Regularity of powers of edge ideals Huy Tài Hà Tulane University Joint with Selvi Beyarslan and Trân Nam Trung Huy Tài Hà Regularity of powers of edge ideals
Problem and Motivation 1 Asymptotic linearity of regularity Stabilization index and free constant Literature Reviews 2 Polynomial ideals Edge ideals Known answers 3 Results Regularity of powers of forests Regularity of powers of cycles Methods and general bounds Huy Tài Hà Regularity of powers of edge ideals
Asymptotic linearity of regularity R a standard graded algebra over a field k m its maximal homogeneous ideal M a finitely generated graded R -module end ( M ) := max { l | M l � = 0 } The regularity of M is reg ( M ) = max { end ( H i m ( M )) + i } . Theorem (Cutkosky-Herzog-Trung (1999); Kodiyalam (1999); Trung-Wang (2005)) Let R be a standard graded k-algebra, let I ⊆ R be a homogeneous ideal and let M be a finitely generated graded R-module. Then reg ( I q M ) is asymptotically a linear function in q, i.e., there exist a and b such that for q ≫ 0 , reg ( I q M ) = aq + b . Huy Tài Hà Regularity of powers of edge ideals
Asymptotic linearity of regularity R a standard graded algebra over a field k m its maximal homogeneous ideal M a finitely generated graded R -module end ( M ) := max { l | M l � = 0 } The regularity of M is reg ( M ) = max { end ( H i m ( M )) + i } . Theorem (Cutkosky-Herzog-Trung (1999); Kodiyalam (1999); Trung-Wang (2005)) Let R be a standard graded k-algebra, let I ⊆ R be a homogeneous ideal and let M be a finitely generated graded R-module. Then reg ( I q M ) is asymptotically a linear function in q, i.e., there exist a and b such that for q ≫ 0 , reg ( I q M ) = aq + b . Huy Tài Hà Regularity of powers of edge ideals
Asymptotic linearity of regularity R a standard graded algebra over a field k m its maximal homogeneous ideal M a finitely generated graded R -module end ( M ) := max { l | M l � = 0 } The regularity of M is reg ( M ) = max { end ( H i m ( M )) + i } . Theorem (Cutkosky-Herzog-Trung (1999); Kodiyalam (1999); Trung-Wang (2005)) Let R be a standard graded k-algebra, let I ⊆ R be a homogeneous ideal and let M be a finitely generated graded R-module. Then reg ( I q M ) is asymptotically a linear function in q, i.e., there exist a and b such that for q ≫ 0 , reg ( I q M ) = aq + b . Huy Tài Hà Regularity of powers of edge ideals
Stabilization index and free constant The constant a is well understood (Trung-Wang (2005)) The free constant b and the stabilization index q 0 = min { q ′ | reg ( I q M ) = aq + b ∀ q ≥ q ′ } are not known. Problem Understand b and q 0 from invariants and properties of I and M Explicitly compute b and q 0 for special classes of ideals and modules. Huy Tài Hà Regularity of powers of edge ideals
Stabilization index and free constant The constant a is well understood (Trung-Wang (2005)) The free constant b and the stabilization index q 0 = min { q ′ | reg ( I q M ) = aq + b ∀ q ≥ q ′ } are not known. Problem Understand b and q 0 from invariants and properties of I and M Explicitly compute b and q 0 for special classes of ideals and modules. Huy Tài Hà Regularity of powers of edge ideals
Problem and Motivation 1 Asymptotic linearity of regularity Stabilization index and free constant Literature Reviews 2 Polynomial ideals Edge ideals Known answers 3 Results Regularity of powers of forests Regularity of powers of cycles Methods and general bounds Huy Tài Hà Regularity of powers of edge ideals
Polynomial ideals R = k [ x 1 , . . . , x n ] and I ⊆ R a homogeneous ideal. Römer (2001); Eisenbud-Harris (2010); Hà (2011); Chardin (2013): if I is equi-generated then b is related to the regularity of fibers of certain projection map Eisenbud-Ulrich (2012); Berlekamp (2012); Chardin (2014): if I is m -primary then q 0 can be related to “partial” regularity of the Rees algebra of I . In practice: regularity of fibers of projection maps and “partial” regularity of Rees algebras are difficult to compute. Huy Tài Hà Regularity of powers of edge ideals
Polynomial ideals R = k [ x 1 , . . . , x n ] and I ⊆ R a homogeneous ideal. Römer (2001); Eisenbud-Harris (2010); Hà (2011); Chardin (2013): if I is equi-generated then b is related to the regularity of fibers of certain projection map Eisenbud-Ulrich (2012); Berlekamp (2012); Chardin (2014): if I is m -primary then q 0 can be related to “partial” regularity of the Rees algebra of I . In practice: regularity of fibers of projection maps and “partial” regularity of Rees algebras are difficult to compute. Huy Tài Hà Regularity of powers of edge ideals
