Pseudo symmetric monomial curves Mesut S ¸ahin HACETTEPE UNIVERSITY Joint work with Nil S ¸ahin (Bilkent University) Supported by Tubitak No: 114F094 IMNS 2016, July 4-8, 2016 Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 1 / 31
Part I Indispensability Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 2 / 31
Semigroup, toric ideal, semigroup ring Let n 1 , . . . , n 4 be positive integers with gcd( n 1 , . . . , n 4 ) = 1. Then S = � n 1 , . . . , n 4 � is { u 1 n 1 + · · · + u 4 n 4 | u i ∈ N } . Let K be a field and K [ S ] = K [ t n 1 , . . . , t n 4 ] be the semigroup ring of S , then K [ S ] ≃ A / I S where, A = K [ X 1 , . . . , X 4 ] and the toric ideal I S is the kernel of the φ 0 → K [ S ], where X i �→ t n i . surjection A − Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 3 / 31
Pseudo symmetric S Pseudo frobenious numbers of S are defined to be the elements of the set PF ( S ) = { n ∈ Z − S | n + s ∈ S for all s ∈ S − { 0 }} . The largest element is called the frobenious number denoted by g ( S ). S is called pseudo symmetric if PF ( S ) = { g ( S ) / 2 , g ( S ) } . S is symmetric if PF ( S ) = { g ( S ) } . Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 4 / 31
S is pseudo symmetric if PF ( S ) = { g ( S ) / 2 , g ( S ) } . Recall the set PF ( S ) = { n ∈ Z − S | n + s ∈ S for all s ∈ S − { 0 }} . S = � 5 , 12 , 11 , 14 � = { 0 , 5 , 10 , 11 , 12 , 14 , 15 , 16 , 17 , 19 } + N and its complement is { 1 , 2 , 3 , 4 , 6 , 7 , 8 , 9 , 13 , 18 } . 1 + 5 , 2 + 5 , 3 + 5 , 4 + 5 , 6 + 12 , 7 + 11 , 8 + 10 , 13 + 5 / ∈ S but n + s ∈ S for all s ∈ S − { 0 } , for n = 9 , 18. So, S is pseudosymmetric. Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 5 / 31
Komeda proved that, the semigroup S is pseudo symmetric if and only if there are positive integers α i , 1 ≤ i ≤ 4, and α 21 , with α 21 < α 1 , s.t. n 1 = α 2 α 3 ( α 4 − 1) + 1, n 2 = α 21 α 3 α 4 + ( α 1 − α 21 − 1)( α 3 − 1) + α 3 , n 3 = α 1 α 4 + ( α 1 − α 21 − 1)( α 2 − 1)( α 4 − 1) − α 4 + 1, n 4 = α 1 α 2 ( α 3 − 1) + α 21 ( α 2 − 1) + α 2 . For ( α 1 , α 2 , α 3 , α 4 , α 21 ) = (5 , 2 , 2 , 2 , 2), S = � 5 , 12 , 11 , 14 � . Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 6 / 31
Pseudo symmetric S Komeda proved that, K [ S ] = A / ( f 1 , f 2 , f 3 , f 4 , f 5 ), where f 1 = X α 1 − X 3 X α 4 − 1 , 1 4 f 2 = X α 2 − X α 21 X 4 , 2 1 f 3 = X α 3 − X α 1 − α 21 − 1 X 2 , 3 1 f 4 = X α 4 − X 1 X α 2 − 1 X α 3 − 1 , 4 2 3 f 5 = X α 3 − 1 X α 21 +1 − X 2 X α 4 − 1 . 3 1 4 Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 7 / 31
S -degrees Let deg S ( X u 1 1 X u 2 2 X u 3 3 X u 4 4 ) = � 4 i =1 u i n i ∈ S . d ∈ S is called a Betti S -degree if there is a minimal generator of I S of S -degree d and β d is the number of times d occurs as a Betti S -degree. Both β d and the set B S of Betti S -degrees are invariants of I S . S -degrees of binomials in I S which are not comparable with respect to < S constitute the minimal binomial S -degrees denoted M S , where s 1 < S s 2 if s 2 − s 1 ∈ S . In general, M S ⊆ B S . Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 8 / 31
Indispensables By Komeda’s result, B S = { d 1 , d 2 , d 3 , d 4 , d 5 } if d i ’s are all distinct, where d i is the S -degree of f i , for i = 1 , . . . , 5. A binomial is called indispensable if it appears in every minimal generating set of I S . Lemma A binomial of S -degree d is indispensable if and only if β d = 1 and d ∈ M S . Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 9 / 31
We use the following Lemma twice in the sequel. If 0 < v k < α k and 0 < v l < α l , for k � = l ∈ { 1 , 2 , 3 , 4 } , then v k n k − v l n l / ∈ S . Proposition M S = { d 1 , d 2 , d 3 , d 4 , d 5 } if α 1 − α 21 > 2 and M S = { d 1 , d 2 , d 3 , d 5 } if α 1 − α 21 = 2. Corollary Indispensable binomials of I S are { f 1 , f 2 , f 3 , f 4 , f 5 } if α 1 − α 21 > 2 and are { f 1 , f 2 , f 3 , f 5 } if α 1 − α 21 = 2. Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 10 / 31
Strongly indispensable minimal free resolutions For a graded minimal free A -resolution → A β k − 1 φ k − 1 → A β k − 2 φ k − 2 φ 2 φ 1 → A β 1 → A β 0 − F : 0 − − − → · · · − − → K [ S ] − → 0 of K [ S ], let A β i be generated in degrees s i , j ∈ S , which we call i -Betti β i degrees, i.e. A β i = � A [ − s i , j ]. j =1 The resolution ( F , φ ) is strongly indispensable if for any graded minimal resolution ( G , θ ), we have an injective complex map i : ( F , φ ) − → ( G , θ ). Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 11 / 31
