Betti Numbers, Grobner Basis and Syzygies for

Betti Numbers, Grobner Basis and Syzygies for Certain Affine Monomial Curves

A thesis

submitted for the degree of 9octot:

of

'!J3ljiIosopfjg

in the faculty of Science

by

lndranath Sengupta

Department of Mathematics

Indian Institute of Science Bangalore

- 560 012

September, 2000

This thesis i s dedicated to

All My Fiends for their love and &ection

Declaration I hereby declare that the work reported in this thesis is entirely original and has been carried out by me under the supervision of Professor Dilip P. Patil in the Department of Mathematics, Indian Institute of Science, Bangalore 560 012. I further declare that this work lias not formed the basis for the award of any degree, diploma, fellowship, associateship or similar title of any University or Institution.

lndranath Sengupta

Certified

Prof. Dilip P. Patil

Table of Contents

Acknowledgements

ix

Abstract

xi

0 Introduction

1

The last Betti number 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Key Lemma and its consequences . . . . . . . . . . . . . . . . . . . . . . . . 1.3 T h e c a s e W = f l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 T h e s e t A : i f W # 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 ThesetA; i f W # f l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 11 13 16 19

A Grobner basis for the defining ideal 21 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 A minimal generating set for the defining ideal . . . . . . . . . . . . . . . . . 21 2.3 Monomial orders and Grobner bases . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Grevlex order and the leading monomials . . . . . . . . . . . . . . . . . . . . 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5 S(bj,Ew) 2.5.A Case(A): i = j, h = i , k # 1 . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.B C a s e ( B ) . i # j , k = i , i # l . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5.C Case (C) : Otherwise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 S(tkl,pi). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6.A C a s e ( A ) . r + i = k . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6.B C a s e ( B ) : r + i = l . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6.C Case ( C ) : r t i # I c and r t i # l . . . . . . . . . . . . . . . . . . . . 35 2.7 S(tkl,$j). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.7.A Case(A): r l + j = k . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.7.B C a s e ( B ) . r l + j = l . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.7.C C a s e ( C ) . r l + j # k a n d r l + j # l . . . . . . . . . . . . . . . . . . . 41 2.8 S(E

ifr=landrl=l,

{p+l,p+rl,r'+l,rl),

ifr=1andr122,

{

i f r > 2 a n d r t = 1,

,

+

r

,

( {r, rl, r + rt - I},

+r }

ifr22andrt22.

Grobner basis and Syzygies.

The description of a standard basis of r has been used in [20], t o construct a minimal generating set g, for the defining ideal p of monomial curve C defined by an almost arithmetic seyuerlce. We give a description of this set G in 92.2 of Chapter 2. In this chapter, using Buchbcrger's Criterion we show that ( see 92.14 ), Theorem. Let p

2 1 and m o , . . . ,m,, n be a minimal almost arithmetic sequence of coprime

positive integers, such that mo < . .. < m, form an arithmetic sequence and n is arbitrary. Let K be a field and p denote the defining ideal of the monomial curve C in A& defined parametrically by Xo = Tn", . . . ,X p = T m p , Y = Tn. Then, the set G is a Grobner basis for p, with respect to grevlex monomial order on KIXo,. . . ,X,, Y]. Let KIX1, . . . ,X,]be the polynomial ring over a field K and I be an ideal in KIX1, . . . ,X,]. Let G := {gl, . . . ,g t ) be an ordered set of generators for I, which is a Grijbner basis, with respect to some fixed monomial order on KIX1, . . . ,X,].Suppose that, for i # j,

Schreyer showed that (see [32; Theorem 10.2.81), the t-tuples

generate Syz(gl, . . . , gt). A classical approach to compute syzygies is via Grobner basis and the above observation of Schreyer. In Chapter 3, we obtain an explicit minimal free resolution for the coordinate ring A of monomial curves in A;, defined by a minimal, arithmetic sequence. We use Schreyer's result and the Grijbner basis computation in Chapter 2, to find a minimal set of generators for the first syzygy module of A. For finding an explicit minimal generating set for the second syzygy, many ideas came from computations using Macaulay. Let us now state the main result of Chapter 3. Let mo < ml < m2 < m3 be a minimal arithmetic sequence of coprime positive integers with common difference d. We write n := rns and mo = 3a + b, where a and b are unique Ph. D.Thesis

Chapter 0 : Introduction

integers such that a 0 and 1 b I 3. A minimal free resolution for A , depend upon the j 3.3.A, values of b = 1, 2 , 3 , as seen below. These three cases are considered separately in E 5 3.3.B and 5 3.3.C respectively. Theorem. Let m~= 3a

+ b, for some integer a 2 0 and b E [ I , 31. Let mo < ml < mz
2, p (DerK (A)) E

{p+p+r, 1

( {r, TI, r

+ r' - 1);

,

ifr >2andr1=l, ifr>2andr1>2.

An additional motivation was provided by a striking similarity that was observed in several cases in [19], [22] and [23] between the behaviour of p (P) and of p ( D ~ ~ K ( A ) ) . Since any three integers are in almost arithmetic sequence with mo < ml < n, we can write down a minimal set of generators for DerK(A) where A is the coordinate ring of an algebroid monomial space curve. We give examples to illustrate our algorithmic procedure.

$1.2 Key Lemma and its consequences

2

Key Lemma and its consequences

The main ingredient in the proof of our main result is the following proposition ( 1.2.1), an explicit description of the standard basis of a numerical semigroup generated by an almost arithmetic sequence and the properties of the non-negative integers A, p , v,u , v, w which were defined in [23; $31. Let I' be a numerical semigroup generated by a sequence m , m l , . . . ,me-1 of positive integers. Let I?+ := I'\ (01, A := { a E Z+/ cu r+ I') and A' := A \ I'. Then we have

+

A be the coordinate ring of the algebroid monomial curve in the afine e-space A: defined parametrically b y Xo = T", . . . , Xe-I = TnLc-1and e 2 3. Then, {T"+'& I a E A' U {0}} is a minimal set o j generators for lhe A-module DerK(A). In particular, p(Derrc(A)) = card (A1)+ 1. (1.2.1) Proposition. Let K be a field of characteristic zero and let

Proof. [16; page 8751.

Therefore, we need to find the set A' and formula for card (A') explicitly. For this, first we will brea,k the set A' into two disjoint pieces A{ and A', ( see (1.2.6) ) and then, considering several cases, compute the cardinalities of A: and A', systematically in 53 - 5. For convenience, we recall some notation, definitions and results from [23].

I' be a numerical semigroup generated by m o , m l , . . . ,me-', a sequence of positive integers. Then the set S,, := {x E I? ( a - mo @ I?) is called the standard basis of I' with respect to m. It is clear that S,, depends on I? and mo, but for simplicity we write S := Sm. It is easy to see that S = {so := 0, s l , . . . , s,,-l}, where sl, . . . , s,,,- E I? are positive integers with the following properties : (1.2.2) Standard basis. Let

(a) si EE i mod rno for every i E [0,mo - I]. (1)) If z E

I' then x

= i mod ?no for a unique i E [0,mo - 11 and z >- s;.

We now assume that the sequence mo:. . . ,m,-a is an arithmetic sequence of positive integers with r n o < . - - < me-z and n := meFl is an arbitrary positive integer. Furthermore, let g c d ( m , . . . ,me-*,n) = 1 and mo, . . . ,me-2, n.is a minimal set of generators for the numerical semigroup I' := C T ZNmi ~ Nn. Then the following explicit description of the standard basis S := S,, was given in 123; $31.

+

(1.2.3) Key Lemma. Let;p := e - 2 and let d be a positive integer with mi = mo + i d for all 0 5 i 5 p. Let n be an arbitrary positive integer with g c d ( m , d , n) = 1. Let r1:= Nmi and I'= I?' Wn. Let S := S,, be the standard basis of I? with respect to mo. For t E JY, let qt € Z, rt E [I,P] and gt E r' be defined by l = qtp rt and g, = qtmP rn,, .

+

Ph. D. Thesis

x;=,

+

+

Chapter 1 : The last Betti number

( 1 ) g,

9

+ gt = Ern0 + gs+t with

E;

=1

or 0 according as r, + rt j p or r,

>

+ rt > p.

( 2 ) Let u := rnin{t E W I gt. # S ) and v := min{b 1 1 bn E I"). Then there exist unique integers w E [O,v - 11, x E [O,u- 11, X 2 1, ,u 0 such that

+

(ii) vn = prno g, ; (iii) g,-, ( v - w ) n = vmo. Moreover) v = X r,-, < r, err,-, 1 T,.

+

>

+ p + E where

E

=1

or 0 according as

(3) L e t V : = [ O , u - l ] x [ O , v - I ] , W : = [ u - x , u - 1 ] x [ v - w , v - 1 1 a n d U : = V \ W . Then S = { g , bn I (s, b) E U ) . In particular, ilf ( s ,b),( t ,c) E U with g, + bn r gt cn mod n - ~then ( s, b) = ( t ,c).

