Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Monomial curves of homogeneous type Raheleh Jafari Kharazmi University The 13th Seminar on Commutative Algebra and Related Topics November 16th-17th, 2016 IPM Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Based on a joint work with Santiago Zarzuela , University of Barcelona Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Table of Contents Motivation: a conjecture of Herzog-Srinivasan. 1 Homogeneous semigroups and semigroups of 2 homogeneous type Small embedding dimensions 3 Asymptotic behavior under shifting 4 Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Table of Contents Motivation: a conjecture of Herzog-Srinivasan. 1 Homogeneous semigroups and semigroups of 2 homogeneous type Small embedding dimensions 3 Asymptotic behavior under shifting 4 Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Table of Contents Motivation: a conjecture of Herzog-Srinivasan. 1 Homogeneous semigroups and semigroups of 2 homogeneous type Small embedding dimensions 3 Asymptotic behavior under shifting 4 Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Table of Contents Motivation: a conjecture of Herzog-Srinivasan. 1 Homogeneous semigroups and semigroups of 2 homogeneous type Small embedding dimensions 3 Asymptotic behavior under shifting 4 Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Let n := 0 < n 1 < · · · < n d be a family of positive integers. Let S = � n 1 , . . . , n d � = { r 1 n 1 + · · · + r d n d ; r i ≥ 0 } be the semigroup generated by the family n . Let k be a field and k [ S ] = k [ t n 1 , . . . , t n d ] ⊆ k [ t ] be the semigroup ring defined by n . We may consider the presentation ϕ 0 − → I ( S ) − → k [ x 1 , . . . , x d ] − → k [ S ] − → 0 given by ϕ ( x i ) = t n i . Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Let n := 0 < n 1 < · · · < n d be a family of positive integers. Let S = � n 1 , . . . , n d � = { r 1 n 1 + · · · + r d n d ; r i ≥ 0 } be the semigroup generated by the family n . Let k be a field and k [ S ] = k [ t n 1 , . . . , t n d ] ⊆ k [ t ] be the semigroup ring defined by n . We may consider the presentation ϕ 0 − → I ( S ) − → k [ x 1 , . . . , x d ] − → k [ S ] − → 0 given by ϕ ( x i ) = t n i . Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Let n := 0 < n 1 < · · · < n d be a family of positive integers. Let S = � n 1 , . . . , n d � = { r 1 n 1 + · · · + r d n d ; r i ≥ 0 } be the semigroup generated by the family n . Let k be a field and k [ S ] = k [ t n 1 , . . . , t n d ] ⊆ k [ t ] be the semigroup ring defined by n . We may consider the presentation ϕ 0 − → I ( S ) − → k [ x 1 , . . . , x d ] − → k [ S ] − → 0 given by ϕ ( x i ) = t n i . Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Let n := 0 < n 1 < · · · < n d be a family of positive integers. Let S = � n 1 , . . . , n d � = { r 1 n 1 + · · · + r d n d ; r i ≥ 0 } be the semigroup generated by the family n . Let k be a field and k [ S ] = k [ t n 1 , . . . , t n d ] ⊆ k [ t ] be the semigroup ring defined by n . We may consider the presentation ϕ 0 − → I ( S ) − → k [ x 1 , . . . , x d ] − → k [ S ] − → 0 given by ϕ ( x i ) = t n i . Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Let n := 0 < n 1 < · · · < n d be a family of positive integers. Let S = � n 1 , . . . , n d � = { r 1 n 1 + · · · + r d n d ; r i ≥ 0 } be the semigroup generated by the family n . Let k be a field and k [ S ] = k [ t n 1 , . . . , t n d ] ⊆ k [ t ] be the semigroup ring defined by n . We may consider the presentation ϕ 0 − → I ( S ) − → k [ x 1 , . . . , x d ] − → k [ S ] − → 0 given by ϕ ( x i ) = t n i . Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Set R := k [ x 1 , . . . , x d ] . For any i ≥ 0, the i-th (total) Betti number of I ( S ) is β i ( I ( S )) = dim k Tor R i ( I ( S ) , k ) . -We call the Betti numbers of I ( S ) as the Betti numbers of S . Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding For any j ≥ 0 we consider the shifted family n + j := 0 < n 1 + j < · · · < n d + j and the semigroup S + j := � n 1 + j , . . . , n d + j � that we call the j-th shifting of S . Conjecture (J. Herzog and H. Srinivasan) The Betti numbers of S + j are eventually peridic on j with period n d − n 1 . Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding For any j ≥ 0 we consider the shifted family n + j := 0 < n 1 + j < · · · < n d + j and the semigroup S + j := � n 1 + j , . . . , n d + j � that we call the j-th shifting of S . Conjecture (J. Herzog and H. Srinivasan) The Betti numbers of S + j are eventually peridic on j with period n d − n 1 . Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Let S = � 4 , 7 � . Then S + 2 = � 6 , 9 � is not a numerical semigroup (gcd ( 6 , 9 ) > 1). Let S = � 4 , 10 , 11 � , then S + 2 = � 6 , 12 , 13 � = � 6 , 13 � . If j > n d − 2 n 1 , then S + j is minimally generated by d elements. Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Let S = � 4 , 7 � . Then S + 2 = � 6 , 9 � is not a numerical semigroup (gcd ( 6 , 9 ) > 1). Let S = � 4 , 10 , 11 � , then S + 2 = � 6 , 12 , 13 � = � 6 , 13 � . If j > n d − 2 n 1 , then S + j is minimally generated by d elements. Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Let S = � 4 , 7 � . Then S + 2 = � 6 , 9 � is not a numerical semigroup (gcd ( 6 , 9 ) > 1). Let S = � 4 , 10 , 11 � , then S + 2 = � 6 , 12 , 13 � = � 6 , 13 � . If j > n d − 2 n 1 , then S + j is minimally generated by d elements. Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding Let S = � 4 , 7 � . Then S + 2 = � 6 , 9 � is not a numerical semigroup (gcd ( 6 , 9 ) > 1). Let S = � 4 , 10 , 11 � , then S + 2 = � 6 , 12 , 13 � = � 6 , 13 � . If j > n d − 2 n 1 , then S + j is minimally generated by d elements. Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding The conjecture has been proven to be true for: d = 3 (A. V. Jayanthan and H. Srinivasan, 2013). d=4 (particular cases, A. Marzullo, 2013). Arithmetic sequences, n i − n i − 1 = n i + 1 − n i , (P . Gimenez, I. Senegupta, and H. Srinivasan, 2013). In general (Thran Vu, 2014) Namely, there exists positive value N such that for any j > N the Betti numbers of S + j are periodic with period n d − n 1 . Raheleh Jafari Monomial curves of homogeneous type
Motivation: a conjecture of Herzog-Srinivasan. Homogeneous semigroups and semigroups of homogeneous type Small embedding The conjecture has been proven to be true for: d = 3 (A. V. Jayanthan and H. Srinivasan, 2013). d=4 (particular cases, A. Marzullo, 2013). Arithmetic sequences, n i − n i − 1 = n i + 1 − n i , (P . Gimenez, I. Senegupta, and H. Srinivasan, 2013). In general (Thran Vu, 2014) Namely, there exists positive value N such that for any j > N the Betti numbers of S + j are periodic with period n d − n 1 . Raheleh Jafari Monomial curves of homogeneous type
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