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Betti Numbers, Grobner Basis And Syzygies For Certain Affine Monomial Curves

Electronic Theses of Indian Institute of Science

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Title Betti Numbers, Grobner Basis And Syzygies For Certain Affine Monomial Curves
 
Creator Sengupta, Indranath
 
Subject Mathematics
Curves
Grobner Basis
Algebreic Geometry
Betti Number
Manomials
Syzygies
Minimal Free Resolution
 
Description Let e > 3 and mo,... ,me_i be positive integers with gcd(m0,... ,me_i) = 1, which form an almost arithmetic sequence, i.e., some e - 1 of these form an arithmetic progression. We further assume that m0,... ,mc_1 generate F := Σ e-1 I=0 Nmi minimally. Note that any three integers and also any arithmetic progression form an almost arithmetic sequence.
We assume that 0 < m0 < • • • < me-2 form an arithmetic progression and n := mc-i is arbitrary Put p := e - 2. Let K be a field and XQ) ... ,Xj>, Y,T be indeterminates. Let p denote the kernel of the if-algebra homomorphism η: K[XQ, ..., XV) Y) -* K^T], defined by r){Xi) = Tm\.. .η{Xp) = Tmp, η](Y) = Tn. Then, p is the defining ideal for the affine monomial curve C in A^, defined parametrically by Xo = Trr^)...)Xv = T^}Y = T*. Furthermore, p is a homogeneous ideal with respect to the gradation on K[X0)... ,XP,F], given by wt(Z0) = mo, • • •, wt(Xp) = mp, wt(Y) = n. Let 4 := K[XQ> ...,XP) Y)/p denote the coordinate ring of C.
With the assumption ch(K) = 0, in Chapter 1 we have derived an explicit formula for μ(DerK(A)), the minimal number of generators for the A-module DerK(A), the derivation module of A. Furthermore, since type(A) = μ(DerK(A)) — 1 and the last Betti number of A is equal to type(A), we therefore obtain an explicit formula for the last Betti number of A as well
A minimal set of binomial generatorsG for the ideal p had been explicitly constructed by PatiL In Chapter 2, we show that the set G is a Grobner basis with respect to grevlex monomial ordering on K[X0)..., Xp, Y]. As an application of this observation, in Chapter 3 we obtain an explicit minimal free resolution for affine monomial curves in A4K defined by four coprime positive integers mo,.. m3, which form a minimal arithmetic progression.
(Please refer the pdf file forformulas)
 
Publisher Indian Institute of Science
 
Contributor Patil, Dilip P
 
Date 2007-03-30T08:02:46Z
2007-03-30T08:02:46Z
2007-03-30T08:02:46Z
2000-09
 
Type Electronic Thesis and Dissertation
 
Format 10081751 bytes
application/pdf
 
Identifier http://etd.iisc.ernet.in/handle/2005/271
null
 
Language en
 
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