+

+

( 4 ) Every element of I? can be expressed uniquely in the form am0

and (s,b) E U .

-

(5) The map (N)" between S and U.

( N ) ~defined b y

aiei

ei

Proof. See [23;$31.

+ gs + bn with a E W

a i m , ~ ~ + is~a nbijection )

I

Now we list some lemmas which are needed in the proof and which are immediate from the rninimality assumption on the sequence mo,ml,. .. ,me-1 and the Key Lemma. (1.2.4) Lemma. Let q := q,, r := r, and qt := q,-,,

T

=

r

- rl, -

r

rt := r,-,.

i f r > rt, ifr 5 7 ~ .

(6) If X = 1 then w # 0. ( 8 ) Let H I := [ u - p , u - 11 x {v - w - I), Hz := [ u - z - p , u - z 11 x { v - 1)) H := HI U H2. Then A' C (gi bn - m,-, I ( i ,b) E V nH ) . In particular, card(At)5 card(VnH) 5 2 p = 2e-4.

+

Ph. D. Thesis

51.2 Key Lemma and its consequences

10

Proof. (1) to (7) arc easy to check and (8) is proved in 123; Lemma (4.2)]. (9) 1~

+ 7n.i +

7nj

-

=h

+ mi+j E l7 by the Key Lem~na.

(1.2.5) Lemma. Lei q := q,, r := r, and q' := qu-z, r' := rU+.

+ (v - w - 1)n - m~$! A' for every i E [O, qp] . (2) If r = 1 ihen gi + (v - w - l ) n - r n o # A' for every i E [O, (q - l)p] . (3) If r' 2 2 then gi + (v - 1)n - mo $! A' for evemJ i E [0, q'p] . (4) I f r ' = 1 and q' > 0 then gi + (v - 1)n - mo $! A' for every i E [0, (q' - l)p] . (1) If r

22

ihen gi

(5) If w

#0

then gi

+ (v - w - 1)n - mo gf A' for every i E [0, u - x - 11 .

+

+

Proof. (1) Let r 2 2 and y := gi (v - w - l ) n - rn with i = up b, such that, 1 and -1 5 a i q - 1. Then y m,+l-b = (a l)m, ml (v - w - 1)n - mo $ ((a 1)p 1,v - w - 1) E U and therefore y 6A'.

+ +

+

+

+ +

0 , ifr'22.

and for i E I, let cri := g(,~-~),+~+ (v - l ) n - m.

Ph. D. Thesis


11

Wenote that ((d-l)p+i, v-I) E U foreveryi E I, ((9-l)p+r+j-1, v-w-1) E U for every j E [I, p] and ((q - 1)p r k - 1, v - 1) E U for every k E [I, p]. Therefore, by the Key Lemma (l.2.3), ai,Pj and yk do not belong to I'for any i E I, j, k E [I, p]. Moreover, ai = gu-z-p-r~+i+(~1)n-mo for all i E I, Pj := gu-p+j-l +(v-w- 1)n-mo ifw=O, for all j E [I, p] and for all k E [I, p], y k = QT+k-1 if z = 0 .

++

A; := At n {ai1 i € I}. Al, := A'

n {pj I j E J).

(1.2.7) Proposition.

(1) A' = A: U A',.

(2) Suppose that W

= 0.

(3) Suppose that W

+

Then

Q). Then

A\ E {ai1 i E I)

and

A/, E

{Pj

I j E: [I, PI)

if

r=l,

{Pj

IjE: b-r+2,~]}

if

r12-

Proof. The proof follows from (1.2.4)(8) and (1.2.5).

51.3

The case W = 0

In this section, we assume that W = 0 and write down the set A' explicitly in (1.3.3). First note that, since W = 0, we have either w = 0 or z = 0 and therefore,

We need the following two lemmas for proving (1.3.3). (1.3.1) Lemma. Suppose that r = 1. Then

+

(1) If X > 1 then g(,-l),+,+k-l

mj

- mo E I' for every j, k

E [I,p], Ph. D. Thesis

51.3 The case

( 2 ) If z = 0 then yk

+ 7nj E J? for

every j, k E [l,p].

(3) If eitherX > 1 o r p > 0 t h e n y k + n E r f o r every k Proof. (1) In view of (1.2.4)(9) we may assume that i

(2) Put 1 := k

(v- 1)n-mo

=0

+j

E

[l,p].

> p. Then, by the Key Lerrrma,

+ j - p. Then 1 - 1 E [O,p- 11 and since z = 0, we have gu-.p+k71+ mj + + (u- 1 ) n - m o = ( v - 2 ) m + m ~ +1 (w - l ) n E 1'.

=gu+l-l

(3) If p > 0 then the assertion is clear, since vn X > 1. Then x > p by (1.2.4)(5) and so

Therefore, since X > 1, in any case y k

- mo E

J?. Now, assume that

,LL

= 0 and

+ n E I?.

(1.3.2) Lemma. Suppose that r >_ 2. Then

(I) y k + m j E r f o r every j E [l,p] and k E [ p - r + 2 , p ] (2) y k + n € I ' f o r e v e y k ~b - r + 2 , p ] . Proof. (1) Follows frorn (1.2.4)(10).

(2) If p > 0 then ihe assertion is clear, since vn - m0 E I?. NOW,assume that p = 0. Then m > p by (1.2.4)(5) arid yk n = (A - 1)mo m k - 1 9,-, wn E I? by (1.2.3)(2).

+

+

(1.3.3) Proposition. Suppose that

W

= 0.

+

+

Then

Proof. Note that (see (1.2.6)) yk @ I? for every k E [I, p]. First suppose that r = 1 and k E [I, p]. If w = 0 then X > 1 and so yk+mj E I'by (1.3.1)(1) and y k + n E I? by (1.3.1)(3). If z = 0 then p > 0 and so yk m j E I? by (1.3.1)(2) and yk n E I? by (1.3.1)(3). This shows that yk E A' for every k E [I, p].

+

+

Now suppose that r 2 2 and k E [P-r+2, p]. Then, x + m j E I' by (1.3.2)(1) and y k + n E 1: by (1.3.2)(2). This shows that, yk E A' for every k E b - r + 2, p]. Now the proof follows m from (1.2.7)(2). Ph. D. Thesis


13

In this section, we assume that W # 0 and write down the set A: explicitly. It is convenient 2 separately (see Propositions (1.4.6) and (1.4.7) to consider the cases r' = 1 and r' below). First note that, since W # 0, we have w # 0 and x # 0. Furthermore, we make some observations in Lemmas (1.4.1) - (1.4.5), which will be used in the proofs of (1.4.6) and (1.4.7).

>

(1.4.1) Lemma. Suppose that r' = 1 and q' = 0. Then

(1) a, + m j E F for every j E [ l , p ] .

+ n E I?.

(2) If p > 0 then a,

(3) If p > 0 then a, E

A:.

Proof. (1) Since r' = 1 and q' = 0, u - x = 1 and so a p + m j = mj-1 + m l + (v- l)n-2m, = ( u - 2)mo (w - l ) n mj-1 E I?, since v 2 and w # 0.

+

+

+

(2) q, n = v n -

>

mo = ( p - l)ino +g,

E

I?.

(3) Follows from (1) and (2).

m

(1.4.2) Lemma. Suppose that r' = 1 and q'

> 0. Then

+ m j E I' for every i,j E [I,p]. + g, - mo E I? for every i E [ l , p- r + 11. (2) If r 2 2 then g,-,-,+i-l (1) ai

(3)

If p > 0 then A/l > {ai I i E 11,p]).

(4) I f p = O a n d r 2 2 thenA', > { a i ) i[~ l,p-r+1]) Proof. (1) Let i , j E [I, p]. Then by (1.2.4)(9) a i + m j E r i f i + j p . Then

and so ai

+ m j E F, since v 2 2 and w # 0,

(2) Since r'

=

I and r

> 2, r,

= r - 1. Therefore, gu-z-p+i-l

+ g, - rn = g,-p+i-l

(3) By ( I ) , ctii + m j E I' for all i , j E [I, p] and ai + n = gu-,,+i-l since p > 0.

+ ( p - l ) m o +g, E r,

(4) By ( I ) , Q i + m j ~ F f o r a l l i ,Ej [1,p]. S i n c e p = 0 , v n - m o = g , - m o ai n = gu-z-p+i-l g, - mo E I?, for every i E [I, p - r 11.

+

+

+

E I?.

a n d s o b y (2),

Ph. D. Thesis

s1.4 ÿ he set A\ if

W#0

(I-4.3) Lemma. Suppose that r' = I and ,u = 0. Then

(1) If

d =0

t / 1 ~ 7 1011,

$ A:.

