Minimal free resolutions for certain affine monomial curves

Transcription

1 Contemporary Mathematics Volume 555, 11 Minimal free resolutions for certain affine monomial curves Philippe Gimenez, Indranath Sengupta, and Hema Srinivasan This paper is dedicated to Wolmer V Vasconcelos Abstract Given an arbitrary field and an arithmetic sequence of positive integers m < < m n, we consider the affine monomial curve in A n+1 parameterized by X = t m,, = t m n In this paper, we conjecture that the Betti numbers of its coordinate ring are completely determined by n and the value of m modulo n We first show that the defining ideal of the monomial curve can be written as a sum of two determinantal ideals Using this fact, we describe the minimal free resolution of the coordinate ring in the following three cases: when m 1 mod n (determinantal, when m n mod n (almost determinantal, and when m mod n and n =4 (Gorenstein of codimension 4 Introduction Let denote an arbitrary field and R be the polynomial ring [X,, ] Consider the -algebra homomorphism ϕ : R [t] givenbyϕ(x i =t m i, i =,,n Then the ideal P := er ϕ R is the defining ideal of the monomial curve in A n+1 given by the parametrization X = t m,, = t m n The-algebra of the semigroup Γ N generated by m,,m n is [Γ] := [t m,,t m n ] R/P, which is one-dimensional and P is a perfect ideal of codimension n Itiswellnown that P is minimally generated by binomials Moreover, P is a homogeneous ideal and [Γ] is the homogeneous coordinate ring if we give weight m i to the variables X i Henceforth, homogeneous and graded would mean homogeneous and graded with respect to this weighted graduation Assume now that the positive integers m,,m n satisfy the following properties: 1991 Mathematics Subject Classification Primary 13D; Secondary 13A, 13C4 Key words and phrases Monomial curves, arithmetic sequences, determinantal ideals, Betti numbers, minimal free resolutions The first author is partially supported by MTM1-79-C-, Ministerio de Educación y Ciencia, Spain The second author thans DST, Government of India for financial support for the project Computational Commutative Algebra, reference no SR/S4/MS: 614/9, and for the BOYSCAST 3 Fellowship This collaboration is the outcome of the second author s visits to the University of Missouri, Columbia, USA during Fall 7 and to the University of Valladolid, SPAIN for two months in 9 He thans both the institutions for their support 1 c 11 American Mathematical Society 87

2 88 PHILIPPE GIMENEZ, INDRANATH SENGUPTA, AND HEMA SRINIVASAN (i gcd (m,,m n =1; (ii <m < <m n and m i = m + id for every i [1,n], ie, the integers form an arithmetic progression with common difference d; (iii m,,m n generate the semigroup Γ := Nm i minimally, where N = {, 1,,}, ie, m j / i n; i j i n Nm i for every i [,n] Definition 1 A sequence of positive intergers (m =m,,m n is called an arithmetic sequence if it satisfies the conditions (i,(ii and (iii above Let us write m = an + b such that a, b are positive integers and b [1,n] Note that b [1,n] and condition (iii on m,,m n ensures that a 1; otherwise m = b and m b = m + bd =(1+db contradicts minimally condition (iii Let (m =m,,m n be an arithmetic sequence We say that the monomial curve in A n+1 parameterized by X = t m,, = t m n is the monomial curve associated to the arithmetic sequence (m and denote it by C(m A minimal binomial generating set for the defining ideal P of C(m was given in [P], and it was rewritten in [MS] toprovethatp is not a complete intersection if n 3 An explicit formula for the type of [Γ] is given in [PS, Corollary 6] under a more general assumption of almost arithmetic sequence on the integers (m =m,,m n and it follows from this result that if (m =m,,m n is an arithmetic sequence then [Γ] is Gorenstein if and only if b = In this paper, we prove in Theorem 11 that ideal P has the special structure that it can be written as a sum of two determinantal ideals, one of them being the defining ideal of the rational normal curve We exploit this structure to construct an explicit minimal free resolution of the graded ideal P, in the cases when m 1orn modulo n, andwhenm modulo n and n 4 Note that a minimal free resolution for P had already been constructed for n = 3 in [S1] usingagröbner basis for P In [S] the following question was posed : do the total Betti numbers of P depend only on the integer m modulo n? In this article, we answer this question in affirmative for the above cases This question will be addressed in general in our wor in progress [GSS] 1 The defining ideal By [P] and[ms], one nows that the number of elements in a minimal set of generators of the ideal P depends only on m modulo n Our first result shows that P has an additional structure that will be helpful in the sequel Theorem 11 Let (m =m,m n be an arithmetic sequence and P be the defining ideal of the monomial curve C(m associated to m Then P = I ( X X 1 1 X a n X b +I X 1 X ( X b Proof It is nown from [P] and[ms], that P is minimally generated by the following set of binomials {δ ij i<j n 1} {Δ 1,j j =,,n+ b}, such that δ i, j := X i X j+1 X j X i+1, i<j n 1

