On the Interval Property Level and Gorenstein in algebra and - PowerPoint PPT Presentation

The Interval Property On the Interval Property Level and Gorenstein in algebra and combinatorics algebras Pure O -sequences Pure FABRIZIO ZANELLO f -vectors (MIT and Michigan Tech) Differential posets AMS Meeting of Lincoln, NE October 15, 2011

References The Interval Property Many colleagues to acknowledge. Discussion will involve: Level and FZ: Interval Conjectures for level Hilbert functions, J. Gorenstein algebras Algebra 321 (2009), no. 10, 2705-2715; Pure O -sequences M. Boij, J. Migliore, R. Mir` o-Roig, U. Nagel, FZ: On the Pure shape of a pure O -sequence, Memoirs AMS, to appear f -vectors (arXiv:1003.3825); Differential posets T. H` a, E. Stokes, FZ: Pure O -sequences and matroid h -vectors, preprint (arXiv:1006.0325); R. Stanley, FZ: On the rank function of a differential poset, in preparation.

Statement We say that a class S of (possibly finite) integer sequences The Interval has the Interval Property if, when h , h ′ ∈ S coincide in all Property Level and entries but one, say Gorenstein algebras h = ( h 0 , . . . , h i − 1 , h i , h i + 1 , . . . ) Pure O -sequences Pure and f -vectors h ′ = ( h 0 , . . . , h i − 1 , h i + α, h i + 1 , . . . ) Differential posets for some α ≥ 1, then ( h 0 , . . . , h i − 1 , h i + β, h i + 1 , . . . ) is also in S , for all β = 1 , 2 , . . . , α − 1 .

Statement We say that a class S of (possibly finite) integer sequences The Interval has the Interval Property if, when h , h ′ ∈ S coincide in all Property Level and entries but one, say Gorenstein algebras h = ( h 0 , . . . , h i − 1 , h i , h i + 1 , . . . ) Pure O -sequences Pure and f -vectors h ′ = ( h 0 , . . . , h i − 1 , h i + α, h i + 1 , . . . ) Differential posets for some α ≥ 1, then ( h 0 , . . . , h i − 1 , h i + β, h i + 1 , . . . ) is also in S , for all β = 1 , 2 , . . . , α − 1 .

A few well-known examples Famous classes of sequences coming from graded algebra The Interval or combinatorics, where the Interval Property is known to Property hold or fail, include: Level and Gorenstein algebras Hilbert functions of standard graded algebras: holds Pure (Macaulay’s theorem). O -sequences Pure f -vectors of simplicial complexes: holds (the f -vectors Kruskal-Katona theorem). Differential posets h -vectors of Cohen-Macaulay (or shellable) simplicial complexes: holds (Stanley’s theorem). Matroid h -vectors: fails. Many examples: e.g. ( 1 , 4 , 4 ) and ( 1 , 4 , 6 ) are matroid h -vectors, ( 1 , 4 , 5 ) is not.

A few well-known examples Famous classes of sequences coming from graded algebra The Interval or combinatorics, where the Interval Property is known to Property hold or fail, include: Level and Gorenstein algebras Hilbert functions of standard graded algebras: holds Pure (Macaulay’s theorem). O -sequences Pure f -vectors of simplicial complexes: holds (the f -vectors Kruskal-Katona theorem). Differential posets h -vectors of Cohen-Macaulay (or shellable) simplicial complexes: holds (Stanley’s theorem). Matroid h -vectors: fails. Many examples: e.g. ( 1 , 4 , 4 ) and ( 1 , 4 , 6 ) are matroid h -vectors, ( 1 , 4 , 5 ) is not.

A few well-known examples Famous classes of sequences coming from graded algebra The Interval or combinatorics, where the Interval Property is known to Property hold or fail, include: Level and Gorenstein algebras Hilbert functions of standard graded algebras: holds Pure (Macaulay’s theorem). O -sequences Pure f -vectors of simplicial complexes: holds (the f -vectors Kruskal-Katona theorem). Differential posets h -vectors of Cohen-Macaulay (or shellable) simplicial complexes: holds (Stanley’s theorem). Matroid h -vectors: fails. Many examples: e.g. ( 1 , 4 , 4 ) and ( 1 , 4 , 6 ) are matroid h -vectors, ( 1 , 4 , 5 ) is not.

A few well-known examples Famous classes of sequences coming from graded algebra The Interval or combinatorics, where the Interval Property is known to Property hold or fail, include: Level and Gorenstein algebras Hilbert functions of standard graded algebras: holds Pure (Macaulay’s theorem). O -sequences Pure f -vectors of simplicial complexes: holds (the f -vectors Kruskal-Katona theorem). Differential posets h -vectors of Cohen-Macaulay (or shellable) simplicial complexes: holds (Stanley’s theorem). Matroid h -vectors: fails. Many examples: e.g. ( 1 , 4 , 4 ) and ( 1 , 4 , 6 ) are matroid h -vectors, ( 1 , 4 , 5 ) is not.