(2) If q' > 0 and r = 1 then ai

# A: for everyi E [ I , p].

Proof. ( 1 ) Since 0 < z S u - 1 , a , + n = u n - m o

r = r t = 1, r, = p. Let i

(2) CYi

E

# r.

[ l ,p ] . Then 0 

=g,-mo

r.

+ i - 1 5 u - 1 and so

(3) Sincer 2 so r, = r - 1. Let i E [joer+2,p]. Then 0 < u - p + i ai n = gu-z-p+i-l 9, - mo = gu-p+i-l - m0 pf I?.

+

+

(1.4.4) Lemma. Suppose that r'

1 < u - 1 andso H

2 2. Then

(-,+i (2) If r, < p then g ,-,-, ( 3 ) If ,U = 0 then

+ g, + ( v - l ) n- mo E I' for

{ {aiIi€k+1,p+rt-I])

every i E

ifeither r = l 25r r t : Inthis c a s e r , = r - r t < p a n d Ip+1,p+rr-11 C [I,+l,2 ~ - ~ , ] . 2 < r < r' : I n this case r, = p + r - r' < p and 2p - r, = p + - r , Therefore, the assertion follows from ( 1 ) and (2).

(1.4.5) Lemma. Suppose that r'

Ph. D. Thesis

2 2 and U,

= 0.

+

+


15

(1.4.6) Proposition. Suppose that r' = 1 . Then

(8 0

if

p=O and q l = O ,

if

p = 0 , q l > O , and r = 1,

{ ~ i l i ~ [ l , p - r + l 27 )

A: =

{QP)

,{ai1 2 E [ L P ] )

p = 0 , q'>O

and r 2 2 ,

if

p > O and q ' = O ,

if

p > 0 and q l > O .

Proof. p = 0 and q' = 0 : In this case, the assertion follows from (1.2.7)(3) and (1.4.3)(1).

, q' > 0 and r = 1 = 0 , q' > 0 and r 2

p =0

: In this case, the assertion follows from (1.2.7)(3) and (1.4.3)(2).

p and (1.4.2)(4).

2 : In this case, the assertion follows from (1.2.7)(3), (1.4.3)(3)

p > 0 and q' = 0 : In this case, the assertion follows from (1.2.7)(3) and (1.4.1)(3). p > 0 and q' > 0 : In this case, the assertion follows from (1.2.7)(3) and (1.4.2)(3). (1.4.7) Proposition. Suppose that r'

A{ =

I

2 2.

Then

{aiIi€[l,+l,p+rl--11)

if

p = O and r = l ,

[l,+l,p+rl-11)

if if

p = O , r > 2 and r > r l ,

if

p = O , 7 - 2 2 and r < r l ,

{Q'iIi€

0 {ffiIi€

{a((

a

b+l,p+rl-r])

iIp+ ~ l , p + r r - I ] ) if

p = O , r 2 2 and r = r l , p>O.

Proof. In both the cases p = 0 , r = 1 and 1 = 0 , r >_ 2 from (1.2.7)( 3 ) and (1.4.4)(3).

, r > r', the assertion follows

p = 0 , r 2 2 and r = r' : In this case, the assertion follows from (1.2.7)(3) and (1.4.5)(2). p = 0 , r 2 2 and r < r' and (1.4.5)(1).

:

In this case, the assertion follows from (1.2.7)(3), (1.4.4)(3)

p > 0 : In this case, the assertion follows from (1.2.7)(3)and (1.4.4)(4). Ph. D. Thesis

$1.5

16

Theset A; if W

51.5

he set

aL if W # 0

# Q)

this section, we assume that W # @ and we write down the set A/, explicitly. It is convenient to consider the following four cases ( see Propositions (1.5.6) - (1.5.9) ) , I11

e Case

e

1 : r = 1 arid r'= 1,

Case 2 :

r = 1 and r' 2 2,

Case 3 :

r 2 2 and r'= I,

Case 4 : r

2 2 and r' 2 2.

We first make some simple observations in the lemmas (1.5.1) - (l.5.5), which are needed for proving (1.5.6) - (1.5.9). Note that, since W # 8,we have w # 0 and x # 0.

+

+

Proof. For j E 11, p] we have pj n = gU-,+j-l (v - w ) n- mo = g,-,+,-,+j-~ mo = (v- c)mo+gz-p+j-l where E = 1 or 2 according as r'+r,-p+j-l > p or

and therefore

pj + n E I?

+

(v - w)nr'+rz-p+j-l 5 p

since v 2 2.

m

(1.5.2) Lemma. Suppose that r = 1. Then

+

m k E I 'for every k E [I,p]. (1) ( 2 ) If either X > 1 or u - x = 1. Then pj

+ mh E 'I for

every j , k

E

[ I , p].

( 3 ) I f x p then E A',. (4) Ifx

( 6 ) I f X = l a n d u - ~ = 1 . Then&

(7) If X

E [ I ,p].

> { p j I j~

[l,p]}.

> 1 then if

+

+

z z p .

Proof. (1) PI mi; = gu-,+* ( v - w - 1 ) n E I' for every k E [ l , p - 11 and g, (v - w - 1 ) n - mo = (A - l ) m o (v - 1)n E I?.

+

+

(2) In view of (1.2.4)(9) we may assume that j

Ph. D. Thesis

+ k > p.

Then

PI + m, =

Chapter 1 :The last Betti number

17

+

and so ,Oj mk E I? if X > 1. Now assume that u - x = 1 and X = 1. Then for j + k > p+ 1, from ( E .1.1) we have

and so ,Oj

+ mk E I',since v 2 2 and w # 0.

(3) and (4) Immediate from (1.5.1).

(5) fillows from ( I ) and ( 4 ) . (6) Follows from ( 2 ) ) (3) and (4).

(7) By ( 2 ) , pj

+

m k

E

r for every j , k E [l, p]. Hence, the proof follows from (3) and ( 4 ) .

(1 5 3 ) Lemma. Suppose that r

.

2 2.

Proof. (1) Let j E - r + 2 , p]. Then by (1.2.4)(10),pj + m k E I'for every k E [ I , p]. Now q' < q implies that x 2 r - 1 = p 1 - (p- r 2 ) and therefore ,Oj n E I? by (1.5.1).

+

+

+

+

+ m k E I? for every k E [ I , p]. The + r' + 1) and therefore by (1.5.1)

( 2 ) Let j E [p - r + r' 1, p]. Then, by (1.2.4)(10),,Oj condition q' = q implies that x = r - r' = p 1 - ( p - r we get ,Oj + n E I?.

+

(1.5.4) Lemma. Suppose that r = 1.

( 1 ) I f z 0 or r'

,Oj

+

every j

[2,p].

E

Since w # O a n d x 2 then pj 6 A' for

+

( 2 ) Since either q' > 0 or r' 2 2 we have ( 1 , v - 1 ) E U.Now, since X (v - 1 ) n - mo 4 I'for every j E [2, p].

ml

+

(1.5.5) Lemma. Suppose that r pj

2 2 , r'

> 2 and q' = q.

@ a;.

Proof. Since j E [p - r (1.2.5)(5).

+ 2,p - r + r'],u - p + j

1 . j E [p- r

=

1, ,Oj

+

mp+2-j=

w

+ 2, p - r + r'] then

- 1 E [ I , u - x - I] and so

Pj 4 A', by Ph. D.Thesis

$1.5 The set

if

W # (b

(1.5.6) Proposition. Suppose that r = 1 and r' = 1. Then

d=O,

( { P j I j ~ [ l , ~ lif} {Dl )

A/, =

{

j

Proof. Since r = 1 and r' = 1, z

O 1). 0 < q' < q and X = 1 : In this case, the assertion follows from (1.2.7)(3), (1.5.2)( 5 ) and (1.5.4)(2). R 0 < q' < q and X > 1 : In this case, the assertion follows from (1.2.7)(3), (1.5.2)(7). (1.5.7) Proposition. Suppose that r = 1 and r'

PI). Pj I j E [r', p])

a: =

1 Proof. Since r = 1 and r'

[

l

p

2 2. Then if

X = l and z 1 and z 1 and z 2 p .

> 2, by (1.2.4)(2) r, = p + 1 - r' and z = p + 1 - r' if z < p.

A = 1 and z 1 and z 1 and z >_ p : In this case, the assertion follows from (1.2.7) (3) and (1.5.2) (7).

X

= 1 and

(1.5.8) Proposition. Suppose that r

2 2 and r'

I

= 1. Then A/, = {,Oj j E

- r + 2, p ] ) .

Proof. We consider the two cases q' < q and q' = q separately.

q' c q : In this case, the assertion follows from (1.2.7)(3) and (1.5.3)(1). q' = q : In this case, the assertion follows from (1.2.7) (3) and (1.5.3) (2). (1.5.9) Proposition. Suppose that r

n;

I j E b-r+2,

PI) {pj I j E [ I , - - r + r 1 + 1 , p ] ) if

{Pj

=

1 2 and r' 2 2. Then 4' < 4 , q'=q.