3 MINIMAL FREE RESOLUTIONS FOR CERTAIN AFFINE MONOMIAL CURVES 893 and Δ 1,j := X b+j Xn a X j, j =,,n+ b The binomials δ ij are precisely the ( n the matrix A = ( X X 1 X 1 X generators of the ideal of minors of 1 On the other hand, the binomials Δ 1,j are the minors from the first and the jth column of the matrix ( X a n X b B = X b Since the rest of the minors of B are already in the ideal I (A, one gets that P = P 1 + P,whereP 1 and P are the determinatal ideals generated by the maximal minors of the matrices A and B respectively The actual generators also depend only on the first term m, the common difference d and the length n of the arithmetic sequence (m This is no surprise as these three numbers do determine the arithmetic sequence However, we also have that the number of minimal generators of the ideal P is ( n + n b + 1 and hence only depends on n and b We conjecture that the following statement is true: Conjecture 1 Let (m =m,,m n be an arithmetic sequence Then all the Betti numbers of the homogeneous coordinate ring [Γ] of the affine monomial curve C(m ina n+1 associated to (m are determined by n and the value of m modulo n In the next section, we discuss the conjecture in cases b = 1, andn and construct a minimal resolution when b = 1orn using the aforesaid determinantal structures In the first special case, b =1,wehavethatP is a determinantal ideal The second special case is b = n, thatiswhen ( X a n X B := and therefore the ideal P is the principal ideal generated by +1 Xn a+1 The third special case is b =, that is when [Γ] is Gorenstein In this case, we wiil give the Betti numbers when the codimension n is 4 The minimal resolution 1 First case: determinantal (m 1 modulo n Assume that b =1 In this case, ( X a n X 1 B := X 1 and hence I (A I (B Thus, P = P is a determinantal ideal of codimension n generated by the maximal minors of the (n +1 matrix B Therefore, the homogeneous coordinate ring [Γ] is minimally resolved by the Eagon-Northcott complex The resolution is given by n+1 R n+1 D n 1 (R 3 R n+1 D 1 (R R n+1 R [Γ],

4 49 PHILIPPE GIMENEZ, INDRANATH SENGUPTA, AND HEMA SRINIVASAN such that D s 1 (R =(S s 1 (R denotes the R-module which is the dual of the symmetric algebra of R and the module S s 1 (R isfreeofrans, with a basis the set of monomials of total degree (s 1 in λ, λ 1 (symbols representing basis elements of R Therefore, up to an identification, D s 1 (R =(S s 1 (R is a free R-module of ran s, with a basis {λ v λv 1 1 v,v 1 N,v + v 1 = s 1} For every s =1,,n,letG s denote s+1 R n+1 D s 1 (R, which is a R-free module of ran s ( n+1 s+1 generated by the basis elements (ei1 e λv is+1 λv 1 1, for every 1 i 1 <i < <i s+1 n +1andv,v 1 N,v + v 1 = s 1 The differentials d s : G s G s 1 are given by the following formulae: d 1 (e i1 e i =X i1 1X i X i 1X i1, 1 i 1 <i n +1, and for s =,,n and 1 i 1 < <i s+1 n +1, s+1 d s ((e i1 e is+1 λ v λv 1 1 = ( 1 j+1 X ij 1((e i1 ê ij e is+1 λ v 1 j=1 λ v 1 1 s+1 + ( 1 j+1 X ij ((e i1 ê ij e is+1 λ v λv j=1 such that summands on the right hand side involve only non-negative powers of λ and λ 1 We have therefore proved the following: Theorem 1 The minimal free resolution of the homogeneous coordinate ring [Γ] of the affine monomial curve C(m in A n+1 associated to the arithmetic sequence of integers (m =m,,m n with the property m 1 modulo n is R n d n R (n 1( n+1 n d n 1 R s(n+1 s+1 d s R (n+1 d 1 R ϕ [Γ] In particular, the Betti numbers are β =1and β s = s ( n+1 s+1 for every s [1,n] Remar The case a =1(andb = 1 corresponds to semigroups Γ of maximal embedding dimension, ie, those semigroups such that the inequality m(γ e(γ is an equality, being m(γ := m = an + b and e(γ := n +1 the multiplicity and the embedding dimension respectively Note that the above argument gives indeed the graded resolution (with respect to the grading given by deg(x i =m i = m + id: Corollary 3 Under the hypothesis of Theorem 1, the minimal graded free resolution of [Γ] is given by

