lqleEJ), T(J)] Then rjorr. S -+k[T(j)s, Tm] is the map defined by rji=ou(X) bj iT(j)+ c..T(j). So ker V: n ker (ij V) 11 2 jzls =15 {fES|f(bj0 T“) + cjorgllm bjn T“) + cjn Tm): 0} 8 — —n {feSlf e J(L.—)} — n 9:1 i=1 i=1 10 So A=S/I 2 im V. We now use V to obtain a matrix representation for each graded piece Ad of the graded ring A. n e. Lexicographically order the monic monomials in S (1 via II Xil > i=0 fi 11 xi1 if e.=f. for i=0,...t, and e i- 0 1 1 n+d a a 1](d)=[ n ]. Let ggd)=Xo0 ----- Xnn (010+....+an=d). Then xggdl>=xxpa° ------ ref“ =<{bloTil)+cloT§l)}ao”"{b1nTil)+ClnT(l)}an: (d) (d) t+1>ft+l, as g1 ,....,gn(d), Where 2 ., bso'r]3)+cso T(3)}a°~~{bs nT(“’)+csn'r§3)}0'“> < g0 CiMTU) ]2T(d-a[ ”01].... d a agoc CU J)[T(3)]d- 011153)] >, S for some constants )c(j(); j: .1..,u(d); r=1,...,sj a=0,...,d. Define [gldl] =[e(1)e(] ,inl], and let I‘d(A) be the n(d)xs(d+1) matrix 1' j ro ’ Cr1 ~l [gist]; . defined by cm (1) (1) (1) 1d ’ C20 ’ ’Cso ’ ,ch ,...,r]g(d>), agar...,.ggd»,....gg 1, that is, V(G) = {0,....,n} is the BEES & of G. The gge se_t of G, E(G) is a subset of the set of unordered pairs of distinct vertices. Let s = |E(G)|. We assume that G has no isolated vertices, so, in particular 3 z n. For each edge e = ij 6 E(G), Let .9; be the ideal in k[Xo,...,Xn] generated by {Xo,...,Xn}\{Xi, Xj}’ and I =e€g(G) 5;. As each .9; = V(Le) for some L8 in P: (in fact, a coordinate axis), A = k[XO,...,Xn]/I is k[V] for some union of lines V. A is usually denoted k[G] to indicate its construction from the graph G. I is generated by monomials, for if f e n 93, then each e€E(G) monomial term of f is in each .93, as these are generated by monomials. We first locate a generating set for 1. Suppose I contains an element of degree 1, say Xi' Then Xie .9é,VeeE(G)=>ij¢E(G) forany j => 1 is an isolated vertex of G. As we assume that G has no isolated vertices, I contains no elements of degree 1. As I is reduced, it contains no elements X? As for other elements of degree 2, 15 16 Xixjeleexie 93 or Xje 93, VeeE(G) «:1 ij ¢ E(G). I will not contain any elements of the form: (1) le, (2) Xli‘ X}, where ij 6 E(G). Finally, if i, j, and k are distinct, each Pe will contain one of Xi’ X]. or Xk’ so all other monomials are in 1. Thus, I is generated by {Xileij t E(G)} U {Xi X]. Xkl it i, la k}, and A is a Stanley—Reisner ring [Ho, §1]. Note that for each graded part Ad of A, a basis is given by the set of all monomials of degree d which do not occur in Id. For example, A2 has a basis {if} u {xileij e E(G)}. Promsition 4.1 (Reisner, [Ho, p. 180]) AIn is Cohen—Maculay if and only if G is connected. BENI- We give a different proof than the the one in [Ho], by computing l(A/q) for an m—primary ideal q. We note first that the prOposition holds in one direction by Hartshorne's result (Theorem 1.3). Assume G is connected. let q = (f1, f2), where g 293 f = x.,f = a.x., where: 1 i=0 1 2 i=0 1 l Oaéaiek, and aiataj foriatj. We have immediately, q0 = (0), so dimkq0 = 0 (11 = (fl, f2), so dimkq0 = O. q2 = ( {xjfi|j=0,....,n; i = 1, 2}). 17 n 11 Suppose £0 cixifl, + iEO dixif2 = 0. Then (ci + diai)x? = 0 for each i, by linear independence of basis elements of A2, so ci = —diai, and we have n n (*) .2 —d.a.x.f + 2 dixif2 = 0. 01111 i l —0 Claim. (10 = (11 = = (ln Ergo! of Claim. If ij 6 E(G), then xixj # 0. So the coefficient of . * . . _ xixj 1n ( ) ls zero, l.e., _diai-djaj + diaj + djai -— 0, so di = dj' The claim now follows because any two vertices of G are connected by a path in G. So ci = 'doa‘i’ and n 11 £0 cixif1 + iEO dixif2 n n = —d (2 a.x.f — 2 X-f ). 0 i=0 1 1 1 i=0 12 So every linear relation on the generators of q2 is a multiple of the relation 11 So dimk(q)2 = 2(n+1) - 1 = 2n + 1. n X i: Next, we not that for i, j, i at j, _ 2 2 _ xixjf1 — xixj + xixj, as xixjxk — 0, for iE i, i, k. and ai at a., _ 2 2 xixjf2 — aixix. + a.xix J 111" so x?xj€q for any i#j. Then q contains x?fl = x3 + 2 x?x. E x? (mod q). ‘ #i ‘1 18 So q3 = m3 = m3. So q is an m—primary ideal. Moreover, dimk AO/qo =1- 0 =1. dimk Al/ql=n+l-2=n-l. dimk A2=n+1+8, as I2 contains every element of degree 2 except x? for each i, and xixj for the s edges ij E E(G). So dimk A2/q2 = n + 1 + s — (2n+1) = s - n. Thus l(A/q) = s, and Am is Cohen-Macaulay (prop. 2.3). . In order to compute the type of Am, we will need some definition from elementary graph theory. Consider a sequence of vertices v1,...,vi such that vjvj+1 e E(G) for j=1,...,i-1. If vj ,1 vk for jatk, the sequence is a path , and if v1=vi, but vjatvk for jatk otherwise, the sequence is a w. G—v denotes the graph obtained from G by deleting the vertex v: V(G-v) = V(G) - {v}, and E(G-v) = E(G) — {uv luv 6 E(G)}. A vertex v is a cut vertex if G—v is disconnected (i.e., not path—connected). G is said to be a M if G has no cut vertices, and the M of G are the subgraphs which are maximal with respect to the pr0perty of being a block. For example, a graph consisting of two vertices and one edge is a block, and a graph mnsisting of a single cycle is a block. the graph depicted in Figure 4.1 is not a block, but it has as blocks the graphs in Figure 4.2. Figure 4.1 19 Figure 4.2 We will make use of the following fact: If |V(G)| 2 3, then G is a block if and only if any two vertices lie on a common cycle (for this and other elementary results, see [CL]). We now return to consideration of the ring A = k[G]. If g 6 A2, we can use the basis for A2 given previously to write g uniquely as 2 g = X gux. + 2 g ieV(G) 1‘ ‘ ijeE(G) where in the second sum, each edge ij is counted only once. To simplify (i: ”"in notation, we can refer to g(i, j) as gij or gji’ unique coefficient of xixj in g. So 2 g = 2 gux. + 2 g..x.x.. ieV(G) 1‘ ‘ ijeE(G) ‘1 ‘ 3 either one denoting the Ergmsition 4.2. Suppose 0, 1, ..., k, 0 is a cycle in G. Then there is no element 2 g = E g..x. + E g..x.x. E q such that. j€V(G) 11 J ij€E(G) ‘1 ‘ J 2 1) gjj = 0. for j = 0, ...., n. 2) g01#0, g0k =0, gjj+l = 0 for j: 1, ...,k-l. 20 Proof. k = 1. The proposition is trivial. n k > 1. Suppose there is such a g, and let g = 2 c.x.f1 - d.x.f2. i=0 J J J J . . 