Proof. q' < q : In this case, the assertion follows from (1.2.7) (3) and (1.5.3) (1). q' = q : In this case, the assertion follows from (1.2.7) (3), (1.5.3) (2) and (1.5.5) Ph. 0.Thesis

Cha~ter1 :The last Betti number

19

$1.6 Theorem

>

(1.6.1)Theorem. Let K be a field of characteristic zero and let mo,. . . ,me-l, e 3 be a sequence of positive integers with gcd(mo,. . . , = 1. Assume that mo, . . . ,me-1 is a

minimal set of generators for the semigroup I' generated by 7720,.. . ,me-1 and some e - 1 terms of mo, . . . , me-l form a n arithmetic sequence. T h e n

Proof. We know by (1.2.1) that, { T a f

& / a E A; U A', U {o)}is a minimal set of gener-

ators for the A-module DerK(A). The sets A', and A', have been described explicitly in (l.3.3), (l.4.6), (l.4.7), (l.5.6), (1.5.7), (1.5.8) and (1.5.9) and the proof follows from these propositions. (1.6.2) Corollary. Let S := K[[Xo,. . . ,Xe-l]] denote the ring of formal power series over a field K of characteristic 0. Let t A denote the type of A and /3e-1 denote the last Betti number in the minimal free resolution of A ( a s a module over S ) . Then,

Be-1

=

t~

I

{p, p

i - 1r r - 1

{ p r - l r - 1

i f r = 1 and r'

> 2,

z f r r 2 andrl=l,

Proof. A is a domain of dimension 1, therefore A is Cohen - Macaulay and it follows by Auslander - Buchsbaum, that the projective dimension of A as an S-module is e - 1. Hence, A) = dimK TO~~-,(A, K ) = ,&-1. we obtain ta = dimrc EX~$(K, Now, to show that tA = ,u(DerK(A)) - 1, we note that t~ = dim~(m-'/A) and m-l DerK(A) (see [27; $31 and [16; page 8751 ). The rest is immediate from the above n theorem. We end this chapter with some illustrative examples. (1.6.3) Examples.

(1) L e t m o = 10,ml = 1 5 a n d n = 7. T h e n u = 2 , v = 5 , w = O , z = l , X = 3 , , u = 2, v = 5 and W = 0. In this case A' = {33) and {T&,T ~ ~ &is )a minimal set of generators. Ph. D.Thesis

31.6 Theorem

20

( 2 ) Let mo = 10,snl = 19, m2 = 28, ms = 37 and n = 35. Then u = 5 , v = 2, w = 1, ,Z = 0 , X = 3, p = 7, v = 10 and Mi = 0. In tliis case A' = { 8 1 ) and T B 2 & ) is a rniriilnal set of generators.

{TS,

( 3 ) Let 7no = 5,7121 = 9 and rl = 6 . Then u = 3,v = 3 , w = 2 , z = 2,X = 3 , p = 0 , u = 3 and W # 0. In this case A' = 113) and {T&, T ' ~ - $ is ) a minimal set of generators.

(4) L e t m o = 1 6 , r n l = 2 1 a n d n = 1 3 . T h e n u = 2 , v = 9 , w = 2 , z = 1 , X = 1 , p = 6 , u = 7 and LV # 0. I11 tliis case A' = {83,88) and { T & , TS4$,T~'-&}is a minimal set of generators. (5) Let mo = 7,rnl = 12,mz = 1 7 a n d n = 11. Then u = 3,v = 3 , w = 2 , z = 1,X = 1, p = 3, Y = 4 and W # 0. In this case A' = ( 2 7 ) and IT$, T 2 8 & }is a minimal set of generators.

Ph. D.Thesis

Chapter 2 A Grobner basis for the defining ideal

2.1

Introduction

Let mo, . . . , me-1 be an almost arithmetic sequence of coprime positive integers, such that e 2 3 and p := e - 2. As before we assume that < . . . < m, form an arithmetic sequence, n := mp+l is arbitrary and mo, . . . ,%+I is minimal.

. . . , X,,Y ,T be indeterminates. Let p denote the kernel of the KLet K be a field and Xo, algebra homomorphism q : K [Xo,. . . , X,, Y] + K [TI, defined by q(Xi) = Tmi,v ( Y )= T". Then, p is the defining ideal for the affine monomial curve C in A; defined parametrically by Xo = Tmo, . . . , X, = Tmp, Y = P . Furthermore, p is a homogeneous ideal with respect to the gradation on K [Xo,. . . , X,,Y] , given by wt (Xo) = mo , . . . , wt (X,) = m, , wt(Y) = n. A minimal set G of binomial generators for p is explicitly constructed in [20], which we recall in the next sectmion.The aim of this chapter is to prove the following theorem (see 52.14).

>

Theorem. Let p 1 and mo, . . . , m,, n be a minimal almost arithmetic sequence of coprime positive integers, such that mo < . . . < m, f o r m an arithmetic sequence and n is arbitrary. Let K be a field and p denote the defining ideal of the monomial curve C in Ak defined parametrically b y Xo = TQ,. . . , X p = T*, Y = Tn,Then, the set G is a Grzbner basis for p , with respect to grevlex monomial order on KIXo,. . . , X,,Y]. Our method for showing that G is a Grobner basis involves a direct computation with the 5'-polynomials and an application of Buchberger's Criterion. The structure of this chapter is as follows : In 52.2 we describe the set G and in $2.3 we record the basic notions of monomial order and Grijbner bases. The monomial order grevlm is defined in 52.4 and $2.5 - 2.13 are devoted to the computation with S-polynomials. We present the proof of the main theorem in 52.14.

52.2 A minimal generating set for the defining ideal

The generators for p are used explicitly in this thesis and therefore we state them as given in [20]. Here, q, q', r , r', u, v,w , x , A, p , V ,W continue to bear the same meaning assigned

$2.2 A minimal generating set for the defining ideal

to them by the Key Leirinla (1.2.3) and (1.2.4).

XiXj- XoXitj if i + j < P, X j- X X i + j if i f j > P tji and tOj= & = 0. In fact, tij# 0 if and only if i, j

tij:=

For i , j E [ O , p ] ,let Notc tilat,

tij=

E (1,p - 11.

Fur i E [0,p - r] , let q5i := X,+iX,V - x$-'x,Y" . For j E [ 0 , p - r'], let

Let I :=

and J :=

$j := x , I + ~ x $-Yx,"-'x~. ~-~

10,P - TI

p#O p=O

if [ i n ~ ~ ( r , - r + l , O ) , p - r ] if

if

W=0,

n - 1 ,- r ) if

W # 0.

I" 0 i

or W = 0 , and W f 0 ,

Then, by [20; (4.5)], the set

forms a niinirn'l set of generators for p. A criterion for complete intersection for C can now be derived easily, as is seen below. Let p(p) denote the minimal number of generators for p. Then, by [20; (4.6)], p(p) is either p(p - 1)/2 + p - r 2 , p(p - 1)/2 p - r' 2 or p(p - 1)/2 2p - r - r' 3. Hence, p ( p ) 2 p(p - 1)/2 2. Therefore, a necessary condition for C to be a complete intersection is p 5 2 This implies that a monomial curve in affine e-space defined by an almost arithmetic sequence is never a complete intersection if e 5. Hence, we consider only the cases e = 3 and e = 4.

+

+

+

+

+

+

>

(2.2.1) Proposition. Let p E {1,2) and mo,. . . , m,+l be a minimal almost arithmetic sequence of positive integers. The following statements are equivalent:

(i) The monomial curve C is a complete intersection. (ii) card(1) + card(J) = 1. (iii) Either r

= p,

W = 0 or r = r, = p, W # 0, p = 0.

Proof. Follows easily from the construction of Ph. 0.Thesis

4 given above.

Chapter 2 : A Grobner basis for the defining ideal

We illustrate the above proposition with some examples. (2.2.2) Examples.

(1) Let ma = 10,ml = 15 and n = 7. Then W = 0. Hence, C is a complete intersection. (2) Let ma = 5 , m l = 9 and n = 6. Then W intersection.

(3) Let ma = 16,ml intersection.

= 21

and n = 13. Then W

# 0,

p = 0. Hence, C is a complete

# Q), p = 6.

(4) Let mo = 8,7711 = 10,mz = 12 and n = 15. Then W complete intersection. (5) Let ma = 7,ml = 12,mz = 1 7 a n d n = 11. Then W Hence, C is not a complete intersection.

52.3

Hence, C is not a complete =

0, r

# 0, r

= 2. Hence, C is a = 1, r, = 1, p = 3.