5 MINIMAL FREE RESOLUTIONS FOR CERTAIN AFFINE MONOMIAL CURVES 915 d n ( n n +1 R( (a + n + dm + d d ( n 1 ( n +1 R( nm + d d ( d n 1 n 1 R( (a + d + n 1m + d r i 1 r 1 <r n 1 n i=1 d n 1 d s 1 r 1 <<r s n 1 r 1 <<r s 1 n ( s 1 R( sm + d ( s 1 s d r i i=1 s 1 R( (a + s 1+dm + d i=1 r i d d s 1 d 1 r 1 <r n d 1 ϕ R [Γ] R( m (r 1 + r 1d ( n 1 R( (a +1+dm d Second case: almost determinantal (m n modulo n In this case, we show that [Γ] can be resolved minimally by a mapping cone using the resolution for R/P 1, which is the homogeneous coordinate ring of the rational normal curve The minimal free resolution for R/P 1 is the Eagon-Northcott complex given by E : n R n (S n (R R n (S (R R R/P 1 Since E s := s+1 R n (S s 1 (R = R s( n s+1, for every s =1,,n 1, E := R = R, theresolutione taes the form R n 1 R (n ( n n 1 R s( n s+1 R ( n R R/P1 The differentials d 1 s : E s E s 1 are defined similarly as d s above, for every s =1,,n 1 = Now consider the short exact sequence of R-modules R/(P 1 :Δ 1, Δ 1, R/P 1 R/P 1 +(Δ 1, where the map R/(P 1 :Δ 1, Δ 1, R/P 1 is the multiplication by Δ 1, The colon ideal (P 1 :Δ 1, is exactly equal to P 1,sinceP 1 is a prime ideal and Δ 1, P 1, being a part of a minimal generating set for the ideal P = P 1 + P Therefore, the above short exact sequence becomes, R/P 1 ( (a + d +1m Δ 1, R/P 1 R/P 1 +(Δ 1,,

6 69 PHILIPPE GIMENEZ, INDRANATH SENGUPTA, AND HEMA SRINIVASAN taing gradation into account We define the graded complex homomorphism ψ : E( (a + d +1m E as the multiplication by Δ 1,, which is a lift of the map R/P 1 ( (a + d +1m Δ 1, R/P 1 In fact, the complex homomorphism ψ is simply multiplication by Δ 1, The mapping cone F(ψ is given by the free modules F s = E s 1 E s = R (s 1(n s R s( s+1 n, for every s =,,n (with E 1 = E n = and the differentials d s : F s F s 1, defined as d s(x, y =(ψ s 1 (y+d 1 s(x, d 1 s 1(y, for every s =1,,n R/P 1 +(Δ 1, is resolved by the mapping cone F(ψ It is minimal because each of the maps in ψ is of positive degree and the resolution E is minimal Hence, the mapping cone F(ψ is the minimal free resolution of the homogeneous coordinate ring [Γ] = R/P We have therefore proved the following: Theorem 4 The minimal free resolution of the homogeneous coordinate ring [Γ] of the affine monomial curve C(m in A n+1 associated to the arithmetic sequence of integers (m =m,,m n with the property m n modulo n is R n 1 d n R (n ( n 1+(n 1 n d n 1 R 1+(n d 1 R ϕ [Γ] In particular, the Betti numbers are β =1, β 1 =1+ ( n for every s [,n] and βs =(s 1 ( n s ( +s n s+1 As in Section 1, one can be more precise and give the graded resolution First note that the graded minimal resolution of P 1 is given by n 1 ( n +1 d 1 n 1 R( nm + d d ( d 1 s s 1 R( sm + d r i d d 1 1 r 1 <<r s n 1 r 1 <r n R( m (r 1 + r 1d d 1 s 1 d 1 1 R R/P 1