A few well-known examples Famous classes of sequences coming from graded algebra The Interval or combinatorics, where the Interval Property is known to Property hold or fail, include: Level and Gorenstein algebras Hilbert functions of standard graded algebras: holds Pure (Macaulay’s theorem). O -sequences Pure f -vectors of simplicial complexes: holds (the f -vectors Kruskal-Katona theorem). Differential posets h -vectors of Cohen-Macaulay (or shellable) simplicial complexes: holds (Stanley’s theorem). Matroid h -vectors: fails. Many examples: e.g. ( 1 , 4 , 4 ) and ( 1 , 4 , 6 ) are matroid h -vectors, ( 1 , 4 , 5 ) is not.

Level and Gorenstein algebras Interval Property first conjectured in combinatorial The Interval Property commutative algebra (FZ, J. Algebra, 2009), for the Level and sets of Hilbert functions of graded artinian level and (in Gorenstein algebras an obviously symmetric fashion) Gorenstein algebras. Pure O -sequences All known techniques (algebraic, combinatorial, Pure homological) to study level/Gorenstein Hilbert functions f -vectors seem to point in the direction of the Interval Property. Differential posets A characterization of such Hilbert functions seems hopeless, and the Interval Property would provide a very natural, strong structural result. Some timid progress, but Property still wide open.

Level and Gorenstein algebras Interval Property first conjectured in combinatorial The Interval Property commutative algebra (FZ, J. Algebra, 2009), for the Level and sets of Hilbert functions of graded artinian level and (in Gorenstein algebras an obviously symmetric fashion) Gorenstein algebras. Pure O -sequences All known techniques (algebraic, combinatorial, Pure homological) to study level/Gorenstein Hilbert functions f -vectors seem to point in the direction of the Interval Property. Differential posets A characterization of such Hilbert functions seems hopeless, and the Interval Property would provide a very natural, strong structural result. Some timid progress, but Property still wide open.

Level and Gorenstein algebras Interval Property first conjectured in combinatorial The Interval Property commutative algebra (FZ, J. Algebra, 2009), for the Level and sets of Hilbert functions of graded artinian level and (in Gorenstein algebras an obviously symmetric fashion) Gorenstein algebras. Pure O -sequences All known techniques (algebraic, combinatorial, Pure homological) to study level/Gorenstein Hilbert functions f -vectors seem to point in the direction of the Interval Property. Differential posets A characterization of such Hilbert functions seems hopeless, and the Interval Property would provide a very natural, strong structural result. Some timid progress, but Property still wide open.

Level and Gorenstein algebras Interval Property first conjectured in combinatorial The Interval Property commutative algebra (FZ, J. Algebra, 2009), for the Level and sets of Hilbert functions of graded artinian level and (in Gorenstein algebras an obviously symmetric fashion) Gorenstein algebras. Pure O -sequences All known techniques (algebraic, combinatorial, Pure homological) to study level/Gorenstein Hilbert functions f -vectors seem to point in the direction of the Interval Property. Differential posets A characterization of such Hilbert functions seems hopeless, and the Interval Property would provide a very natural, strong structural result. Some timid progress, but Property still wide open.

A = R / I is a standard graded artinian algebra if The Interval Property R = k [ x 1 , . . . , x r ] , I ⊂ R is a homogeneous ideal, √ Level and I = ( x 1 , . . . , x r ) , deg ( x i ) = 1. Gorenstein algebras The Hilbert function of A is h ( A ) = ( 1 , h 1 , . . . , h e ) , Pure O -sequences with h i = dim k A i (suppose h e � = 0, h e + 1 = 0). Pure f -vectors The Socle of A is soc ( A ) = 0 : ( x 1 , . . . , x r ) ⊂ A . Differential The Socle-vector of A is s ( A ) = ( 0 , s 1 , . . . , s e ) , where posets s i = dim k soc ( A ) i . A is level (of type t ) if s ( A ) = ( 0 , . . . , 0 , t ) . A is Gorenstein if s ( A ) = ( 0 , . . . , 0 , t = 1 ) .

A = R / I is a standard graded artinian algebra if The Interval Property R = k [ x 1 , . . . , x r ] , I ⊂ R is a homogeneous ideal, √ Level and I = ( x 1 , . . . , x r ) , deg ( x i ) = 1. Gorenstein algebras The Hilbert function of A is h ( A ) = ( 1 , h 1 , . . . , h e ) , Pure O -sequences with h i = dim k A i (suppose h e � = 0, h e + 1 = 0). Pure f -vectors The Socle of A is soc ( A ) = 0 : ( x 1 , . . . , x r ) ⊂ A . Differential The Socle-vector of A is s ( A ) = ( 0 , s 1 , . . . , s e ) , where posets s i = dim k soc ( A ) i . A is level (of type t ) if s ( A ) = ( 0 , . . . , 0 , t ) . A is Gorenstein if s ( A ) = ( 0 , . . . , 0 , t = 1 ) .