Generic doublings of almost complete intersections of codimension 3
aa r X i v : . [ m a t h . A C ] J un GENERIC DOUBLINGS OF ALMOST COMPLETE INTERSECTIONS OFCODIMENSION 3
JAI LAXMI
Abstract.
We study Gorenstein ideals of codimension 4 derived from generic doublingsof almost complete intersection perfect ideals of codimension 3. We also investigate spinorcoordinates of such Gorenstein ideals with 8 and 9 generators. For an ideal J of commutativering R , the R/J module
J/J is called conormal module and R/J -dual of
J/J is callednormal module. We study properties of conormal and normal modules of almost completeintersection perfect ideals of codimension 3. Introduction
The problem of classifying Gorenstein ideals of codimension 4 was around ever since Buchs-baum and Eisenbud classified Gorenstein ideals of codimension 3 in [1]. The first resultsrelated to the structure of codimension 4 case were obtained by Kustin and Miller [10]. Westudy Gorenstein ideals of codimension 4 obtained from generic doubling of almost completeintersection perfect ideals of codimension 3.Christensen, Veliche and Weyman in [4] gave structures of two types of generic form ofalmost complete intersection ideals of codimension 3, of Cohen Macaulay even and oddtype, see [4, Prop. 2.2,2.4]. Moreover in [4, Theorem 4.1], they show that any minimal freeresolution of almost complete intersection perfect ideal J n over a commutative local ring R , of codimension 3 and Cohen Macaulay type n , is a specialization of one of the type,either Cohen Macaulay even or odd type stated in [4, Prop. 2.2,2.4]. So it is enough tostudy generic doublings of two types of generic form of almost complete intersection idealsof codimension 3, of Cohen Macaulay even and odd type. We denote generic form of almostcomplete intersection ideals of Cohen Macaulay type n given in [4, Prop. 2.2,2.4] as J n overa polynomial ring R in generic variables.In Section 2 we recall structure of generic form of almost complete intersection ideals ofcodimension 3, of Cohen Macaulay even and odd type, see [4, Prop. 2.2,2.4].We set generic almost complete intersection ring of type n as S n = R/J n with a canonicalmodule ω S n . Section 3 deals with equivariant generators of Hom S n ( ω S n , S n ). The S n -module J n /J n is called conormal module and S n -dual J n /J n is called normal module. In Section 4 westudy properties of conormal and normal modules of J n . Matsouka [9] proved that there isan embedding ϕ n : ω S n → S n such that S n /ϕ n ( ω S n ) ≃ J n /J n . Using embedding ϕ n we showthat S n -module Hom S n ( ω S n , S n ) has 4 generators and conormal module of J n is a reflexive Mathematics Subject Classification.
Key words and phrases.
Cohen Macaulay rings, Gorenstein ring, spinor coordinates, spinor structures,almost complete intersection, reflexive modules, conormal modules,normal modules. n -module. Moreover we observe that Hom S n ( ω S n , S n ) is a maximal Cohen Macaulay S n module.In Section 5 we construct generic doublings of S n of Cohen Macaulay odd and even typeusing generators of Hom S n ( ω S n , S n ). Such generic doublings give Gorenstein rings of codi-mension 4. By [5, Theorem 4.2], there exist spinor structures on generic doublings of S n . Weinvestigate spinor coordinates of Gorenstein ideals with 9 generators obtained from genericdoublings of S . There are two examples in Gorenstein ideals with 9 generators in [5, Ex-ample 5.3] where none of the spinor coordinates are among minimal generators of ideals. InProposition 6.1 we show that there are 5 spinor coordinates among minimal generators ofGorenstein ideals obtained from doublings of S . In Section 7 we study spinor coordinatesof generic doublings of S .2. Structure of almost complete intersection rings