7 MINIMAL FREE RESOLUTIONS FOR CERTAIN AFFINE MONOMIAL CURVES 937 Corollary 5 Under the hypothesis of Theorem 4, the minimal graded free resolution of [Γ] is given by d n n 1 ( n +1 R( (a + n + d +1m + d d ( n 1 ( n +1 R( nm + d d ( d n R( (a + n + dm + d r i 1 r 1 <<r n 1 n d n 1 d s 1 r 1 <<r s n ( s 1 d R( sm + d r i ( s 1 r 1 <<r s 1 n d R( (a + s + dm + d r i d s 1 d 1 r 1 <r n R( m (r 1 + r 1d R( (a +1+dm d 1 R ϕ [Γ] 3 Third case: Gorenstein of codimension 4 (m modulo n and n =4 When b =, the ideal P is Gorenstein of height n and hence the first and the last Betti numbers are nown This provides a very interesting family of Gorenstein ideals of height n generated by (n 1(n +/ binomials When n =,P is a complete intersection and when n =3,P is the ideal of 4 4pfaffiansofa5 5 sew symmetric matrix For n = 4, the ideal P is a height 4 Gorenstein ideal minimally generated by 9 elements and therefore has Betti numbers 9, 16, 9 and 1 In fact, we can show that with this weighted grading, the shifts in a graded resolution of R/P are as follows: Theorem 6 Given two non-negative integers a and d, consider the monomial curve C(m in A 5 associated to the arithmetic sequence (m =4a +, 4a ++ d, 4a ++d, 4a ++3d, 4a ++4d Setq := a +1 Then the defining ideal P of C(m is a height four Gorenstein ideal whose minimal graded free resolution is

8 894 PHILIPPE GIMENEZ, INDRANATH SENGUPTA, AND HEMA SRINIVASAN R( q(q +d +9 9d ( 9 ( 7 R( 8q d R( q(q +d +5 d R( q(q +d +5 5d =7 =3 ( 7 R( 6q 4d R ( 6q d R( 6q 8d R( q(q +d +3 d =5 ( 4 R ( q(q +d +3 d R( q(q +d +3 5d = ( 6 ( R( 4q d R( 4q 4d R( q(q +d +1 d P = Proof We now that P is Gorenstein of height 4 and by Theorem 11, it is generated by the minors of A and B, where ( X X 1 X X 3 A = X 1 X X3 and ( X a 4 B = = X 4 X X 1 X X X 3 X 4 Moreover, the 9 minimal generators of P are the six minors of A whose degrees are m + d, =, 3, 4, 4, 5, 6form := 4a + = (a +1 = q, and the 3 minors of B involving the first column which are of degrees m (a + d +1+d, =, 1, Note that for both determinantal ideals I (A andi (B, the Eagon-Northcott complex provides a minimal resolution The 8 determinantal relations from the Eagon-Northcott resolution of I (A and the 6 determinantal relations from the Eagon-Northcott resolution of I (B involving the first column will necessarily be among the 16 minimal syzygies of P The degrees of the other two relations are determined by the symmetry of the resolution Thus the remaining two relations must be of degrees q(q +3+(q +d and q(q +3+(q +4d This determines the rest of the degrees in the resolution We note that when d = 1, this ideal can never be determinantal even though it is height 4 and generated by 9 elements That is, this ideal is not the ideal of minors of a 3 3 matrix for sum of the degrees of such an ideal must be even For in this case, the ideal is generated by 9 elements whose degrees add up to 7d + m Since m is even, this number is odd whenever d is odd Hence if d is odd, it cannot be a determinantal ideal However, it is not clear if it can be determinantal for even values of d Of course, the conjecture only says that the Betti numbers are determined by b References [GSS] P Gimenez, I Sengupta and H Srinivasan, Minimal graded free resolution for monomial curves defined by an arithmetic sequence (11, in progress [MS] A K Maloo and I Sengupta, Criterion for complete intersection of certain monomial curves, in: Advances in Algebra and Geometry (Hyderabad, 1, , Hindustan Boo Agency, New Delhi, 3

9 MINIMAL FREE RESOLUTIONS FOR CERTAIN AFFINE MONOMIAL CURVES 959 [P] D P Patil, Minimal sets of generators for the relation ideals of certain monomial curves, Manuscripta Math 8 ( [PS] D P Patil and I Sengupta, Minimal set of generators for the derivation module of certain monomial curves, Comm Algebra 7(11 ( [S1] I Sengupta, A minimal free resolution for certain monomial curves in A 4, Comm Algebra 31(6 ( [S] I Sengupta, Betti numbers of certain affine monomial curves, In EACA-6 (Sevilla, F-J Castro Jiménez and J-M Ucha Enríquez Eds, ISBN: Department of Algebra, Geometry and Topology, Faculty of Sciences, University of Valladolid, 475 Valladolid, Spain address: pgimenez@agtuvaes School of Mathematical Sciences, RKM Viveananda University, Belur, India Current address: Department of Mathematics, Jadavpur University, Kolata, WB 7 3, India address: senguptaindranath@gmailcom Mathematics Department, University of Missouri, Columbia, MO 6511, USA address: SrinivasanH@missouriedu