2 ’1 Th co = l o = o s o . = o o o '— en g” 0 mp les cJ dJaJ (as x‘I at 0) So g jgo deJ(an1 f2), and g : diai "" d-a- + d-&- - d-ai 1 J J J J = (di - dj)(ai - aj), for each i j E E(G). ij Then ai—ajgtO for iatj, but, for j=1,...,k—1 gjj+1=0' So dj = dj+l’ contradiction. . and similarly dk = do, so (10 = (11. But g01 a! 0, a n Pronosition 4.3. Let h = E bixi 6 (q: m)l. If bi = b. = 0 for i=0 0 o ioj0 E E(G), then bl = 0 for all vertices l in the block B containing the edge in0 . PM. If V(B) = {i0, j0 }, there is nothing to show. So assume |V(B)| > 2, and let 1 E V(B) be any third vertex. Suppose bl at 0. Claim 1. There is path v, v0, v1 in B such that bv = 0, b = 0, b ,1 0. v0 v1 m. Because B is a block, we can choose path 111, ...., uk from j0 = 111 to l = uk, which does not contain i Let P be the path i0, 0' ul, ...., uk. Let v1 be the first vertex along P such that bv ,t 0, and 1 let v and v0 be the two preceding vertices. There is a path P2 in B from v1 to v which does not contain v0, say P2 is the path v1,v2, ...., kk—l’ vk = v. Then v0, ...., vk, v0 is a cycle in B. Relabel G, if necessary, so that vj = j for 21 j = 0,..., k. Claim 2. g = xoh satisfies the conditions of prOposition 4.2. n 2199:. Let g = x h = E ..x.x.. Here x h = 2 b.x x.. If 0 ij€E(G) glj 1 j 0 i=0 1 o 1 i a! 0 34 j, then gij = 0, so gjj+l = 0 for j = 1, ..., k - 1. Moreover, bk = 0, so g0k = 0, and b1 3% 0 so g01 ,e 0. So condition (2) of 4.2 is satisfied. If j at 0, then gjj = 0. As b0 = 0 implies g00 = 0 also, condition (1) is also satisfied. Then 4.2 implies xoh ¢ q2, contrary to h E (q: 111). therefore bl = 0 for all 1 E V(B). I Prgpgsitign 4.4. Suppose i is a cut vertex of G, and V0 the set of vertices in some connected component of G - i. Let g = E (ai — a.)x.. then g E (q: m)1. 0 21991. We show xkg E q, for k = 0, ...., n. Caag 1. k 9! V0, k at 1. Here kj ¢ E(G) for j 6 V0, so xkxj = 0 and xkg = 0 E q. Cgez. kEV. xg=x 2 (a.—a.)x.= E (a.—a-)xx.. If 0 k kjEvo 1 jj jEVO l jkj p f V0 U{i} then xkxp = 0, so if p f V0, (ai -— ap)xkxp = 0. Thus xkg = je§f(G) (ai — aj)xkxj =aixkf1 - xkf2 E q. Caaa 3. k = i. x.g=fg—Ex Eq. ' 1 11¢ng ' Prgpgsition 4.5. If G is a block, then (q: m)1 = q1. 22 n M. Let h = 2 bixi E (q: M). Relabel so that 01 e G(G). ° 0 1: Define I b1 - b0 h = h-bofl—(——a -a )(aofl—f2) o 1 n = 2 ex. i=0 “ Then co = cl = 0, and h’ E (q: m), so h’ = 0 by pr0p. 4.3. 80 h E q. . Let C = {i E {0, ..., n}| i is a cut vertex}. C may be empty. For each i E C, let Vi1""’vip(i) denote the connected components of G - i. Let W g A1, be the vector space generated by {f , f} U 2 (a. - a.)x.| i E C, p = 1, ..., p(i)}. 1 2 {jEVip l J J Ergpggitign 4.6. (q: m)1 = W. Raga; By pr0p. 4.4., it is enough to show (q: m)1 E W. If G is a block, this inclusion follows immediately from pr0p. 4.5. So assume G is not a block and relabel so that 0 is a cut vertex. For any n h = 2 bixi e A1, let 2: (h) = {i e V(G)| bi = 0}. Let (21(h)) be the i=0 subgraph generated by (71(h)); i.e., V((?£(h))) = V(G) n %(h), and E((WID) = {e = ii 6 E(G)| iri 6 7411)}- Let 210(h) denote the set of all vertices in the connected component of (?l(h)) which contains 0 (thus ?l(h) = (b if b0 9% 0). Suppose the proposition is false. Let h E (q: m)1\ W, h as above. We can replace h mod W by h’ = h — bof1 and so assume that 23 h0 = 0. So the following set is nonempty 95’: {h E (q: m)1\ Wlb0 = 0}. Choose h E oi’ so that 2(O(h) is a maximal (w.r.t. inclusion) element of 210(1)|l E of}. If flo(h) = V(G), then h = 0 E W, so we can choose v E V(G)\?to(h) and let u0 = 0, 111, ....,u If necessary, replace v by the first ui on the path such that ui ¢ %0(h). In=v beapathfromOtov. Relabel G so that ui = i, for i = 1, ..., m. If bm = 0, then 11m = v E 24h). Since there is a path in %(h) from 0 to v, v E 2t0(h), a contradiction. So bm ,t 0. Clam. m - l is a cut vertex. P_rggi. @334. m = 1. There is nothing to show. W. m > 1. If m - 1 is not a cut vertex, then G — (m—l) is connected, so there is a path from m — 2 to m which does not contain m - 1. But then m -2, m -1, and m lie on a common cycle and thus in the same block. Now bm—2 = b = 0. So bm = 0 by prop. 4.3. This is a m-l contradiction. Therefore m — l is a cut vertex of G. Let V0 be the set of all vertices in the component of G — (m — 1) which contains m. P_rQQf. Suppose not. Then (%O(h)) U (V0) is connected. So there is a path P in (%O(h)) U (V0) from m — l E %0(h) to m E V0, say P is v0 = m - 1, vl,..., vt = m. Then v0, ...., vt, v0 is a cycle in G, so In - 1, v1, and m are in the same block of G. As 24 m — 1 1! V0, (111 - 1)v1 E E((ito(h))) and v1 E 710(h). So bm—1= b = 0 v 1 forces bm_1 = bV1 = 0 forces bIn = 0, by pr0p. 4.3, a contradiction. E ( ) bm Let g = a — a. x., and h’ = h - _ g. Then jEV0 m—l J J am—l 3m g E W, so h’ E h mod W, and h’ E 97. But by the claim, 210(h) g 210(h’), and the coefficient of x is zero in h’, so m flo(h) S 7(0(h’), contrary to choice of h. So 4.6 holds. . In the following lemma, b(G) will denote the number of blocks of G. Lemma 4.7. Let G be a graph with cut vertices v1, ..., v s 2 1. Then there exists V1, ..., V such that t 1) Each Vi is the vertex set of some connected component of G — vj for some j. 2) For each j = 1, ..., s, G - v. has all of its connected J components in {V1, ..., Vt} except for one component. 