We recall structures of almost complete intersections of codimension 3 given by Chris-tensen, Veliche, and Weyman in [4]. By [4, Theorem 4.1], any minimal free resolution ofalmost complete intersection perfect ideal J n over a commutative local ring R , of codimension3 and Cohen Macaulay type n , is a specialization of one of the type, either Cohen Macaulayeven or odd type stated in [4, Prop. 2.2,2.4].2.0.1. Structure of almost complete intersection for n = 2 m + 1 . We record results from [4,Prop. 2.2]. Let K be a field. Consider a (2 m + 1) × (2 m + 1) generic skew symmetric matrix C = ( c ij ), and 3 × (2 m + 1) generic matrix U = ( u kl ) with 1 ≤ k ≤ ≤ l ≤ m + 1.Let R be a polynomial ring over K on the entries of C and U . Set F = R m +1 and G = R .Set J m +1 = h x , x , x , x i where x = C m − ∧ u ∧ u ∧ u , x = C m ∧ u , x = C m ∧ u and x = C m ∧ u where u i are ith row of U . Then0 −→ F d −→ F ∗ ⊕ G ∗ d −→ R ⊕ G d −→ R → R/J m +1 → R/J m +1 where d = (cid:2) x x x x (cid:3) , d = (cid:20) CU (cid:21) ,d = w w . . . w m +1 v , v , . . . v , m +1 x x v , v , . . . v , m +1 − x x v , v , . . . v , m +1 − x x Here w i = ± Pf(ˆ i ) and v i,γ = P j,k ± ∆ j,kα,β Pf(ˆ i, ˆ j, ˆ k ) where Pf(ˆ i, ˆ j, ˆ k ) is Pfaffians of C withomitted i, j, k rows and columns, and ∆ j,kα,β is 2 × U involving α, β rows and j, k columns. Also γ is complement of α, β in { , , } . The Schubert varieties are normaldomain, so almost complete intersection R/J m +1 are also. 2.0.2. Structure of almost complete intersection for n = 2 m . We record results from [4, Prop.2.4]. Let K be a field. Consider a 2 m × m generic skew symmetric matrix C = ( c ij ), and3 × m generic matrix U = ( u kl ) with 1 ≤ k ≤ ≤ l ≤ m . Let R be a polynomialring over K on the entries of C and U . Set F = R m and G = R .Set J m = h x , x , x , x i where x = C m , x = C m − ∧ u ∧ u , x = C m − ∧ u ∧ u and x = C m − ∧ u ∧ u . Then0 −→ F d −→ F ∗ ⊕ G ∗ d −→ G ∗ ⊕ R d −→ R → R/J m → R/J m where d = (cid:2) x x x x (cid:3) , d = (cid:20) CU (cid:21) ,d = w w . . . w m x x x v { , } , v { , } , . . . v { , } , m − x − v { , } , − v { , } , . . . − v { , } , m − x v { , } , v { , } , . . . v { , } , m − x . Here w i = P j,k,l ± ∆ j,k,l Pf(ˆ i, ˆ j, ˆ k, ˆ l ) and v { α,β } ,i = P j u γ,j Pf(ˆ i, ˆ j ) where Pf(ˆ i, ˆ j, ˆ k, ˆ l ) is Pfaffianof C with i, j, k, l rows and columns omitted, and ∆ i,j,k is a 3 × U involving i, j, k columns. Also γ is complement of { α, β } in { , , } . The Schubert varieties are normaldomain, so almost complete intersection R/J m are also.3. Generators of
Hom S n ( ω S n , S n )In this section we discuss generators of Hom S n ( ω S n , S n ). Assume notation stated in Section2. Let S n = R/J n with a canonical module ω S n . Since S n is normal domain, ( S n ) p is regularfor all prime ideals p of height one. Thus ( S n ) p is complete intersection for height one primeideals p . Then by [7, Theorem 1] ω S n is reflexive S n module and J n /J n is a torsion free S n module.Let us study generators of ω S n . For a matrix A , A t denotes the transpose of A . Wehave v ,j x + v ,j x + v ,j x − w j x = 0 and w j x − v { , } ,j x + v { , } ,j x − v { , } ,j x = 0 for n = 2 m + 1 and n = 2 m respectively for 1 ≤ j ≤ n since d d = 0. Set L n = h x , x , x i .Then w j ∈ ( L n : J n ) for all j . Since ω S n is the first Koszul homology module on generators x , x , x and x of J n , see [8], we get ω S n ≃ ( L n : J n ) /L n .Set matrices in S n as ϕ m +1 = ¯ w ¯ w . . . ¯ w m +1 ¯ v , ¯ v , . . . ¯ v , m +1 ¯ v , ¯ v , . . . ¯ v , m +1 ¯ v , ¯ v , . . . ¯ v , m +1 , ϕ m = ¯ w ¯ w . . . ¯ w m ¯ v { , } , ¯ v { , } , . . . ¯ v { , } , m − ¯ v { , } , − ¯ v { , } , . . . − ¯ v { , } , m ¯ v { , } , ¯ v { , } , . . . ¯ v { , } , m where ¯ denotes going mod J n .Let { e , e , e , e } be a basis of S n . Then by [9, Prop. 1] there is a short exact sequence0 −→ ω S n ϕ n −→ S n π n −→ J n /J n → . (3)3ith π n ( P i =1 t i e i ) = P i =1 T i x i where T i is a representative of t i in R .We discuss the equivariant form of generators of Hom S n ( ω S n , S n ). These generators playcrucial role in construction of generic doublings of almost complete intersection. Proposition 3.1.