Monomial orders and Grobner bases

In this section, we mention some of the basics of monomial orders and Grobner bases. Several good references are available for complete details ( see [I], [7], [8], [9], [261, [29]). Here we follow [7]. Let K [XI,. . . , Xn] denote the K-algebra of polynomials over a field K. Let X" := denote a typical monomial in KIX1,. . . , X,], where a := ( a l , . . . , a,) E Nn. (2.3.1) Definition.

Xi

A monomial order on K [XI,.. . , Xn]is a relation > on the set of monomials

X" in KIX1,. . . , X,] or equivalently on the exponent vectors a, E Nn, satisfying:

( I ) The relation > is a total ordering relation. (2) Whenever X" > XP and X' is any monomial, for a,P, y E Nn, then X"X7 = Xa+r > xP+r = ~ 0 x 7 . (3) The relation > is a well ordering, i e . , every non-empty collection of monomials has a smallest element under >. Let > be a monomial order on K [XI,. . . , X,].Henceforth, K [XI,. . . , X,] will always denote the polynomial ring together with some fixed monomial order >.

+

Let f E KIX1, . . . , Xn] be non-zero with f = cX" C cpxP, where c, cp E K and X" is the largest monomial appearing in f with respect to the ordering >. Then, the product c,X" is called the leading term of f , c the leading constant of f , and X" the leading monomial of f with respect to >. These are usually denoted by Lt, ( f ) , LC, ( f ) and Lm, ( f ) respectively. We simply write Lt ( f ) , LC(f ), Lm( f ) , when no confusion can occur. Ph. D. Thesis

52.3 Monomial orders and Grobner bases

24

> be a monomial order on K I X I , ... , Xn] and I 2 K [ X I , . . . ,X,]be an ideal. Suppose, G = {gI,.. . ,gt) 5 I is a finite set of polynomials. The set G is called a Grubner basis or standard basis for I, with respect to >, if for every non-zero f E I, Lt(gi) [ Lt(f), for some 1 gr,vl,x P, if wt(cr) > wt(P), or if wt(a) = wt(P) and in the difference a - P E Ze,the rightmost non-zero entry is negative. We write Yap+'

nyz0X?

>grevlex

yPptl

IX=o x?, if (ap+l,.. . , QO)

(&+I,

- . ,PO).

It is easy to verify that the order ">,re,1e," is indeed a monomial order on KIXo, . . . , Xp,Y]. Henceforth, R always denotes the polynomial ring with the grevlex monomial order. Since we will be working throughout with the same monomial order we will write > instead of >1,, . For the leading monomial of a polynomial in R, we will simply write Lm in place of Lm,. At this point, we note an important fact. Since all elements in the set 5! are homogeneous binomials with respect t o the gradation given by wt(Xi) = m; and wt(Y) = n, so are all the S-polynomials and all their successive reductions. Therefore, in deciding the leading terms of the elements of and the S-polynomials, the respective weights on the indeterminates do not play a role. This is the reason why we do not have to consider separate cases depending upon the position of the integer n in the almost arithmetic sequence mo, . . . ,m,, n. We list a couple of simple observations in the following Lemma, which will be helpful in finding the leading monomials. Remark.

Ph. D. Thesis

$2.4 Grevlex order and the leading monomials

26

(2.4.3) Lemma. Let

(i) If index(a)

a, P E Ne, with wt (a) = wt (P). Then

> index(P), then a > P.

(ii) Ij index(n) = index(0) and ba.se(a)

< base(,@, then a > p.

Proof. f;'ollows from the definition of grevlm. (2.4.4) Lemma. (i) Lm(&) = XiXj, 1

 i + j - . p = index (XPXi+jtj,p).Now we use (2.4.3)(i).

+
p and 2i I p ,

(b) Xoti, i+l - X&i-p,

29


# j , k = i, i # l . Proposition. Let i,j, 1 f [I, p - 11 be distinct integers. Then,

'$2.5.8 Case (B): i (2.5.2)

(i)

S (tij,ti^)

=

XZC i j - Xj X O X ~ + E-Xx~~ X i + j X l XpXi+l-pXj - XoXi+jX1

i f i + j < _ pa n d i f l i p ,

if i + j 5 p and i

+ 1 > p,

XoXjXi+z-XpXIXi+j~p i . f i + j > p a n d i + I p and i 1 > p .

+

=

( max(XoXi+iXj , XoXi+jXi) (ii) Lm (S(Sij,[a)) =

I

i j i - t - j s pandi+l i p ,

XPXi+,-pXj xpX~Xi+j-p

(maw(XpXjXi+r-p,XpXIXi+jjp) i]i+ j > p and if1 > p .

( (a) xob+z,j- XoCii-j, (iii) S(&, CiJ =

( b ) Xp[i+~-p, j - Xo[i+j,~ (c)

X~f,i+~-p,l - XPtj,i+l

ifi+jI p a n d i + l ( p ,

+ j 5 p a n d i + 1 > P, if i + j > P and i + 1 < P,

if i

Proof. (i) Note that lcm(Lm(&j), Lm(&l))= X i X j X l and the rest follows from the defini-

tions. (ii) Follows easily from (2.4.3)(i). (iii)(a) Let i

+ j p . Then,

Ph. D. Thesis

(c) Lct i + j > p and i + 1 for this case.

< p.

The same proof as in (b), with j and 1 inerchanged, holds

(d) Let i + j >I-,and i f 1 > p Then, $(tij,

6)=

Xp(XjXi+l-p

- XlXi+j-p)

= X p [ j , ~ + lp XpJl,+jp

by Lemma (2.4.5) .

Hence the proof. s2.5.C

Case (C): Otherwise.

This case leads us to four subcases:

# i, (ii) i = j, k < 2 , k # i, I # i, (iii) i < j, k = 1, k # j, k # i, (iv) i < j , k < 1, {i, j} n {k, l } = 0. (i) i = j , k = 1, k

In these four subcases (i) - (iv), the leading monomials of tij and tklare coprime, and hence the proof of the following proposition is immediate by Lemma (2.3.5). We fuc a notation first, which helps us to state the proposition neatly. For a,,O E [ l , p - 11, let 0 ifa+pIp, Moreover, &(a,a ) is denoted by &(a). 1 ifa+p>p.

Proof. Follows from (2.3.5). Ph. D. Thesis

I


Let us recall that, XkXl - X0Xk+,

&:=

+ 1 < p, if k + 1 > p ,

if k

XkXl - XpXk+l-p

31

and Lm(tjkl) = XkXl , for 1

< k < 1 < p - 1,

and pi := Xr+iX; - x ~ - ' x ~ and YLm(qi) ~ = X,+iX;,

for i E I.

We consider the following cases which are exhaustive: (A) r

+ i = k,

(B) r + i = l , (C) r + i J : k a n d r + i # l .

Actually, we only have t o prove Cases (A) and (C). It will be apparent that, in proving Case (A), we will never use k 5 I . Therefore, the same proof is applicable for Case (B) as well because &l = &. s2.6.A

Case (A) : r

+i = k

Here again we will consider two subcases, viz., r (2.6.1) Proposition. Let i E

I, 1 5 k

+ i + I 5 p and r + i + I > p.

< 1 < p - 1 , such that r + i = k

and r

+ i + 1 I p.

Then, (i) S(

i

)

= =

Xp4 E ~ z XLPi

x,X-~X~X~Y" XiXIYw

) (ii) Lm(S(Jk1, ~ i) =

- xoxr+i+lX;t.

ifX=l

andi#O,

XoXr+i+lX;t otherwise.

Proof. (i) If r+i = k, then lcm(Lm(tjkl), Lm(cpi)) = Xr+iXIX; , and therefore, the expression for S(&, pi) is clear from the definition.

(ii) To evaluate the leading monomial, we consider the following cases: (a) X = 1 and i # 0, (b) X = 1 and i = 0, (c) X = 2 a n d i # O , (d) X = 2 and i = 0, Ph. D. Thesis

(e) A 2 3.

I t is clear that index (X;X,.+i+lX;) = 0 and base (XoXy+i+lX;) = 1 , since r Also note that, index (XiXIYW)= min(i, I). Suppose that X = 1. Then,

+ i + 1 > 0.

111case (a), index (XiX1Yw)> 0 = index (XoXr+i+lXi) and we use (2.4.3)(i). In case (b), S(&l,pi) =

(XiYw- XT+i+lXi).

Now index (X,,i+lX,P) = r + i + 1 > 1 = index (XIYw)and we can use (2.4.3)(i) again.

Let X

> 2. Then, i n d e x ( ~ $ - ' ~ ~ X r=~ 0" ) and b a s e ( ~ $ - ~ ~ i ~=L ~ " ) {;-I

Therefore, in cases (d) and (e),the proof follows from (2.4.3)(ii)..

arid

index (X,+i+lXp4) = r + i

+ 1 > max{i, 1)

2 min{i, 1) = index (XiXIYw).