3) ViI U V. for i=1, ....,t j1. i=1 5) t = b(G) — 1. REEL We prove 4.7 by induction on the number of blocks of G. As 9 2 1, the smallest case is when G has two blocks. Letting V1 be the vertex set of either component of G — vl satisfies the conditions. Suppose G has m blocks and the lemma holds for all G with m - 1 blocks. Choose a block B which contains only one cut vertex v of G. Let G’ be the subgraph of G obtained by deleting all edges and vertices of B, except v. Choose Vi, ..., Vé satisfying 1 - 5 for G’. 25 Let V1=V(B—v), and for i=2, ...,t+1 let Vi_1, if v E Vi-l Vi = . It follows immediately that 1—4 hold Vi_1 U V1 if v E V§_1 for V1, ..., Vt+1’ and t + l = b(G’)-1 +1 = b(G’) = b(G) — 1, so 5 holds also. I For the next prOposition, it should be noted for each i = 1,..., t, there is a unique cut vertex vj such that Vi is a component of G - vj. The proof is as follows. Let v1 and v2 be cut vertices with vl at v2. Let C be a connected component of G - v1, D a connected component of G — v2, and suppose C = D. Choose w E V(C) so that wv1 E E(G). If w 1! v2, then wv1 E E(D). But wv1 ¢ E(C), so C a! D, a contradiction. Suppose now that w = v2. then v2 E V(C) - V(D), again contrary to C = D. Therefore, C = D is impossible. EeraitiQa 4.§. dimk((q:m)1/q1) = b(G) — 1. 21991. If b(G) = 1, this is pr0p. 4.5. So assume b(G) > 1. We use pr0p. 4.6 and show that dimk W/ql = b(G) - 1. Let V1, ...,, Vt be as in lemma 4.7. Let v1.0) be the cut vertex corresponding to Vi’ and define gi = 2 (av__ - ak)xk. Then g1,....,gt l(EVi j(l) E W. Fix i. The connected components of G - v. 1 are among V1, ...., Vt’ except for one, say V0. We can assume G — vi = V0 V VI v---- V Vm by relabeling V1,...,V if needed (mgt). Let h = 2 (a -aj)x.. t . v. J jEV0 1 26 Suppose the vertex sets of the other components are V1,...,Vm. Then 1 1 (’3 ) ii E ‘5 2 ( ) a - — g. =a x - ax — a —a x = V11 2 j=1J Vi 1:0 1‘ k=0 J‘J‘ i=1 revj Vi k k n E(a — )x-E (a —a)x=23 (a -a)x=h.Sohis k=0 vi 8“ 1‘ kEV(G)—VO Vi 1‘ 1‘ kEVO V' k k in the span of {g1,....,gt, fl, f2}. But h was any generator of W not in this set, so {g1,....,gt, fl, f2} spans W. So dimkW/q1 5 t = b(G) — 1. By 4.7 (3), g1,....,gt are linearly independent, and by 4.7 (4) (and choice of f1, f2), (gl,...,gt ) n (f1, f2) = 0, so gl,....,§t are linearly independent over W/ql' So dimkW/qlz b(G) — l. I Using prop. 4.8 it is now easy to determine the type of Am' fIfhgirem 4.2. Let A = k[G], for some connected graph G, and let m be the homogeneous maximal ideal. Let |V(G)| = n+1, and assume n22. If |E(G)| = s, then the Cohen-Macaulay type of Am is r=s-(n+1)+b(G). For example, the graph in Figure 4.1 has 10 edges, 8 vertices, and 4 blocks, so the type is 6. ErQQf 9f 4.2. As |E(G)| > 1 q # m, so dimk(q: m/q)0 = 0. By prop. 4.8, dimk(q: m/q)1 = b(G) — 1. Finally, by the proof of 4.1, qi = mi for i 2 3, so (q: m)2 = A2, and thus dimk(q: m/q)2 = dimkA2/q2 = s — n (again from the proof of 4.1). Moreover, dimk(q: m/q)i = 0 for 123. So a, 2 r =.3 l((_2), then r=n—1. I Example 4.11. A graph is mama; if it can be embedded in s2 (This does agt imply that the corresponding union of lines embeds in P2). By Euler's formula, |V(G)] - |E(G)| + |R(G)] = 2, where R(G) is the set of regions determined by any planar embedding of G. 80 if G is a planar graph, then r = |R(G)| + b(G) — 2. In particular, if G is a cycle on n+1points, r=1. I W- A W graph on n+1 points is the graph which contains all possible ["31] edges. If G is a complete graph, then ,- [“5”] -.,...1- ["21] 1:] = [Sl- Examplgilfi. Let G be the graph obtained by adding 1 edges to a cycle on 11 +1 points, 0 g i g [’2‘] - 1. Then |V(G)| = n + 1, |E(G)| =n+1+i, and b(G)=1, so r=1+i.Soagraphon n + 1 points can have any type r, 1 g r 58]. I If G = G1 U G2, where V(Gl) n V(G2) consists of a single vertex, and |V(G1)| > 2, |V(G2)| > 2, then Theorem 4.9 implies r = II + r2 + l, where ri is the type of Gi' This makes it easy to prove the following. Ergmsiiion 4.14. If G is a graph on n + 1 > 2 vertices, then r 5 [3]. If n + 1 > 3, then equality holds only when G is a complete graph on n + 1 vertices. 28 m. We induct on the number of blocks of G. 11 b(G) = 1, then |E(G)| g [“42“] so r g [g], by 4.9. Suppose b(G) = m > 1 and that the proposition is true for b(G) < m. Then G = C1 U G2, where G1 is a block with t vertices, G2 is a graph on n — t + 2 vertices, b(G2) = m - 1, and V(Gl) n V(G2) consists of a single vertex. Suppose t>2 and n—t+2>2,i.e.,2 1, thus |V(G)| > 2. Any connected graph satisfies |E(G)| 2 |V(G)] - 1, with equality precisely when G is a tree. 80 if r = 1, — 1 + b(G) S 1, 29 b(G) 5 2. If b(G) = 2, |E(G)| = |V(G)| — 1, and G is a tree with two blocks, i.e. G has two edges. If b(G) = 1, then |E(G)| = |V(G)]. Removing one edge yields a tree, which must be a path, as b(G) = 1. 