Consider resolutions (1) and (2). Set the transpose of ϕ n as H n . Then im( H n ) ⊆ Hom S n ( ω S n , S n ) .Proof. By a change of basis, a free presentation of ω S is( F ⊕ G ) ⊗ S n d t ⊗ S n −−−−→ F ∗ ⊗ S n → ω S n → . (4)The equivariant form of the generators of Hom S n ( ω S n , S n ) is the kernel of the map F ⊗ S n d ⊗ S n −−−−→ ( F ∗ ⊕ G ∗ ) ⊗ S n . For n = 2 m , consider the module H m generated by the image of m V F ⊗ G F ⊗ m − V F ⊗ ( F ⊗ G ) where we identify the factors m − V F and F ⊗ G with sub representations in R , andby the image of m V F ⊗ V G F ⊗ m − V F ⊗ ( V F ⊗ V G ), where we identify the factors m − V F and V F ⊗ V G with sub representations in R.Set basis of F and G as B F = { f , . . . , f m } and B G = { e , e , e } respectively. Let P r + s be a symmetric group on { , , . . . , r + s } . Define map∆ : r + s ^ F → r ^ F ⊗ s ^ F as ∆( f ∧ · · · ∧ f r + s ) := X σ ∈ P r,sr + s ( − sgn σ f σ (1) ∧ · · · ∧ f σ ( r ) ⊗ f σ ( r +1) ∧ · · · ∧ f σ ( r + s ) where P r,sr + s = { σ ∈ P r + s | σ (1) < · · · σ ( r ); σ ( r + 1) < · · · < σ ( r + s ) } . Then we have m ^ F ⊗ G ∆ ⊗ −−→ F ⊗ m − ^ F ⊗ G ⊗ ∆ ⊗ −−−−→ F ⊗ m − ^ F ⊗ ( F ⊗ G )such that(∆ ⊗ f ∧ f ∧ · · · ∧ f m ⊗ e i ) = X σ ∈ P , m − m ( − sgn σ f σ (1) ⊗ f σ (2) ∧ · · · ∧ f σ (2 m ) ⊗ e i , (1 ⊗ ∆ ⊗ − sgn σ f σ (1) ⊗ f σ (2) ∧ · · · ∧ f σ (2 m ) ⊗ e i )= X τ ∈ P , m − m − ,τ ( σ (1))= σ (1) ( − sgn ( τσ ) f σ (1) ⊗ f τσ (2) ∧ · · · ∧ f τσ (2 m − ⊗ f τσ (2 m ) ⊗ e i where τ σ (2) < τ σ (3) < . . . < τ σ (2 m − m ^ F ⊗ ^ G ∆ ⊗ −−→ F ⊗ m − ^ F ⊗ ^ G ⊗ ∆ ⊗ −−−−→ F ⊗ m − ^ F ⊗ ( ^ F ⊗ ^ G ) 4∆ ⊗ f ∧ f ∧· · ·∧ f m ⊗ e ∧ e ∧ e ) = X σ ∈ P , m − m ( − sgn σ f σ (1) ⊗ f σ (2) ∧· · ·∧ f σ (2 m ) ⊗ e ∧ e ∧ e . (1 ⊗ ∆ ⊗ − sgn σ f σ (1) ⊗ f σ (2) ∧ · · · ∧ f σ (2 m ) ⊗ e ∧ e ∧ e )= X τ ∈ P , m − m − ,τ ( σ (1))= σ (1) ( − sgn( τσ ) f σ (1) ⊗ f τσ (2) ∧ · · · ∧ f τσ (2 m − ⊗ f τσ (2 m − ∧ f τσ (2 m − ∧ f τσ (2 m ) ⊗ e ∧ e ∧ e where τ σ (2) < τ σ (3) < . . . < τ σ (2 m ).For indexing set L ⊂ { , . . . , n } , we donote Pfaffian of C involving L rows and columnsof C as Pf( L ). For σ ∈ P , m − m set v { α,β } ,σ (1) = X τ ∈ P , m − m − ,τ ( σ (1))= σ (1) ( − sgn( τσ ) u γ,τσ (2 m ) Pf( d σ (1) , \ τ σ (2 m )) , w σ (1) = X τ ∈ P , m − m − ,τ ( σ (1))= σ (1) ( − sgn( τσ ) Pf( d σ (1) , \ τσ (2 m − , \ τσ (2 m − , \ τσ (2 m ))∆ τσ (2 m − ,τσ (2 m − ,τσ (2 m )1 , , where γ is the complement of { α, β } in the set { , , } , and ∆ j,k,l is a 3 × × (2 m ) matrix with rows u , u , u of matrix U involving columns j, k, l . Then the matrixpresentation of H m is the transpose of ϕ m For n = 2 m + 1 consider the module H m +1 generated by the image of m +1 V F F ⊗ m V F where we identify the factor m V F with subrepresentation in R , and by the image of m +1 V F ⊗ V G F ⊗ m − V F ⊗ V F ⊗ V G where we identify the factors m − V F and V F ⊗ V G withsub representations in R . Set basis of F and G as B F = { f , . . . , f m +1 } and B G = { e , e , e } respectively. We have the following maps: m +1 ^ F ∆ ⊗ −−→ F ⊗ m ^ F (∆ ⊗ f ∧ · · · ∧ f m +1 ) = X σ ∈ P , m m +1 ( − sgn σ f σ (1) ⊗ f σ (2) ∧ · · · ∧ f σ (2 m +1)2 m +1 ^ F ⊗ ^ G ∆ ⊗ −−→ F ⊗ m ^ F ⊗ ^ G ⊗ ∆ ⊗ −−−−→ F ⊗ m − ^ F ⊗ ( ^ F ⊗ ^ G )(∆ ⊗ f ∧ f ∧ · · · ∧ f m +1 ⊗ e i ∧ e j ) = X σ ∈ P , m m +1 ( − sgn σ f σ (1) ⊗ f σ (2) ∧ · · · ∧ f σ (2 m +1) ⊗ e i ∧ e j (1 ⊗ ∆ ⊗ − sgn ( σ ) f σ (1) ⊗ f σ (2) ∧ · · · ∧ f σ (2 m +1) ⊗ e i ∧ e j )= X τ ∈ P , m − m ,τ ( σ (1))= σ (1) ( − sgn( τσ ) f σ (1) ⊗ f τσ (2) ∧ · · · ∧ f τσ (2 m − ⊗ f τσ (2 m ) ∧ f τσ (2 m +1) ⊗ e i ∧ e j τ σ (2) < τ σ (3) < . . . < τ σ (2 m −
1) and τ σ (2 m ) < τ σ (2 m + 1).Set v σ (1) ,γ = X τ ∈ P , m − m ,τσ (1)= σ (1) ( − sgn( τσ ) ∆ τσ (2 m ) ,τσ (2 m +1) α,β Pf( [ ( σ (1) , \ τ σ (2 m ) , \ τ σ (2 m + 1))where γ is complement of { α, β } in the set { , , } , where τ σ (2) < τ σ (3) < . . . < τ σ (2 m − τ σ (2 m ) < τ σ (2 m + 1). Set w σ (1) = Pf( d σ (1)). Then the matrix presentation of H m +1 isthe transpose ϕ m +1 .Since d d = 0, this forces im( H n ) ⊆ ker( d ⊗ S n ) which is same as im( H n ) ⊆ Hom S n ( ω S n , S n ). (cid:3) Normal and conormal module of almost complete intersections
For an S n -module M , M ∗ = Hom S n ( M, S n ), and Ass ( M ) denotes the set of associatedprimes of M . We denote ht( p ) as the height of prime ideal p . For ideal J n , the S n -module J n /J n is called conormal module of J n , and ( J n /J n ) ∗ is called normal module. An S n -module M is said to be reflexive if the natural map j : M → Hom S n (Hom S n ( M, S n ) , S n )which sends m ϕ ∈ Hom S n ( M, S n ) to the map sending ϕ ∈ Hom S n ( M, S n ) to ϕ ( m ) ∈ S n is an isomorphism.In the next lemma we give a minimal number of generators of ( ω S n ) ∗ . We also show thatthe conormal module of J n is a reflexive S n -module. Lemma 4.1.