Hence, the proof in case (c), follows from (2.4.3)(i). (iii) First, we note that r f i

+ I 5 p implies i + 1 'pi) = X ~ - ' x i X ~-yXoXT+,+, ~ Xi

+

Yw(XiX- XoXi+r) - XoXr+i+iXz XoXi+rYw if X = 1 and i 1 0 ,

(+ i

-Xo(x++,X,P - X,"-'xi+lyW)- X $ X j + l ~ w

=

x,X-'xixl~"

YW&l- X0(XT+i+1X,4- XiflYW)

= = Ph. D. Thesis

+

if A = 1 and i # 0,

(xixi- X O X ~ + ~otherwise. ) -XGY~+Lx$-'yW

A-1

w

Xo Y

Jil

- XO(P~+~-

otherwise.

)


Suppose X = 1, i # 0. Then

33

# 0 and

Hence, S(Jkl,pi)

-+g

0. In the other case, we observe that,

Hence, S(&I,pi)

- 4 8

0.

+ i + 1 < p. Let us now consider r + i + 1 > p. (2.6.2) Proposition. Let i E I , I 5 k 5 1 p.

Then,

X,XIYw (ii) Lm(S(&i, yi) ) =

ifX=1 andr+l < p ,

X,Q+' Xv+i+l-p otherwise .

Proof. (i) The expression for S(Ekl,pi) is clear from the definition. (ii) We consider the following cases for the evaluation of the leading monomial:

(4 2 2,

+ 1 = p, (c) X = 1 and r + 1 # p. Since r + i + 1 - p i p, we have index (XpP+lXr+i+l,) (b) X = 1 and r

=T

+ i + 1 - p > 0.

Also note that,

Therfore, in case (a),the proof follows by (2.4.3)(i). Ph. D.Thesis

h t X = 1 and

T

+ 1 = p. Then,

Sillw index (X;+l) = p > 1 = index (XLYw),the proof in case (b) follows by (2.4.3)(i). Suppose that X = 1 and T I f p. Note that, r 1 < p if and only if r i 1 - p < i. 'l:urtlicrnme, r + i + 1 - p < 1, since r i = Ic 5 p - 1. Hence, r 1 < p if and only if r - t i 1 - p < min(i,l). Now we can use (2.4.3)(i),for the proof in case (c).

+

+

(iii) First assume that X = 1 and r car1 write

Note that, r that,

Ph. D. Thesis

+

+

+

+ +

+ I < p. In this case Lm(S(&, pi)) = XiXiYW,and we

+ 1 p imply that i 2 1. Therefore, tia# 0.

Now, we note


We observe that, Lm(S(&, w)) = Lm(XT+i+i-,

(E.2.1)

9,-T)

>

x;-' X ~ X I Yby~ ,(ii).

Furthermore, Lm(S(Ek1,pi)) >

If 2 and r' + j + 1 > p, it follows that, index ( X $ " X , / + ~ + ~ - ~ Y ~ -=~ )r' + j + 1 - p > O = index (X,Y-'X~X~).

(ii) Since Y

Now the proof follows from (2.4.3)(i).

We observe that, (E.2.5)

Lm(Xt$,-,!)

=

Lm(S(&, $j)) = XtX,q'+l y v - w

Furthermore, Lrn(S(tk1,$j)) > X:-'

If

[t,,+/

= [jz = 0 then

X j x ,~ by (ii). Hence,

(E.2.5), otherwise (E.2.5) and (E.2.6), show that S((kl,$j)

+g

0.

Let x - 1 < p - r', then J = [0, x - 11. Next, we have to consider cases depending upon w h e t h e r j f l - p ~ J o r j + l - p @ J. Notethat, j~ J = [ 0 , z - 1 1 a n d l L l 0. (b) L e t x - l < p - r ' a n d j f l - p ~

J = [ O , x - I ] . Then,

In this case {j,l ) n (0, p) = 0, and therefore, tjl # 0. Now we note that,

Ph. D.Thesis

Ph. D. Thesis


Ej2.7.C Case (C) : r'

+j

#k

(2.7.4) Proposition. Let j E J

41

+j # 1 and 1 < k 5 I 5 p - 1, such that r' + j

and r'

# k and r'

+j

# 1.

Then,

Proof. In this case gcd(Lm(&),Lm($j)) = gcd(XkXz, Xr+iX,Q)= 1, and hence the proof follows by Lemma (2.3.5). a

The leading monomials of Jkl and 0 are coprime and hence it is rather simple to show that S(& , 0) --)B 0, as we can directly appeal to Lemma (2.3.5). (2.8.1) Proposition. Let k , 1 be integers such that 1 5 k

5 1 < p - 1. Then,

Proof. We know that gcd(Lm(Jkl),Lm(0)) = gcd(XkXl,Yv) = 1,and hence the proof follows by Lemma (2.3.5). Ph. D. Thesis

Let us recall that, yi := Xr+J; - x $ - ~ x ; Y w and Lrn(yi) = XT+,X; 1 :=

[(A P - TI

if

[~iiau(r,-r+i,O),p-r]

if


Let i, j

E

p#O p=O

,

for i E I, where

or W = 0 , and W # 0 ,

I be distincl. Then,

Proof. (i) Note that, lcm(Lm(yi), Lm(yj)) = XT+iXr+jX,P. The rest follows from the definitions. (ii) It is obvious. (iii) We can write

[niax(r,-r+ l,O),p-T]

if

p#O

or

if

p=0

and W # Q ,

W=0,

and 1Clj

..- X , + ~ X $ Y ~ - " - x,Y-'x,

Ph. D. Thesis

and Lrn($,) = X , r + j ~ , P ' ~ v -,wfor j E J, where


We assume that W

43

# 0, i.e., w # 0 and z # 0, because otherwise J = 0.

We consider the following two cases. (A) r + i = r l + j ,

(B) r + i

# r l +j .

s2.10.A

Case (A) : r


(i) S ( a ,$j)

Let i E I and j

E

=

Yw-""ipi - x,P-" $j

-

x"-1x9-9'~. 0 P J 0

(ii) Lrn(S(pi,$j)) (iii) S(yi, $j)

+ i = r1+ j.

=

J , such that r

+ i = r' + j

. Then,

xX-'xi yw .

= X;-'XJ".

(a)

-x$+~-'x"-P'~~, P - ,y;-lxi@

(b)

-X,X+P-~XP-~'-~E P

Proof. (i) We know that q'

i , p + ~ - ~'

if r1 < r ,

x~X-~X~B

5 q. Therefore, if r + i

=

r'

if r1 > T

+ j, then lcm(Lm(cpi) , Lrn(gj)) =

XT+iXIX~Yw-w , and hence, the expression for S(pi,7,bj) is clear from the definition.

(c) r1 > r . (a) r < r : In this case, v - 1 = X + p. Moreover, r + i = r1+ j implies that i < j and therefore j > 0. We have (ii) We consider the cases:

(a) r1< r , (b) r'

= r,

# 0, then, index (XiYv) = i > 0, and the proof follows from (2.4.3)(i). If i = 0 and p # 0, then, index (XiYv) = 0 and base (X;Yw) = 1. Hence the proof follows

If i

from (2.4.3)(ii). If i = 0 and Q,

= 0,

then, S(pi,$I,) = index (YV)= p

x,"(X;-q1xj

- Y v ) .Therefore,

+ 1 > min{j, p} = index (x,P-~'x~).

We now apply (2.4.3)(i). ( b ) r l = : Weknowthatv-l=X+p-landr+i=r'+jimpliesthat,i=j.

Wehave

Ph. D. Thesis

It is clear that, index (x;-P' x:) =

citllcr case, i~ldex( Y v ) = p (2.4.3) (i).

111

{ p0

if p # O , if p = 0 .

+ 1 > index (x;-q1X[).

(c) r' > r : Zn this case also, v - 1 = X i > j 2 0. We have

Hence, the proof follows from

+ p - 1. Furthermore, r + i = + j implies that, T'

s ( p i , q j ) = xo*-'(x;x;-q'x, -XiYV) and index ( X i Y v ) = i > rnax{j, 0)

2 index ( x [ x ~ - ~ ' x ~ ) .

'Therefore, we can apply (2.4.3)(i) . (iii)(a) Let T' < r. Then, v - 1 = X

+ p and we can write

We note that,

(b) If r'

> r, then v - 1 = X + ,u - 1, and we can write

The rest is similar as in (a).

Ph. D. Thesis

I


45

First, we will prove the following lemma, which will help us in evaluating the leading monomial of certain S-polynomials. (2.10.2) Lemma.

every i

E

I,j

E

Suppose that, W J.

# 0,

p =0

and r' 2 r . Then,I

n J = 0 and j p. Then, I = [max(r, - r + 1, 0), p - r] = - r' + 1, p - r],

Proof. Using Lemma (1.2.4), we get r,

and J

=

[O, min(z - 1, p - r')] = [O, p

(2.1 0.3) Lemma.