80 G is a cycle. I It should be noted that Hochster [Ho, p. 194] has given formulas for the betti numbers for all Stanley—Reisner rings. The aim of this chapter has been to show that in the case of graphs, the type (i.e., the last betti number) is given by a simple formula which can be established using more elementary methods than those used in [Ho]. CHAPTER 5 SIMPLY CONNECTED UNIONS OF LINES Pencils of projective lines are the simplest non-trivial examples of Cohen—Macaulay unions of lines [GW, 4.1]. In this chapter we review these examples and apply some results of [GW], to extend them to a larger class of examples. We say that union V of 3 lines is s_i_rap_ly 991M231 if it is connected and there is no sequence of distinct lines in V, L1,...,Lt, with t 2 3, such that Li n Li+1 at (1), L1 n Ltat (I), and Li n Lj= d), for 1< |i - j] < t - 1. That is, a simply connected union is one which contains no cycles. The goal of this chapter is to characterize those simply connected V which are Cohen—Macaulay, and to obtain some results regarding the type. If V consists of lines through a single projective point (i.e., V is a pencil), then V is simply connected, and V is always Cohen—Macaulay. A proof of the first part of the following prOposition was given in [GW]. 8 Prgpgsition 5.1. ([GW, 4.1]). Let V = U Li be a union of lines i=1 through a point v in P”. Then V is Cohen—Macaulay, and the type of V is the same as the type of a union of points in Pn-l. REEL By a linear change of coordinates, we may assume v = (1: 0:....:0). Then each line is determined by v and one other point vi = (0: ailz....:ain). As before, let .9; = J(Li). Then .9; = (fi1""’fin-1)’ with deg fij = 1, fij E k[X1,....,Xn], as no other linear polynomial vanishes at v. 30 31 Let s = k[x0,....,xn], T = k[X1,....,Xn] “s; = :91“ T, I = .9; and “300 i 1 s I = n 73° As before, A = k[V], so that A = 8/1. i=1 Claim. A = T/I[x0]. Bum. Let V: T -+ 8/1 be the canonical map. Then ker V = InT=(n3;)nT=n(3i’nT)=n3i’=I. So ET/I-lS/I isa monomorphism. For each i, fil’m’fin—l are linearly independent in 8 over k, thus algebraically independent over k, and thus algebraically independent in T over k. So ht 3; = n — 1 for each i, and dim T/I = I. But dim 8/1 = 2, and the canonical map T/I [x0] --1 S/I is subjective, so it is transcendental over T/I, by consideration of 0 dimensions, and so this map is also injective. This proves the claim. Each 7; has height 11 — l and is generated by linear forms, so it is the ideal of a point in Pn—l. As v; = (ailz...ain) E 7(2), this implies T/I = k[U vi] is the coordinate ring of a union of points, and thus contains a regular element g. As x is regular by choice of coordinates, {x0, g} o is a regular sequence in A, so A is Cohen—Macaulay. By [HK, Satz 1.22], the type of Am is the same as the type of (A/(x0))_, where m m is the image of m in A/(xo). That is, the type is the same as that of (T/I)_. I m In order to study simply connected unions of lines, we use the notion of Cohen—Macaulification of the graded ring A. r ii .2. ([GW, 1.71). Let C=UM‘“gK, where M is the homogeneous maximal ideal of A. Then G is a Cohen—Macaulay ring, 32 finitely generated as an A—module, and C = M—n for 11 >> 0. Moreover, every Cohen—Macaulay ring containing A and finite over A contains C. _Rgmarig. C is called the Cohen—Macaulification of A, as it is the natural generalization to the graded ring A of the notion of Cohen-Macaulification of a local ring, as introduced by Grothendreck. Clearly A is Cohen—Macaulay if and only if A = C. The main goal of this chapter is to determine when A = C in the case of a simply connected V. We will use the following construction of C given by Geramita and Weibel. As before, we let the vertices of V be the points of the form Li n LJJ‘ (1), for i 9% j. Label these points P1,....,P and let .2 = .7(P.) I" J J for i = 1,...t. Let J]. = n{.9i|.9§ C 23}, so that S/Jj (where = k[XO, ..... ’an) is the coordinate ring of the lines of V which pass through Pj' The set of ideals {Jj} n {9} U {.23} is partially ordered by inclusion and corresponds to a directed diagram 1‘ of the rings {S/Jj} U {8/53} U {S/.%} as in Figure 5.1. S/J S/J "MN? ”N. N M2 21/ \e ./ S/Ql S/Q2 S/Qt Figure 5.1. The diagram 1‘ 33 The maps of I‘ are defined as follows. For .9; g .2], aji is the canonical map a. .' S/Jj -o 8/9, and 'Bji is the canonical map ,6]. i: M 3/9; -+ .Zj. We will assume throughout the rest of the chapter that V is connected, and consists of more than one line. Thus every line passes through at least one vertex, and the initial rings of the diagram I‘ will be the rings S/J.. J The pullback (or inverse limit) of the diagram I‘ is the ring: {(11, ...,,f)€II S/Jjle(f)=B(fl) j-l whenever the maps A and B are defined and have the same image}. Theorem 53 ([GW, 84]) The pullback ring of the diagram I‘ is the Cohen—Macaulification of A. We first prove as a corollary a slightly stronger version of the theorem. Prgmsition 5.4. Let I" be the directed diagram of the rings {S/Jj} U {S/fli’} (i.e., I" is a subdiagram of I‘). Then the pullback ring of I" is the Cohen—Macaulfication of A. Prmf. Let C’ be the pullback ring of I". Clearly C Q C’ as I" is a subdiagram of I‘ with the same initial rings. So it suffices to shown C' Q C. Let(f, ..,ft)EC’_ IIS/Jj, andsuppose(fl, ft)¢C. Ifjatk, l_<_j 1, with m 35 as large as possible. As V is simply connected, m > 3. Let Pi = Li n Li+1 for 1 S i 5 m—l. Then I’m—1 contains Pm—l and Pm—2’ and, by assumption, there is at least one other line LO which contains Pm—l and some other vertex. L0 = Li for i < m — 1, otherwise V would contain a 100p. So wlog we may assume LO at Lm. Thus there is a vertex Pm at Pm_1 on Lm, and PIn = Lm n Lm+1 for some Lm+1. This contradicts the maximality of m. I The following proposition characterizes those simply connected V which are Cohen—Macaulay. Proposition 5.7. Let V be a simply connected union of lines. Then A = k[V] is Cohen—Macaulay if and only if the degree 1 graded part of A is the pullback of the degree 1 graded parts of the rings of F2. m. (G). Suppose A is Cohen-Macaulay. Then A = C, which is the pullback of 1‘2, by prOpositions 5.4 and 5.5. In