Consider the exact sequence (3). Then → Hom S n ( J n /J n , S n ) π ∗ n −→ S n ϕ ∗ n −→ Hom S n ( ω S n , S n ) → is exact. Moreover, Hom S n ( ω S n , S n ) is the image of the transpose of ϕ n and J n /J n is areflexive S n -module.Proof. Using exact sequence (3) and Hom S n ( − , S n ) we get0 → Hom S n ( J n /J n , S n ) → S n → Hom S n ( ω S n , S n ) → Ext S n ( J n /J n , S n ) → S n ( J n /J n , S n ) = 0. Resolutions (1) and (2) are of the form0 −→ R d −→ R n +3 d −→ R d −→ R → S n → . (7)Set ( − ) ! = Hom R ( − , S n ). We obtain0 → ker( d !2 ) → S n d !2 −→ S n +3 n d !3 −→ S nn → coker( d !3 ) → R ( − , S n ) on resolution (7). We see that ker( d !2 ) = Ext R ( S n , S n ). From theshort exact sequence 0 → J n → R → R/J n → R ( J n , S n ) ≃ Ext R ( S n , S n ). Note that any f ∈ Hom R ( J n , S n ) vanishes on J n .Thus Ext R ( S n , S n ) ≃ Hom S n ( J n /J n , S n ). We get an exact sequence0 → Hom S n ( J n /J n , S n ) → S n d !2 −→ S n +3 n d !3 −→ S nn → ω S n → . (9)Then ( J n /J n ) ∗ is a third syzygy module. Consider a short exact sequence0 → ( J n /J n ) ∗ −→ S n d !2 −→ im( d !2 ) → . (10)Then im( d !2 ) becomes a second syzygy module. By [6, Section 2] Ass (im( d !2 )) ⊂ Ass ( S n )and depth(im( d !2 ) p ) ≥ min(2 , depth(( S n ) p )) for all primes p of S n . If p ∈ Ass ( S n ), then ( S n ) p is complete intersection. For primes p of height 1, ( S n ) p is a regular local ring since S n is anormal domain. Thus ( ω S n ) p ≃ ( S n ) p and ( J n /J n ) p is free ( S n ) p -module for ht( p ) ≤
1. Thus0 −→ ( ω S n ) p ( ϕ n ) p −−−→ ( S n ) p → ( J n /J n ) p → S n ( J n /J n , S n )) p = 0 for ht( p ) ≤
1. By Proposition 3.1 im( d !2 ) ⊂ ( ω S n ) ∗ then we get the following commutative diagram0 / / ( J n /J n ) ∗ p / / ( S n ) p ( d !2 ) p / / (im( d !2 )) p / / (cid:15) (cid:15) / / ( J n /J n ) ∗ p / / ( S n ) p / / ( ω ∗ S n ) p / / . By Snake’s lemma (im( d !2 )) p ≃ ( ω S n ) p for ht( p ) ≤
1. For primes p of height ≥ d !2 ) p ) ≥ S n ) p ) ≥
2. Thus ω ∗ S n ≃ im( d !2 ). By the change of basis wesee that d !2 is the transpose of ϕ n . Then im( d !2 ) ⊂ ω ∗ S n which forces ω ∗ S n = im( d !2 ), and thisgives exact sequence (5).Applying Hom S n ( − , S n ) on exact sequence (5), we obtain0 → Hom S n ( ω ∗ S n , S n ) → S n → Hom S n (( J n /J n ) ∗ , S n ) → Ext S n ( ω ∗ S n , S n ) → . But Ext S n ( ω ∗ S n , S n ) = 0 as ω ∗ S n reflexive S n -module. We obtain a commutative diagram0 / / ω S n ϕ n / / φ ωSn (cid:15) (cid:15) S n / / J n /J n / / φ Jn/J n (cid:15) (cid:15) / / ω ∗∗ S n ϕ ∗∗ n / / S n / / ( J n /J n ) ∗∗ / / ϕ ω Sn is an isomorphism since ω S n is reflexive, and φ J n /J n is injective as J n /J n is torsionfree module. By Snake Lemma φ J n /J n is an isomorphism. Thus J n /J n is reflexive S n -module. (cid:3) Corollary 4.2.