Let i E I , j

E

=p

- r'].

Hence the proof.

J . Then,

- ~,X-'x~x~/+~y" ) = ~,X-'X~X,/+~Y" r+z.x3.xq p -qf

"-1x

Lm(Xo

Proof. Let S := X,Y-'X~+~X~XZ-" - x;-'x~x,I+~Y" . We consider the cases: (i) r'

< r and (ii) r' 2 r.

(i) T' < r : In this case, v - 1 = X

+ p. We have

and index ( x[+'x,+~x~xPQPQ~' ) = 0 , base ( X[+'X,+~X~X~-~' ) 2 p subcases :

+ 1.

Consider the

(4 i # 0 , (b) i = 0 and p f 0 , (c) i = 0 and p = 0.

In subcases (a) and (b), the proofs are essentially the same as those in "r' < r" case in Proposition (2.10.1) (ii). Let us therefore consider (c).

(c) Suppose i

Since j < r'

=0

and p = 0. Then,

+ j , we can write

index (Xrt+jYV)= r'

+j

> j 2 rninjr + i, j}

= index (x~+~x~x,P-")

and we apply (2.4.3) (i). Ph. D. Thesis

(ii r

r : We know that u - 1 = X + p - 1, and therefore,

x;-' ( X ; x,+~ xjx;-P' - XiXTt+jYV). index ( X , X , ~ + ~)Y=~min{i, r' + j ) . Let us consider the following subcases : S =

Note that,

* p # 0 , i # 0 : In this case, index ( X i X T f + j Y v= ) minji, r'

+j }

>0

= index (X,'X~+~X~X,"-'')

and apply (2.4.3)(i). ,u 2 2 , i = 0 : I11 this case, index (XiXTt+jYv)= 0 = index ( X [ X ~ + ~ X ~ X : and -~')

I-Ience, the proof follows from (2.4.3)(ii). * p,= 1 , i = 0 : We have

Clearly, this expression is the same as (E.2.9). ,u = 0 : In this case, j j and i > j, therefore, index (XiXT,+jYv)= ~nin{i,r' + j }

Since r'

> j 2 min{r

+ i, j }

= index

(x~+~x~x,"-")

Tlie proof follows from (2.4.3) (i) . Here we present a lemma, which will help us in the proofs of the main propositions.

Ph. D. Thesis


Proof. (i) We can write

-

1-Xi(Xr/+jXr-T/

r

-Xi&'+j,r-r/

+

- XpXV+j-p) X O X , + ~ X-~XPXiX~+j-p if r

+ Xo(&+i,j - ti,r+j)

ifr+j S p ,

+

if r - X p (XiXr+j-p- X O X , + ~ + ~ - ~XoXT+;Xj ) -XitT/+j, - x ~ X r + i +j-pXp = if T + j p .

+j > p .

 p.

(ii) We have

Hence the proof. We now consider two subcases, vix., r' propositions. Let us recall that q' 5 q. (2.10.5) Proposition. Let i E

(i) Let q'

I, j

E

+j

= p and r'

3, such that r + i # r'

+j

+j

#

and

p, in the following two

T'

+j

= p.

< q. Then,

(ii) Let q' = q. Then,

(iii) Lm(S(tpi, $ j ) ) =

(a) x,"-'x;Y" (b)

if 9' < q,

X : - ' X ~ X ~ / + ~ Y "if 9' = q Ph. D. Thesis

if q' < q and r' 2 T ,

Proof. (i) In this case, lc~n(L~n(qi), Lm(gj)) = X,+iX,4Yv-". defiliitioris.

The rest follows from the

(ii) Note that, lcm(Lm(pi), Lrn($j)) = X,+;X,4+'YV-". Now it is easy to derive the expression for S(pi,+ j ) . (iii)(a) We consider the cases:

( I ) r' < r > (2) r'

2 r.

( 1 ) T' < r : In this case, v - 1 = X + p. We have

> +

and index (X;"X;-"X~) = 0 , base (X(+'X,'-~'X~) p 1. The proof in this case is essentially the same as that in " r' < T " case in Proposition (2.1O.l)jii). (2) 7"

2r

:

We know that v - 1 = X

+ p - 1 and therefore,

Note that, index (XiYv)= i and base (XiYv)= 1. Let us consider the following cases : * p # 0 , i # 0 : In this case,

and apply (2.4.3) (i) . a p 2 2 , i = 0 : In this case, index (XiYv) = 0 = index ( X [ X T + i ~ ,xq-q'-l) P-T p ktse (XiYv) = 1

i 1.1

and

5 base (XIX,+i~p-T.x;-q'-l).

Hence, the proof follows from (2.4.3)(ii). p = l , i = O : Wehave

and index (Yv) = p Ph. D.Thesis

+ 1 > min{r + i,p - r') = index ( X , + i ~ p - T l ~ ; - ~ l - l ~ .


We apply (2.4.3) (i) . a p = 0 : Note that, by Lemma (2.10.2)) i = 0 is not possible. We have

Now, by Lemma (2.10.2) we have, p - r' = j < i. Hence, index (XiYu) = i > p - r' 2 min{r

+ i, p - r ' )

= index

(x,+~x,-,/x;-~~-~ > a

The proof follows from (2.4.3) (i). (b) The proof is given in Lemma (2.10.3).

(iv) (a) In this case v - 1 = X

+ p, and we can write

by Lemma (2.10.4)(i), since p = r'

Furthermore, Lm(S(pi,$ j ) )

If ti,,-+

=

S ( c ~ i$,j )

--+S

Jr+i,p-r(

>

+j

r . Then, v - 1 = X + p - I. We can therefore write

= -XoX+p-1 Xpq-91-1 (G,p+r-7' -

p-Tl) - X:-'X~B

,

by Lemma (2.4.5). Ph. D. Thesis

The last part is similar to that in part (a).

by Lemma (2.10.4) (i), since p = r'

+j

< r + j. The last part is similar to that in part ( a ) . ~

J , such that r + i If r'


Let i E I , j

(ii) Lm(S(cpi,gj)) =

x;-'x~/+~x~ YV.

(iii) S(~pi, $j) =

I

E

+

+j

and r'

(b) - X,*+'-'X~-~+'&,+~-~XoX+P X pq-q! [r+i,j -X~+"-'X~X;-~'[~I+~,~-~/ - X,"-'X,+~X~ 0

+ j # p.

Then,

P, if

T'

proof. (i) In this case, Icm(Lm(cpi),Lm($j)) = Xr+iXr/+jX,PYV-",and therefore, t l ~ eex

pression for S(cpi, $ j ) follows from definitions. (ii) See Lemma (2.10.3).

Ph. D. Thesis

2hapter 2 : A Grobner basis for the defining ideal

51

Yow, we can apply Lemma (2.10.4), to get the desired expressions. Note that,

Furthermore, Lm(S(yi,Qj)) > x,V-'

X~+~X~X:-", by (ii). Hence,

where ( = 0 if r' < r , and C = 1 if r1 2 r. We apply (3.2.12) and (E.2.13), as in the earlier I cases, to show that S(qi,Qj) --+g 0.

The leading monomials of pi and 8 are coprime, therefore we can appeal to Lemma (2.3.5), to conclude that S(vi,8) +g 0.

(2.1 1 .l)Proposition. Let i E I . Then, (i) S ( p i , 9 ) = Y vpi - XT+iXp.

and

S ( y i , 8)

-+g

0.

Proof. We know that gcd(Lm(pi),Lm(0)) = g ~ d ( x , + ~ X,Yv) , Q = 1, and hence the proof I follows by Lemma (2.3.5). Ph. D. Thesis

For co~~vcnience, let us recall that,

qj := XTl+jX,q ' y V --~ XgY-lXj

We assume that W # 0, i.e., w

q ' a d Lm($j) = XT/+jXP

~ ~ for - ~ j,

E J , where

# 0 and z # 0, because otherwise J = @.

(2.12.1) Proposition. Let i, j E J be distinct. Then,

Proof. (i) Note that, lcm(Lm($i), Lm($j)) = X ~ ~ + ~ X ~ ~ +. The ~ X rest $ Y follows ~ - ~ from the definitions. (ii) It is obvious. (iii) We can write

We recall that, $j := Xrl+jXP'YV-w

J

:=

- x,"-'X,

[o,min(z-1,p-T')]

and Lm(dj) = Xrr+jX$Yv-u if

W = 0,

if

w#@.

and Y v- X [ X , . - ~ / X ~ ~ if r' < T ,

0 := Ph. D. Thesis

Y v- X [ X ~ + , . , I X ~ ' - ~ if r'

2T,

and Lm(0) = Yv .

, for j

E

J , where


We assume that W the cases :

# 8,ie.,w # 0 and x #

53

0, because otherwise J = 0. Let us consider

(A) r1 < r, (B) r'

2 r.