particular, A is a pullback in degree 1. (F). We induct on the number of vertices t. Suppose V contains t vertices and A is a pullback in degree 1. If t = 1, then A is Cohen-Macaulay, by pr0p. 5.1. Suppose t > 1 and the prOposition is true when V contains at most t — 1 vertices. By lemma 5.6, choose P1 satisfying the condition of 5.6. Let L1 be the unique line containing Pl as well as some other vertex P2 (L1 exists as t > 1). Let A1 be the directed diagram {S/le J 1‘ 1} U {S/PilLi contains at least two vertices}, Let A2 be the directed diagram {S/Jl} U {S/Pl}. Thus 1"2 = A1 0 A2. Let Vl be the union of those lines of V which contain at least one vertex not equal to P1, and let V2 be the union of all lines of V which 36 contain P1. As k[V2] is Cohen—Macaulay, k[Vz] is the pullback of A2. We show next that k[Vl] is the pullback of A1. Let deg fi = 1, i = 2,...,t, and suppose that (f ""’ft) E H S/Ji i¢1 is in the pullback of Al. A3 Vl n V2 consists of a single line (f2. that A is a pullback in degree 1, there is some f E A1 such that f 5 f2 f2...,ft) e 11 S/Ji is in the pullback of T2. So, by the assumption mod J1 and f5 fi mod Ji for i = 2,...,t. If we replace f by the image of f in k[Vl], then these congruences still hold. Thus k[Vl]l is the pullback of the degree one parts of the diagram A1. As Vl contains t - 1 vertices, and is simply connected, it follows by inductive assumption that k[v A 1] is Cohen-Macaulay, and thus that k[Vl] is the pullback of 1. In order to show that A is the pullback of the Let L =VlflV 1 2' diagram F2 = A1 U A2, it now suffices to show that A is the pullback of the diagram in Figure 5.2. k[V]] k[VQJ ’1 ’2 k[Ll] Figure 5.2 37 As A is the pullback in degree 1, and the maps irl and x2 are subjective, we have: (*) edim k[V] = edim k[Vl] + edim k[V2] - edim k[Ll] Let mi = edim k[Vi], for i = 1, 2. For any projective variety W, edim k[W] = dimk Span (W), where Span (W) denotes the linear span in A?” of the cone of W (by [K, p. 165]). So it follows from (*) that dimk Span (V) = dimk Span (Vl U V2) = dimk Span (V1) + dimk Span (V2) — dimk(Span (V1) n Span (V2)) = dimk Span (V1) + dimk Span (V2) - dimk(Span (L1). Thus, dimk Span ((Vl) n Span (V2)) = dimk Span (L1), and so Span (V1) n Span (V2) = Span (Ll)' It follows that we can change coordinates on V as follows. First, we may assume by an initial change of coordinates that V is minimally embedded; i.e. edim (V) = n + 1. [To do this, change coordinates so that the affine subspace Span (V) consists of all points with non—zero coordinates only in the first 11 + 1 coordinates. Then (if v g P’f) k[V]/(X .,Xm) ~ k[V]] Next, change coordinates so that n+1’" = J(L1) = (x2,...,xn), J(Span (v1)) = (xml,...,xn ) J(Span (V2)) =(X2,..., xm1—1)° Thus J(V1) 2 J(Xm1,...,Xn ), J(V2) 2 (X2,..., Xml—l)’ and then k[LIJ = k[XO, X1]: k[Vl] = k[Xo, X1"""xm1"1]’ k[V = k[Xo, x1, xml,...,xn]. 2] 38 [Note that there are algebraic relations among the generators of the last two rings] As J(Vi) g .7 (V), there are canonical surjective maps pi: k[V] —1 k[Vi]. Let (f, g) E k[Vl] x k[V2] be in the pullback of the diagram in Figure 5.2. Choose F, G so that F E k[X XIn _1] g k[V], G E k[Xo, 1 0,..., X1, Xml""’xn] g k[V], and such that pl(F) = f, p2(G) = g. Let F = Fl + H1 and G = G1 + H2, where HI = F(xo, x1, 0,...,0), H2 = G(xo, x1, 0,...,0). Then xl(p1(Hl)) = 21(f) = 2(g) = 2(p2(H2)). As pi 0 xi: k[V] -. k[Ll] is the identity when restricted to k[xo,x1], it follows that HI = H2. Moreover, F1 + H1 + G1 E A = k[V], and 122(F1) = 0. p1(Gll = 0. so (p1(F1 + H1 + G1), p2(F1 + H1 + G1)) = (f, g). S0 A is the pullback of the diagram in figure 5.2., and therefore also the pullback of T2. So A is Cohen-Macaulay. . The preceding proof allows an alternative characterization of which simply connected unions of lines are Cohen—Macaulay. For i = 1,...,t, let Vi = 7’ (Ji)‘ Thus Vi consists of all lines through the vertex Pi' ThQrem Q8. Let V be a simply connected union of lines. Then edim k[V] S E edim k[Vi] - 2(t-1), where t is the number of vertices of V. Moreover, equality holds if and only if A = k[V] is Cohen—Macaulay. ELQQI- We induct on t. For t = 1, V = V1, and there is nothing to show. Let V have t > 1 vertices and assume the corollary is proven when V has less than t vertices. Let P1 be as in the proof of pr0p. 39 _ 1 _ 5.7. Let V2—V, and V1--Ui>1 the proof of 5.7, A is Cohen-Macaulay vi, and L1 = v1 n v2. then, as in PC A is the pullback of the diagram in Figure 5.2 We show first that the inequality holds. We always have edim k[V] 5 edim k[Vl] + edim k[V2] — edim k[Ll] = edim k[Vl] + edim k[V2] - 2. Then, by inductive assumption, there follows edim k[V] 5 2 edim k[Vi] — 2(t — 2) i>1 + edim k[V2] — 2. As V2 = V1, we conclude edim k[V] g z: edim k[v‘] — 2(t — 1). i>1 Suppose A is Cohen-Macaulay. By pr0p. 5.7, A1 is the pullback of the degree 1 parts of the rings of F2. So by the proof of 5.7, k[V1]1 is the pullback of the degree 1 parts of the rings of A1, and thus is Cohen—Macaulay, by 5.7. So, by inductive hypothesis, edim k[Vl] 5 2: edim k[Vi] - 2(t — 2) i>1 But also, edim k[V] = edim k[Vl] + edim k[V2] - edim k[Ll] = edim k[Vl] + edim k[V2] - 2 = z edim k[Vi] + edim k[Vl] — 2(t — 2) — 2 i>1 . = 23 edim k[V‘] — 2(t — 1) 121 . For the converse, suppose that edim k[V] = 2 edim k[V]] - 2(t — 1). 