Normal module ( J n /J n ) ∗ and ( ω S n ) ∗ are maximal Cohen Macaulay modules.Proof. We see that ( J n /J n ) ∗ is a maximal Cohen Macaulay module since it is a third syzygymodule of maximal Cohen Macaulay module ω S n by (9). 7ote that depth(( ω S n ) ∗ ) ≥ min(depth( J n /J n )+3 , depth( S n )) since ( ω S n ) ∗ is a third syzygyof J n /J n . By the depth criteria on exact sequence (3), depth( J n /J n ) = depth( ω S n ) −
1. Thisforces depth( ω ∗ S n ) = depth( S n ). Thus ( ω S n ) ∗ is a maximal Cohen Macaulay module. (cid:3) Next we study resolution of conormal module J n /J n . Remark 4.3.
Consider resolutions (1) and (2). Take maps θ m +10 = w w . . . w m +1 v , v , . . . v , m +1 v , v , . . . v , m +1 v , v , . . . v , m +1 , θ m = w w . . . w m v { , } , v { , } , . . . v { , } , m − v { , } , − v { , } , . . . − v { , } , m v { , } , v { , } , . . . v { , } , m . By a simple homological algebra result there exists θ : F ∗ → L F such that R / / S n / / R n / / θ n O O ω S n / / φ n O O . Then the mapping cone with respect to θ : F ∗ → L F yields resolution of J n /J n which neednot be minimal. Since depth( J n /J n ) = depth( S n ) − , by Auslander-Buchsbaum formula,the projective dimension of J n /J n is . Generic doublings of almost complete intersection
In this section we discuss generic doublings of almost complete intersection. Let us recallgeneric doublings of S n . In proof of Lemma 4.1 we see that S n is generically Gorensteinwith canonical module ω S n . Then by [2, Prop. 3.3.18] ω S n is an ideal of S n and S n /ω S n is aGorenstein ring of codimension 4.Let F be a minimal free resolution of S n , and h , h , h , h be generators of Hom S n ( ω S n , S n ).Consider a bigger ring e R = R [ α , α , α , α ]. Set ψ n = P i =0 α i h i . Denote e S n = e R/J n e R , and e F as a resolution of e S n . Then e F ∗ is a minimal resolution of ω e S n since e S n is perfect. We lift map ψ n : ω e S n → e S n to a map of complexes ψ : e F ∗ → e F . Then the mapping cone with respect tomap ψ denoted as Cone( ψ ) yields a resolution of Gorenstein ring e S n /ω e S n of codimension 4.In such case, we say that resolution of e S n /ω e S n is constructed by generic doubling.5.1. Generic doubling for n=2m+1.