Polynomial ideals R = k [ x 1 , . . . , x n ] and I ⊆ R a homogeneous ideal. Römer (2001); Eisenbud-Harris (2010); Hà (2011); Chardin (2013): if I is equi-generated then b is related to the regularity of fibers of certain projection map Eisenbud-Ulrich (2012); Berlekamp (2012); Chardin (2014): if I is m -primary then q 0 can be related to “partial” regularity of the Rees algebra of I . In practice: regularity of fibers of projection maps and “partial” regularity of Rees algebras are difficult to compute. Huy Tài Hà Regularity of powers of edge ideals
Edge ideals of graphs V = { x 1 , . . . , x n } ← → R = k [ x 1 , . . . , x n ] G = ( V , E ) a simple graph The edge ideal of G is � { x i , x j } ∈ E � � � I ( G ) = x i x j . Example x 1 ✈ x 5 ✈ ❅ � ❅ � ❅ � ✈ x 4 x 2 ✈ � ❅ � ❅ � ❅ x 3 ✈ x 6 ✈ I ( G ) = ( x 1 x 2 , x 2 x 3 , x 2 x 4 , x 4 x 5 , x 4 x 6 ) ⊆ k [ x 1 , . . . , x 6 ] . Huy Tài Hà Regularity of powers of edge ideals
Edge ideals of graphs V = { x 1 , . . . , x n } ← → R = k [ x 1 , . . . , x n ] G = ( V , E ) a simple graph The edge ideal of G is � { x i , x j } ∈ E � � � I ( G ) = x i x j . Example x 1 ✈ x 5 ✈ ❅ � ❅ � ❅ � ✈ x 4 x 2 ✈ � ❅ � ❅ � ❅ x 3 ✈ x 6 ✈ I ( G ) = ( x 1 x 2 , x 2 x 3 , x 2 x 4 , x 4 x 5 , x 4 x 6 ) ⊆ k [ x 1 , . . . , x 6 ] . Huy Tài Hà Regularity of powers of edge ideals
Asymptotic linearity of edge ideals Know: for q ≥ q 0 , reg ( I ( G ) q ) = 2 q + b . Problem Relate q 0 and b to combinatorial data of the graph G For special classes of graphs, compute q 0 and b explicitly. Huy Tài Hà Regularity of powers of edge ideals
Asymptotic linearity of edge ideals Know: for q ≥ q 0 , reg ( I ( G ) q ) = 2 q + b . Problem Relate q 0 and b to combinatorial data of the graph G For special classes of graphs, compute q 0 and b explicitly. Huy Tài Hà Regularity of powers of edge ideals
Known answers Herzog-Hibi-Zheng (2004) + Fröberg (1990): if the complement graph G c is chordal then I ( G ) q has a linear resolution for all q ≥ 1; that is b = 0 and q 0 = 1 Ferró-Murgia-Olteanu (2012): if I ( G ) is an initial or final lexsegment edge ideal then b = 0 and q 0 = 1 Alilooee-Banerjee (2014): if G is a bipartite graph and reg ( I ( G )) = 3 then b = 1 and q 0 = 1. Problem Characterize graphs for which b = 0 . Huy Tài Hà Regularity of powers of edge ideals
Known answers Herzog-Hibi-Zheng (2004) + Fröberg (1990): if the complement graph G c is chordal then I ( G ) q has a linear resolution for all q ≥ 1; that is b = 0 and q 0 = 1 Ferró-Murgia-Olteanu (2012): if I ( G ) is an initial or final lexsegment edge ideal then b = 0 and q 0 = 1 Alilooee-Banerjee (2014): if G is a bipartite graph and reg ( I ( G )) = 3 then b = 1 and q 0 = 1. Problem Characterize graphs for which b = 0 . Huy Tài Hà Regularity of powers of edge ideals
Problem and Motivation 1 Asymptotic linearity of regularity Stabilization index and free constant Literature Reviews 2 Polynomial ideals Edge ideals Known answers 3 Results Regularity of powers of forests Regularity of powers of cycles Methods and general bounds Huy Tài Hà Regularity of powers of edge ideals
Forests and induced matching number Definition Let G = ( V , E ) be a graph. A matching in G is a collection of pairwise disjoint edges An induced matching in G is a matching { e 1 , . . . , e s } ⊆ E such that these are also the only edges in the induced subgraph of G over the vertices � s i = 1 e i The induced matching number of G is the maximum size of an induced matching in G The graph G is a forest if it contains no cycles. Huy Tài Hà Regularity of powers of edge ideals
Induced matching number Example Consider the graph G as follows: x 1 ✈ x 5 ✈ ❅ � ❅ � ❅ � ✈ x 4 x 2 ✈ � ❅ � ❅ � ❅ x 3 ✈ x 6 ✈ { x 1 x 2 , x 4 x 5 } forms a matching, but not an induced matching in G The induced matching number of G is 1. Huy Tài Hà Regularity of powers of edge ideals
Regularity of powers of forests Theorem (Beyarslan, —, Trung (2014)) Let G be a forest and let ν denote its induced matching number. Then b = ν − 1 and q 0 = 1 . That is, for all q ≥ 1 , reg ( I ( G ) q ) = 2 q + ν − 1 . Huy Tài Hà Regularity of powers of edge ideals
Regularity of powers of cycles Theorem (Beyarslan, —, Trung (2014)) � n � Let G be an n-cycle and let ν = denote its induced 3 matching number. Then b = ν − 1 and q 0 = 2 . In fact, � ν + 1 n ≡ 0 , 1 ( mod 3 ) if reg ( I ( G )) = ν + 2 if n ≡ 2 ( mod 3 ) , and for all q ≥ 2 , we have reg ( I ( G ) q ) = 2 q + ν − 1 . Huy Tài Hà Regularity of powers of edge ideals
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