121 Then edim k[V] = l: edim k[Vi] — 2(t — 2) + edim k[Vl] — edim k[Ll] 1> 40 Also edim k[V] S edim k[Vl] + edim k[V2] — edim k[Ll]. If equality does not hold, then edim k[Vl] + edim k[V2] — edim k[LIJ > 2 edim k[V'] — 2(t — 2) + edim k[Vl] — edim k[Ll] i>1 But, as V VI, this implies edim k[Vl] > 2 edim k[Vi] — 2(t - 2) i>1 2 = which is impossible. So (1) edim k[V] = edim k[Vl] + edim k[V2] — edim k[Ll], and also (2) edim k[Vl] = )3 edim k[Vi] - 2(t — 2). i>1 By inductive hypothesis, (2) implies that k[Vl] is Cohen—Macaulay. In particular, k[Vl]1 is a pullback of Al in degree 1. It then follows from (1), that k[V]1 is a pullback of diagram 5.1 in degree 1, and thus a pullback of F2 in degree 1. So, by prOposition 5.7, A is Cohen-Macaulay. I Mark. These results are false if V is not simply connected. Let V be a 2 x 2 configuration of lines on a nonsingular quadratic surface. Then k[V] is Cohen—Macaulay (theorem 1.4). Also edim k[V] = 4. However, 23 edim k[Vi] — 2(t —1) = 4(3) — 2(3) = 6 ,1 4. We conclude the chapter by showing how to compute the Cohen—Macaulay type of Am in the case where V is simply connected and A is Cohen-Macaulay. As noted in Chapter 2, we can compute the type by computing the graded analogue of the socle. For each i = 1,...,t, let ri be the Cohen—Macaulay type of the ring k[Vi], and r the Cohen—Macaulay type of k[V]. 41 Brgpgsition {2.9. If V is simply connected and Cohen—Macaulay, then Prmf. We induct on T. For t = 1, there is nothing to show. As in the proof of pr0p. 5.7, we reduce to consideration of the diagram in figure 5.2. By inductive assumption k[Vl] has type 2 ri, and it suffices to ZSiSt show that the type of k[V] is the sum of the types of k[Vi], i = l, 2. For ease of notation, let ri be the type of k[Vi]. Assume coordinates have been changed as in the proof of pr0p. 5.7. Choose a system of parameters (fl, f2} for A, with deg fi = 1, i = 1, 2. Let Fi be the preimage of fi in k[XO,....,Xn]. Then W 2M and J(Ll)2nPi so WW=M As (F1, F2) and J(L1) are generated by degree 1 forms, (F1, F2) + J (L1) is the ideal of a linear variety, and thus prime. So (F1, F2) + J(L1) = M; i.e., (F1, F2) + (X2,...,Xn) = M. So X0 = alFl + blF2 — G1, X1 = a2Fl + sz2 - 62, for some ai, bi E k, Gi E (X2,...,Xn) . Replace Fi by F; = aiFl + biF2, so as to assume wlog that F1 = X0 + G1, F2 = XI + GZ’ with Gi 6 (X2,...,Xn) . [Note that {F’, Fé} is still a system of parameters because X0 + G1 1‘ (X1 + G2) 3L1 b1 (12 b2 9‘ F1’ F2 6 (F’, Fé) ) 411:1, F2) + fl Pi = M.) G #0 42 Let g1,...,g e k[Vl] such that g1 + H,...,g + a is a basis for the 1'1 r1 socle of k[V1]/§, and h1,...,hr2 6 k[V2] such that h1 + H,...,hr2+ E1- is a basis for the socle of k[V2]/Ei. Suppose gi = Fi(xo’ x1) + Gi(xo,...,xn), where each monomial term of Gi contains one of x2,...,xn as factor. then replace gi by gi - Fi(f1, f2) E gi mod (f1, f2), so that we may assume wlog that each monomial term of gi contains one of x2,...,xn as a factor. Similarly, we may assume that each monomial term of each hi contains one of x2,...,xn as a factor. Then 11(gi) = 12(hj) = 0 E k{L1] for all i, j, so (gi, 0), (0, hj) E k[V] for all i, j. M. (gi, 0) is in the socle of k[V]/q for each i. m. It suffices to show that (xk, xk)(gi, 0) = (0, 0) = 0 E k[V]/q, for k = 0,...,n. Thus it suffices to show xkgi = 0 in k[Vl]/§. But this true because g1 is in the scole of k[Vl]/§. QLai_m_2. (0, hj) is in the socle of k[V]/q for each j. ELQQI. Similar. Clearly the elements {(gi, 0)} U {0, hj)} are linearly independent over k. In order to show r = r1 + r2, it now suffices to show that this set generates the socle of k[V]/E. Let k1,...,kr E k[V] such that (k1, k1)"”’(kr’ kr) generate the socle of k[V]/H. As in the previous argument, let k1 = Fl(x0, x1) + Gl(xo,...,xn), where each monomial term of G1 contains one of x2,...,xn as a factor. Then replace k1 by k1 — Fl(f1, f2) 5 k1 mod (f1, f2) so that we may assume wlog that each monomial term of k1 contains on of x2,...,xn as a factor. As before, it follows that (kl, O), (0, kl) e k[V] for each i. 43 For each j = 1,...,n, (klxj, klxj) = (0, 0) = 0 E k[V]/q. Consequently, klxj = o in k[Vll/H and in k[V2]§. Thus k1 is in the socles of both k[v11/a and k[v21/a. Let r1 1' 2 k] = 1'21 bjhj e k[V2]/q. r r l 2 Then 0‘ i k) = = 2 a'(g’i 0) + 2 b-(O, 11') SO {(3, 0)} U {(0, h)} l 1 i=1 1 1 j=1 j j 1 j generates the socle of k[V]/q as claimed. . Note that by prOposition 5.1, the Cohen—Macaulay type for a union of lines through a single vertex can be reduced to the computation of the type for a union of points obtained by intersecting with a hyperplane. Thus prOp. 5.9 reduces the problem of computing the type of a simply connected union of lines to computations of the types of unions of points. As any union of points in Fri—1 can be coned to a union of lines through a single vertex in P“, prOp. 5.9 is in some sense the strongest result we can expect for an arbitrary simply connected Cohen—Macaulay union of lines. Finally, note that pr0p. 5.9 is false if V is not simply connected. Example 5.19. Let V consist of 3 lines in P2 with 3 vertices. Then k[Vi] has type 1 for each i = l, 2, 3, as Vi is a hypersurface. But V is also a hypersurface, so k[V] has type 1 )6 3. CHAPTER 6 HILBERT FUNCTIONS One necessary condition for A = k[V] to be Cohen—Macaulay is that its Hilbert function be twice differentiable. In this chapter, this condition will be discussed, and an example constructed which shows that the condition is not sufficient. dimkAi, ' _ Let P (i) be the Hilbert function of A, P (i) = . A A . 