Consider a bigger ring e R = R [ α , α , α , α ]. Set g j = α w j + P i =1 α i v i,j for 1 ≤ j ≤ n and ψ n = (cid:2) g g · · · g n (cid:3) . Then by Proposition 3.1map ψ n ∈ Hom e S n ( ω e S n , e S n ). Moreover ψ n is injective. Consider matrices A = − α α − α and M = (cid:2) U A (cid:3) . Define 8 n := · · · α α α M , , ;1 ,n M , ,n · · · M , m,n − α − M , ,n − M , ,n · · · − M , m,n − α M , ,n M , ,n · · · M , m,n − α , ψ n = − ( ψ n ) t , ψ n = − ( ψ n ) t . Then we get the following commutaive diagram F : 0 / / e F d / / e F ∗ ⊕ e G ∗ d / / e R ⊕ e G d / / e R / / e S n / / F ∗ : 0 / / e R d ∗ / / − ( ψ n ) t O O e R ∗ ⊕ e G ∗ d ∗ / / − ( ψ n ) t O O e F ⊕ e G ψ n O O d ∗ / / e F / / ψ n O O ω e S n / / ψ n O O . Hence the mapping cone with respect to ψ : F ∗ → F isCone( ψ ) : 0 −→ e R δ n −→ e F Le R ∗ ⊕ e G ∗ δ n −→ e F ∗ ⊕ e G ∗ Le F ⊕ e G δ n −→ e R ⊕ e G Le F ∗ δ n −→ e R → e R/I n → I n = J m +1 e R + im( ψ n ), δ n = (cid:2) d ψ n (cid:3) , δ n = (cid:20) d ψ n − d t (cid:21) , δ n = (cid:20) d − ( ψ n ) t − d t (cid:21) and δ n = (cid:20) − ( ψ n ) t − d t (cid:21) . Then Cone( ψ ) is a Gorenstein ring of codimension 4 with resolution of theform: Cone( ψ ) : 0 → e R δ −→ e R n +4 δ −→ e R n +6 δ −→ e R n +4 δ −→ e R → e R/I n → . (13)5.2. Generic doubling for n=2m.
Consider a bigger ring e R = R [ α , α , α , α ]. For1 ≤ j ≤ m set g j = α w j + α v { , } ,j − α v { , } ,j + α v { , } ,j . Then by Proposition 3.1, ψ n = (cid:2) g · · · g m (cid:3) is a map from ω e S n to e S n . Moreover ψ n isinjective. Define q i := u i α + u i α + u i α . Choose matrices A n = α − α − α α − α α , B n = (cid:2) B · · · B n (cid:3) , where B i = q i u i α − u i α u i α such that ψ n = (cid:2) B n A n (cid:3) . Set ψ = − ( ψ ) t and ψ = − ( ψ ) t . F : 0 / / e F d / / e F ∗ ⊕ e G ∗ d / / e G ∗ ⊕ e R d / / e R / / e S n / / F ∗ : 0 / / e R d ∗ / / − ( ψ n ) t O O e G ⊕ e R d ∗ / / − ( ψ n ) t O O e F ⊕ e G ψ n O O d ∗ / / e F ∗ / / ψ n O O ω e S n / / ψ n O O . ψ : F ∗ → F isCone( ψ ) : 0 −→ e R δ n −→ e F Le R ⊕ e G δ n −→ e F ∗ ⊕ e G ∗ Le F ⊕ e G δ n −→ e G ∗ ⊕ e R Le F ∗ δ n −→ e F → e R/I n → I n = J m +1 e R + im( ψ n ), δ n = (cid:2) d ψ n (cid:3) , δ n = (cid:20) d ψ n − d t (cid:21) , δ n = (cid:20) d − ( ψ n ) t − d t (cid:21) and δ n = (cid:20) − ( ψ n ) t − d t (cid:21) . Therefore Cone( ψ ) is a Gorenstein ring of codimension 4 with resolution ofthe form: Cone( ψ ) : 0 → e R δ n −→ e R n +4 δ n −→ e R n +6 δ n −→ e R n +4 δ n −→ e R → e R/I m → . (15)6. Spinor coordinates of (1,9,16,9,1)
We study spinor coordinates of Gorenstein ring with 9 generators obtained from genericdoubling of almost complete intersection ring S . We consider resolution given in (13) for n = 5. In [5, Theorem 2], there is a hyperbolic basis of e R say { e , . . . , e , e − , . . . , e − } withhyperbolic pairs { e i , e − i } . Denote columns of δ with respect to e i and e − i as i and ¯ i respec-tively. In [5, Theorem 2] spinor coordinates are denoted as ( e a ) K where K ⊂ {± , . . . , ± } of cardinality 8 with odd number of ¯ i . In Proposition 6.1 we see that there are 5 spinorcoordinates among minimal generators of ideal I . Proposition 6.1.