52.13.A

Case (A) : I-'< r

(2.13.1) Proposition. Let r'

and S($j,6) -+g

< r and j

E

J . Then,

0

Proof. (i) Note that lcm(Lm(h),Lm(0)) = l c m ( ~ , . ~ + ~ ~YY) f ~= " -X~~,! + ~ X ~ YNOW ".

the proof follows from the definition of S($j, 6) and the from fact that v = X + p + 1 if r' < r. (ii) We know that X If p # 0, then,

2 1 and therefore, index (X~+'X~Y")= 0. index (X[Xr-rtXrt+jX,Q)= 0 ,

and since r - r' > 0,we have base (X:XT-TtXrt+jXi) = p < X + p 5 base (Xo"'x~Y"). Hence, we can apply (2.4.3)(ii). Suppose p = 0. Then, S(llj, 0) = Xr-T/X++jX,P- x,*Xjyw. Again r - r' > 0 implies that, index (X,-,, X,t+jX,Q) = min{r - r', r'

+ j} > 0 = index (x,"xjyw).

Now the proof follows from (2.4.3)(i). Ph. D. Thesis

(iii)(a) Since r' < r , it follows that r, = r - r' and hence I = [O,p- r]. Let us now assume p - r , i.e., that T' + j 2 r. We also know that r' j 5 p. Therefore, 0 I r' - r j r' - r j E I . We can write

+

+

We observe that,

(b) Let r'

+ j < r and r + j < p. Then,

Note that,

Hence, S($j, 0) --+G 0.

(c) Let r'

Ph. D. Thesis

+ j < r and r + j

> p. We have

+
r and j

J . Then,

[ T J + ~ , ~ - ~ I

Case (B) : r'


>r

Let r'

E

Proof. (i) We just need to recall that v = X

+ 1,if r'

2 r. The rest

follows from the

definitions. ( i i ) We have

S($j,0 ) = x

( ( x ~ / ~ x ~ +- ~X ,-" ~- ' X~~xY ~~) -. '

We know that X 2 1. If X > 1, then,

index (xTt+j XP+,-,,Xi-') > 0 = index

(x,"-' xjYW)

and we apply (2.4.3)(i) Ph. D. Thesis

52.14 The Main Theorem

56

 j = index ( X j Y w).

Suppose X = 1. Note that j E J implies that j index (X,./+&Y,-,t

Now the proof follows from (2.4.3) (i).

The last part is si~nilarto that in (2.13.1)(iii)(b).

52.14 The Main Theorem (2.14.1) Theorem. Let p >_ 1 and ma,. . . ,nap,%be a minimal almost arithmetic sequence of coprime positive integers, such that mo < . . . < m, form an arithmetic sequence and rL is arbitray. Let K be a field and p denote the defining ideal of the monomial curve C i n A:< defined parametrically b y XO= Tn",. . . , X, = T m p , Y = T". Then, the set G is a Grobner Oasis for p, with respect to grevlex monomial order on K [Xo,. . . , X,, Y].

.

- 52.13, we have shown that S(gil g j ) -+g 0 , for every distinct gi, yj E G. IIence, by (2.3.4), G is a Grabner basis for p, with respect to grevlex monomial order.

Proof. In $2.5

$2.1 5 Syzygies

(2.15.1) Definition. Let S be a commutative ring with identity, A4 an S-module and let ( g ,. . . ,g, ) be an ordered 2-tuple of elements gi E M.Let 8 : St 4 M be the S-module map defined as .9((al,.. . ,at)) := aigi. Tilen, ker(@):= {(a,, . . . ,at) E St I t= 1 aigi = 0)

xt

is called the syzygg module of ( gl , . . . ,9r ) and is &moted by Syz( 91, . . . , gt ). Ph. D. Thesis


57

Here, we present the version of Schreyer's theorem for S a polynomial ring over a field and M an ideal in S. (2.15.2)Theorem. Let KIX1, . . . ,X,]be the polynomial ring over a field K and I be an ideal in KIX1,. .. , X,]. Let G := {gl,. . . ,gt) be an ordered set of generators for I , which is a Grobner basis, with respect to some fixed monomial order on KIX1,. . . ,X,]. Let Lt(gi) = - 1cm(Xai, xhj) for 1 i ,j t . Suppose that, for i # j, ciXQi, ci E K and aij := Lt($i)

<
P ( p + 1) - nlo + 1, since 1)m0 t P(p 1) 1 2 1 and hence t 1-2 1, i,c., t 2 0. Now, if t = 0, then from equations (E.3.1) and (E.3.2) we get P - - qi - 1 = 0 arid PO, + 1) = i. These imply that, ri = i - pip = ( a + 1 ) ~P > P, which is absurd. Hence, t 2 1.

i 5 7m - 1. SO, (t

+

+ +

+

+

From equation (E.3.2), we get

4- 1) - i = ta(p

+ 1) + tb and therefore,

PO, + 1) 2 t a b + 1) + t b > ta(p+

Ph. D. Thesis

1) >_ a(p + I),

Chapter 3 : A minimal free resolution

(vi) We see that u - x (vii) We always have r'

=b

61

and hence the result follows.

> r = 1, so u = p +

X = p + 1.

m

For our convenience, let us rewrite the minimal generating set G , which we have already mentioned in 52.2. The only difference here is that, we consider the integers in a minimal arithmetic sequence and therefore the constants q, q', r, r', u, v , w, z , A, p, v are replaced by the values written in terms of a, b and d, as derived in the above Lemma. Let mo < . . < m, < n be a minimal arithmetic sequence, with gcd(mo,. . . ,m,, n) = 1. Write mo = a ( p + 1) + b, where a and b are unique integers such that a > 0 and b E [I,p + 11. Let K be a field and XO,. . . ,X,, Y, T be indeterminates. Let p denote the kernel of the Kalgebra homomorphism 77 : K[Xo,. . . , X,, Y] --+ K [ T ] defined , by q(Xi) = T"', q(Y) = T". For i,j E [ I ,p - 11, & = X i X j - x ~ - ~ x ~ +, ~ where - , , ax=~0 or 1 according as i or i + j > p .

+j < p

For i E [O,p - 11, g5i = Xi+lxp - X ~ Y .

We know that (2 is a minimal generating set and also a Grobner basis for p, with respect to grevlex monomial ordering on K[X0,. . . , X,, Y]. In Chapter 2, 52.2, we presented a criterion for complete intersection for monomial curves defined by a minimal almost arithmetic sequence of integers. It has also been mentioned that, a necessary condition for complete intersection is p 2. When the sequence of integers is a minimal arithmetic sequence, then the criterion takes a rather simple shape, which we present here. We only have to consider the cases p = I and p = 2.

is a complete intersection if

+

Proof. First, suppose that mo is even. Then mo = 2a 2 for some integer a 2 0. Now, by Lemma (3.2.2), r = 1 = p and 2 = 0. In this case W = 0 and r = p. Hence, by Proposition (2.2.1)(iii), C is a. complete intersection. Ph. D. Thesis

g3.3 Minimal free resolution

62

+

Conversely, suppose that mo is odd. Then mo = 2a l for some integer u >_ 0. By Lemma (3.2.2) we see that r = 1, w = 1, t = 1, p = a + d. Since W # 8 and p # 0, none of the conditions of Proposition (2.2.1)jiii) is satisfied and therefore, C is not a complete I intersection. (3.2.4) Theorem. Let

. . . ,ma be a minimal arithrnetk sequence of positive integers. Let

7 ~ , ,

K be a field. The afine monomial curve C i n the e-space A& is never a complete intersection. Proof. We have p = 2. By Lemma (3.2.2)) we get r = 1. Hence, by Propositiou (2.2.1), it u

can never be a complete intersection.

53.3 Minimal free resolution

We fiu the notations and assumptions clearly, which will be followed for ihe rest of bhis chapter. (3.3.1) Assumptions and Notations.

Let I< be a field and let R a

Y]be the polynomial ring over K.

:= K I X o ,XI, Xz,

Let mo < ?nl < mz < ?n3be a minimal arithmetic sequence of coprime posi Live ir~tegers with common difference d. We write n := ms and mo = 3a b, where u and b are unique integers such that a 0 and 1 b 3.

<

+

Let p c R be the defining ideal and A := K [ T q ,T m l ,T q ,Tn]= R/p be tllc coordinate ring of the affine monomial curve C in &,defined parametrically by X o = Tmo, X I = Tml,Xz = Tm2,Y = F . By abuse of notation, we will often say C is a monorriial curve defined by the integers mo < ml < rnz < m3, but this would mean that C is defined by the above parametrization. Let

fl,

.

. . . , fm, be finitely many elements of P.

For 1 < - i < - m and 1 fi

= (fli)..

. ,} ( f l ,. . . , fi

(