0, 1 < 0 Let ai = P A(i). Then the first sequence of differences is {bi}iel’ where b. = a. — ai_1, and the second sequence of differences is {ci}iel’ where differentiable. As in Chapter 2, let Q A(z) denote the Poincaré series of A, If ci 2 0 for all i, we say that PA(i) is twice (I) . Q A(z) = 2 P A(i)zl. As noted in the proof of proposition 2.2, if A is i=0 Cohen-Macaulay, then we can choose f1, f2 6 Al such that the set of images {f1, f2 } is a system of parameters for A m’ in which case QA/f1A(Z) = QA(Z) - QA(_1)(Z). So iEO bizi = QA/f1A(Z)' Similarly, with q = (f1, {2), we have QA/q(z) = QA/f1A(Z) — Q(A/f1A)(—1)(Z)’ so °° i . i . . . = = , > . = > as Hilbert functions are nonnegative. As ci = 0 for i < 0, it follows that ci 2 0 for all i and P A(i) is twice differentiable. We conclude the following: Proposition 5.1. ([GMR]) Let A = k[V] be the coordinate ring of a union V of projective lines in Pi. If A is Cohen-Macaulay, then the Hilbert function P A(i) is twice differentiable. 44 45 Proposition 6.1 gives a useful criterion for showing that certain unions of lines are not Cohen—Macaulay. Here is a simple application. Example 6.2. Let L1 = 7’( 0, X1), L2 = V(XZ, X3), V 2 L1 U 3 _ _ L2 g Pk' then .XV) — (X0, X1) n (X2, X3) — (X0X2’ X0X3, X1X2, X1X3). P A(i) is then easily computed, 1, i = 0 2(i+1), i < 0 Then b1: 3, b - 2, and c2 = —1, so V is not Cohen—Macaulay. Unfortunately, the condition of proposition 6.1 is not sufficient. 6 Example 6.3. Let A = k[V], where V = U Li is the union of the i=1 following six lines: L1 = 71x2, x3) = V(Xli X3) = 7(X3, x1 ’ X2) = rix3, x1 + x2) = “X0, X2) = 7(x1, x0 — x t‘r‘r‘r‘t" cams-wt»: 3) Then A is not Cohen—Macaulay, but P A(i) is twice differentiable. M. We show first that A is not Cohen-Macaulay. Figure 6.1 illustrates the intersection relations among the Li: 46 Figure 6.1 Here P1 = xx], x2, x3), P2 = V(XO, x2, x3), and P3 = 74x0, x1, 4 Let V1 = U L., V2 = L1 U L5, and V3 = L2 U 1’6 Then each i=1 ‘ Vi is contained in a hyperplane Hi, where H1 = 7(X3), H2 = 7/(X2) 3 and H3 = V(Xl). So edim k[Vi] = 3 for i = 1, 2, 3, and thus £1 edim k[Vi] - 2(3 - 1) = 5, while edim k[V] = 4 < 5. As v is simply connected, V is not Cohen—Macaulay, by Theorem 5.8. We now show that the Hilbert function is twice differentiable. We begin by computing the Hilbert function for C, the Cohen—Macaulfication of A, as this provides an upper bound for the Hilbert function of A. By prop. 5.5., C is the pullback ring of the diagram in Figure 6.2 k[V2] k[Vl] k[V3l “2,1 k[L Figure 6.2 47 Fix the degree = i for all the graded rings in the diagram of Figure 6.2. Let 7i be the map 7,: 14v?)i e k[vl]i e k[v3]i s k[L1]i e k[L2]i, defined by 7i(f, g. h) = (02,16) - 014(3). 03,201) - 013(3)) Let (T, H) E k[Ll]i 6 k[L2]i, and choose preimages f E k[V2]i and h E k[V3]i (using the fact that 021 and 03,2 are surjective). Then 7i(f, 0, h) = (f, If), so 7i is also surjective. For each i, C. = ker 71’ so 1 3 2 (*) P (i) = 23 P .(i) — 2 P (i). C j=1 k[V]] k=1 k[Lk] 1 _ 1 _ J(v ) _ (x1 X2(Xl — X2) (x1 + X2), X3). and k[V ] _ k[XO, X1, x2, 1 . . X3]/J(V ) g k[XO, x1, x21/(xl x2(x1 — x2) (x1 + x2», which IS the coordinate ring of a hypersurface in Pi of degree 4. So we have the following exact sequence, where S = k[XO, X1, X2], 0 -+ S(-4) -+ S -1 k[V1]-o 0. . Pk[V1](i) = Ps(i) — PS(_4)(i). As Ps(i) = [‘32] and PS(_4)(i) = ['52], we conclude that [“2”], if ogigs P i = k[V1]() 4i — 2, if i 24 Similarly, k[V2] is isomorphic to the coordinate ring of a hypersurface of degree 2 in 1212‘, so (i) = PS(i) — PS(_2)(i) Pk[V2] 48 [‘32], if ogigl — 2i +1, if i 2 2 V3 also consists of two lines in a plane, so P 3 (i) = P 2 (i). k[V] k[V] Finally PML l(i) = i + 1, as Lk is isomorphic to 1211‘. k It now follows from equation (*) that PC(0) = 3%] — 2(1) = 1, pC(1) = 3g] - 2(2) = 5, pC(2) = [g] + 2(5) — 2(3) = 10 130(3) = [3] + 2(7) — 2(4) = 16 and, for i 2 4, PC(i) = (4i - 2) + 2(2i + 1) —- 2(i + 1) = 6i - 2. We have established: QLai_m_1. PC(0) = l, PC(1) = 5, and PC(i) = 6i — 2, for i > 2. We next show: Claim 2. C is generated by elements of degree 1. Proof of Claim 2. Choose lines L’,....,Lé in P: as follows: For i 1, 2,...,5, L; = 7(J(Li)k[X0,....,X4], X4), L6 = ”(X11 X0 ’ X31 X0 ' X4) The same intersection relations hold among L; as bold among the Li' Let 1’ 4 2/ 3/ V = U L5, V =Li U L’, and V =Lé U Lé. Then, for i = 1, 2, i=1 ‘ k[vi'] g k[Vi] via the map induced by s1(xi) = xi, for i = o, 1, 2, 3, and 11(X4) = 0. k[V3I] g k[V3] under a different map. Note first that 49 J(v3') = (x1, x3, x4) n (x1, x0 — x3, x0 - X4) = (x1, x4 - x3, x3) n (x1, x4 - x3, x0 — x3) = (x1, x4 — x3, x3(x0 - x3)). J(v3) = (x1, x3) n (x x — x 1’ o 3) = x3(x0 - x 3)- Let r2: k[X0,....,X4] -1 k[X0,....,X3] be defined by «2(Xi) = Xi’ for . 3’ 3 1 = 1, 2, 3, and 12(X4) = X3. Then 12(J(V )) = J(V ), and 7r2 induces the desired isomorphism. Note finally that 7'1 induces isomorphisms k[Li] g k[Li], for i = l, 2. We have now constructed the following commutative diagram. k[vz'] Figure 6.3 Commutativity is easily checked, as most of the maps are projections. As the maps induced by lrl and 12 are isomorphisms, it follows easily that k[V] and k[V’] have isomorphic Cohen—Maculifications. As edim k[vi'] = edim k[Vi], for i = 1, 2, 3, and edim k[V]] = 5 with v' simply connected, it follows from Theorem 6.8 that k[V’] is 50 Cohen—Macaulay. Thus k[V’]g C, and C is generated by elements of degree 1, as claimed. Proof of Claim 3. Only the last equality is non-trivial, and Pk[V](2) g 10 is obvious. Suppose Pk[V](2) < 10. then V lies in a hypersurface w of degree 2. suppose w 2 H1. Then w n H1 2 v1 which is a reducible curve of degree 4, contrary to Bézout's theorem. Thus W 2 H1. As deg W = 2, we must have W = H1 U H4 for some hyperplane H4. But then H4 2 L5 11 L L5 U L6 = (I). So there is no such W and Pk[V](2) = 10. suffices to show that PJ(V)(3) = 4. Let f6 J(V)3, and W = V(f). 