There are spinor coordinates of resolution (13) for n = 5 which areamong the minimal generators of ideal I .Proof. We use Macaulay 2 [11] to compute 8 × δ . Denote M KL as 8 × δ involving K rows and L columns. Then M , , , , , , , , , , , , , , = ± g x , M , , , , , , , , ¯2 , , , , , , = ± g x , M , , , , , , , , , ¯3 , , , , , = ± g x , M , , , , , , , , , , ¯4 , , , , = ± g x , M , , , , , , , , , , , ¯5 , , , = ± g x . Then by [5, Theorem 2] spinor coordinates are(˜ a ) ¯1 , , , , , , , = ± g , ( e a ) , ¯2 , , , , , , = ± g , ( e a ) , , ¯3 , , , , , = ± g , (˜ a ) , , , ¯4 , , , , = ± g , (˜ a ) , , , , ¯5 , , , = ± g . These 5 spinor coordinates are part of minimal generating set of I but not contained in J .Other non zero spinor coordinates are of the form ± α i x j ± α k x l and Pf(ˆ i ) x j where Pf(ˆ i ) isthe Pfaffian of C obtained by omitting ith row and column. These other spinor coordinatesare not among the minimal generating set of I . (cid:3) Remark 6.2.
Consider the following examples:
Let R be a polynomial ring in variables on the entries of × generic matrix X . Theideal I is generated by × minors of matrix X . Let R be a polynomial ring in variables with ideal I generated by the equation of Segreembedding of P × P × P into P .In both examples, I is Gorenstein ideal of codimension with generators where none ofthe spinor coordinates are among the minimal generators of I by [5, Example 3] .Suppose 1) and 2) are specializations of resolution (13) of I . Since specialization is a ringhomomorphism which maps minors of the matrix to the minors of matrix, and thus mapsspinor coordinates of one resolution to spinor coordinates of the other. In (1) and (2) thereare none of the spinor coordinates are among minimal generators of I . This forces noneof the spinors coordinates of resolution of I in (13) to be among minimal generators of I which is contradiction by Proposition 6.1. Spinor coordinates of generic doubling of (1,4,7,4)
In this section we discuss spinor coordinates of resolution (15) of Gorenstein ring with 8generators obtained from generic doubling of (1 , , , I .By [5, Theorem 2] there exists a hyperbolic basis of e R in resolution (15) for m = 2, say { e , . . . , e , e − , . . . , e − } with hyperbolic pairs { e i , e − i } . Denote columns of δ with respectto e i and e − i as i and ¯ i respectively. In [5, Theorem 2] spinor coordinates are denoted as( e a ) K where K ⊂ {± , . . . , ± } with cardinality of K is 7 with even number of ¯ i . We findthe number of spinor coordinates are among the minimal generators of ideal I . Proposition 7.1.
For m = 2 in resolution (15), at least spinor coordinates among theminimal generators of I .Proof. We use Macaulay 2 [11] to calculate 7 × δ mentioned in resolution (15)for m = 2. Denote M KL as 7 × δ involving K rows and L columns. Then M , , , , , , , ¯2 , ¯3 , ¯4 , ¯5 , ¯6 , = ± g x , M , , , , , , , ¯2 , ¯3 , ¯4 , ¯5 , , ¯7 = ± g x , M , , , , , , , ¯2 , ¯3 , ¯4 , , ¯6 , ¯7 = ± g x , M , , , , , , , ¯2 , ¯3 , , ¯5 , ¯6 , ¯7 = ± g x . Then by [5, Theorem 2] spinor coordinates with respect to above K columns are:(˜ a ) ¯1 , ¯2 , ¯3 , ¯4 , ¯5 , ¯6 , = ± g , (˜ a ) ¯1 , ¯2 , ¯3 , ¯4 , ¯5 , , ¯7 = ± g , (˜ a ) ¯1 , ¯2 , ¯3 , ¯4 , , ¯6 , ¯7 = ± g , (˜ a ) ¯1 , ¯2 , ¯3 , , ¯5 , ¯6 , ¯7 = ± g , These 4 spinor coordinates are among minimal generators of ideal I but not contained in J . (cid:3) cknowledgement The author thanks Jerzy Weyman for helpful discussions. The author acknowledges sup-port of Fulbright-Nehru fellowship.
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