Betti i-degrees of S = � 5 , 12 , 11 , 14 � 1 − Betti degrees : { 22 , 24 , 25 , 26 , 28 } 2 − Betti degrees: { 36 , 37 , 38 , 39 , 40 , 46 } 3 − Betti degrees : { 51 , 60 } . Note that { 51 , 60 } − 42 = { 9 , 18 } . Recall that ( α 1 , α 2 , α 3 , α 4 , α 21 ) = (5 , 2 , 2 , 2 , 2). Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 12 / 31
In, Barucci-Froberg-Sahin (2014), we give a minimal free resolution of K [ S ], for symmetric and pseudo symmetric S and prove that it is always strongly indispensable for symmetric S. It follows that it is strongly indispensable for pseudo symmetric S iff the differences between the i − Betti degrees do not lie in S, for only i = 1 , 2. Using this, we obtain Main Theorem 1 Let S be a 4-generated pseudo-symmetric semigroup. Then K [ S ] has a strongly indispensable minimal graded free resolution if and only if α 4 > 2 and α 1 − α 21 > 2. Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 13 / 31
Part II Cohen-Macaulayness of the Tangent Cone and Sally’s Conjecture Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 14 / 31
If ( R , m ) is a local ring with maximal ideal m , then the Hilbert function of R is defined to be the Hilbert function of its associated graded ring � m r / m r +1 . gr m ( R ) = r ∈ N That is, H R ( r ) = dim K ( m r / m r +1 ) . Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 15 / 31
The Main Problem: Determine the conditions under which the Hilbert function of a local ring ( R , m ) is non-decreasing. A sufficient condition: If the tangent cone is Cohen-Macaulay, H R ( r ) is non-decreasing. But this does not follow from Cohen-Macaulayness of ( R , m ). Sally’s Conjecture (1980): If ( R , m ) is a one dimensional Cohen-Macaulay local ring with small embedding dimension d := H R (1), then H R ( r ) is non-decreasing. Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 16 / 31
Literature: d = 1, obvious as H R ( r ) = 1 d = 2, proved by Matlis (1977) d = 3, proved by Elias (1993) d = 4, a counterexample is given by Gupta-Roberts (1983) d ≥ 5, counterexamples for each d are given by Orecchia(1980). Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 17 / 31
The local ring associated to the monomial curve C = C ( n 1 , . . . , n k ) is K [[ t n 1 , . . . , t n k ]] with m = ( t n 1 , . . . , t n k ), and the associated graded ring gr m ( K [[ t n 1 , . . . , t n k ]]) is isomorphic to the ring K [ x 1 , . . . , x k ] / I ( C ) ∗ , where I ( C ) is the defining ideal of C and I ( C ) ∗ is the ideal generated by the polynomials f ∗ for f in I ( C ) and f ∗ is the homogeneous summand of f of least degree. In other words, I ( C ) ∗ is the defining ideal of the tangent cone of C at 0. Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 18 / 31
Herzog-Waldi,1975 Let C = C (30 , 35 , 42 , 47 , 148 , 153 , 157 , 169 , 181 , 193) ⊂ A 10 and ( R , m ) be its associated local ring. Then the Hilbert function of R is NOT non-decreasing as H R = { 1 , 10 , 9 , 16 , 25 , . . . } . Eakin-Sathaye,1976 Let C = C (15 , 21 , 23 , 47 , 48 , 49 , 50 , 52 , 54 , 55 , 56 , 58) ⊂ A 12 and ( R , m ) be its associated local ring. Then the Hilbert function of R is NOT non-decreasing as H R = { 1 , 12 , 11 , 13 , 15 , . . . } . Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 19 / 31
4-generated case: The conjecture has been proven by Arslan-Mete in 2007 for Gorenstein local rings R associated to certain symmetric monomial curves in A 4 . The method to achieve this result was to show that the tangent cones of these curves at the origin are Cohen-Macaulay. More recently, Arslan-Katsabekis-Nalbandiyan, generalized this characterizing Cohen-Macaulayness of the tangent cone completely. Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 20 / 31
Criterion for Cohen-Macaulayness Let C = C ( n 1 , . . . , n k ) be a monomial curve with n 1 the smallest and G = { f 1 , . . . , f s } be a minimal standard basis of the ideal I ( C ) wrt the negative degree reverse lexicographical ordering that makes x 1 the lowest variable. C has Cohen-Macaulay tangent cone at the origin if and only if x 1 does not divide LM ( f i ) for 1 ≤ i ≤ k , where LM ( f i ) denotes the leading monomial of a polynomial f i . Mesut S ¸ahin (HACETTEPE UNIVERSITY) Pseudo symmetric monomial curves 21 / 31
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