6’ which is impossible, as As in the previous claim, it follows by Bézout's theorem that "W 2 H1. So W 2 H1 U Wl for some hypersurface Wl of degree 2. As before W1 3 L5 U L6" But L5 U 16’ is a union of two skew lines in Pi, and as such, is isomorphic by a change of coordinates to the union of two skew lines given in example 6.2. In that example, we found P A(2) = 6. Thus there are 10 — 6 = 4 linearly independent elements of degree 2 vanishing on two skew lines. So there are only four linearly independent elements of degree 3 vanishing on V. Therefore Ple](3) = 16. glam. k[V]i = Ci’ for i _>_ 2. WM- By claims 1, 3, and 4, Plelm = Pc(i) for i = 2, 3. As k[V] C C, k[V]i = Ci for i = 2, 3. If f6 Ci for i 2 4, then, by claim 2, f = 2 gj, where each gj is a product of degree 1 elements of C. Thus, each gj is a product of elements of C2 and C3. As each of these factors is in k[V], so is gj, and therefore, f e k[V]i. . . 1 if i = 0 M' Pklvlm : 6i-2, if i z 1 ' Proof of Claim 6. For i = O and 1, this follows by claim 3. For i 2 2, this follows by claims 1 and 5. Now, letting ai = Ple](i), we have a first sequence of differences {bi}?=1 with b0 = 1, b1 = 3, bi = 6, for i 2 2. The second sequence of differences is cl = 1, c2 = 2, c3 = 3, ci = 0, for i 2 4. So Pklvlfi) is twice differentiable. . Example 6.3 allows us to answer a related question posed by Geramita, Maroscia and Roberts, relating to differentiable 6—sequences. asequences can be defined as follows [St, p. 60]. If h and i are positive integers, then b can be written uniquely in the form l?) h = n. — 1 n. + L1- 1 ]+....+ [i1], where ni > ni_1 >....>n. 2 j 2 1. Define ni 1 ni_1 + 1 nj 1 . i + 1 + i + j + 1 . An fieguence 18 a sequence {ai}°i°:1 such that a0 = 1 and ai+1 5 a?) for i 2 1. It can be shown that the O-sequences are precisely the sequences of values of Hilbert functions of Standard G—algebras R with k0 = k[St, Theorem 2.2] Thus Pk[V](i) defines an asequences. An 0—sequence is a 2jifferentiable asequence if its first and second sequences of differences are also 0—sequence. Geramita, Maroscia and Roberts have asked [GMR, remark 7.13] whether a union of lines in Pi whose Hilbert function is a 2—differentiable asequence must be Cohen—Macaulay. It is easy to check that the first and second sequences of differences in example 6.3 are 0—sequences, so the answer is negative. CHAPTER 7 EXISTENCE THEOREMS In order to be Cohen—Macaulay, a union of lines must be connected, and the Hilbert function of its coordinate ring must be a twice differentiable asequence. These global necessary conditions suggest that "most" unions of lines are not Cohen—Macaulay. In this chapter we will show that any union of lines V is contained in a union of lines W which is Cohen—Macaulay. That is, it is always possible to add lines to V so as to make some larger union W Cohen—Macaulay. This suggests: (1) that the class of Cohen—Macaulay unions of lines may contain much more than just the classes of examples discussed in previous chapters, and (2), that there may be no simple local necessary condition (i.e., condition on the vertices) for a union of lines to be Cohen-Macaulay. The second observation, if true, would contrast the Cohen-Macaulay prOperty sharply with seminormality, as a union of lines cannot be seminormal if the lines through each vertex are not linearly independent (by [DR, 2, 3.5]). The results in this chapter hold not only for unions of lines, but also for all unions of linear varieties of unmixed projdim 2 1, so we will prove the results in this form. We will need the following prOposition about minimal generating sets of ideals of definition of certain unions of points in PE. Brgmsitign 2.1. Let f1,...,f8 E k[Xo,...,Xn] be polynomials of degree 1, with s > n, with the following property. For every choice of 1 5 i1 < , f. ,....,f. < in+1 S s are linearly independent over R. For each such 11 1 n+1 52 53 choice of i1,...,in+1 define _ H .. gi1"”’in_l ‘ j¢{i1,...,in_1} 1 Then (a) V = 7/( . n . fi ,...,fi )) 5 Pi: is a set of [3] points. 1511<....<1nSs 1 n (b) V is not contained in any hypersurface of degree < s - n + 1. (c) The gi ""’i are linearly independent over k. (d) If 1 n—l I = ({gil"”’i }), then I = J(v). Proof. (a) it follows immediately from the linear independence condition that each of the [3] ideals (fi ,....fi ) is the ideal of a point in 1 11 P11: and that any two of these ideals are distinct. (b) Let H = V(h) with deg h 5 s - n, and suppose H 3 V. Then, for any choice of 1 _<_ i1 < ....< in—l S s, H contains each of the s — n + 1 points V(fil,....fin_l, fj)’ where j j! {i1,...,in_1}, so by Bézout's theorem, H contains the line 7’ (fi ,....,fi ). So 1 n—l h e (1 (fi ,...,fi ). Repeat the argument. For any choice of 1_<_i1<.... [3], it follows that the points of V are in generic [3] - position [(301, pr0p. 3]. Moreover, J(V) n Sd = I n Sd , as I g J(V) and o o d +n . . s _ 3+1 3 _ o __ S _ dlmk(I n Sdo) 13, by (c), [n—l] _ [n ] - [n] _. [n ] [n] _ dimk(J(V) n Sdo). It now follows by [G02, prOp. 4]. that J (V) = I. I We now construct Cohen-Macaulay unions of linear varieties which are higher dimensional analogues of the unions of points in prop. 7.1. 55 Prop. 7.2. Let f1,....,fs E k[Xo,...,Xn] be homogeneous of degree 1. Let t < s, 2 5 t _<_ n, and suppose that for every choice of l 5 il < < it+1 S 3, fi ,...,fi are linearly independent over k. For each choice of 1 t+l 1< i < < i < 3 define g. = II L. Then: ' l t+l ' ” l ’ ’1 . . . J 1 t—l j¢{11,...1t_1} (a) V = 7’( (1 (fi ,...,fi )) g PE is a union of [i] lgi1<...1. Let W = 7’( n (fi ,...,fi )). By 1$i1<...