A Partial Characterization of Virtually Cohen-Macaulay Simplicial Complexes Nathan Kenshur, Feiyang Lin , Sean McNally, Zixuan Xu , Teresa Yu UMN REU July 24, 2019 (UMN REU) Project 6 July 24, 2019 1 / 20
Outline 1 Preliminaries 2 Property of Virtual Resolutions 3 The Intersection Method 4 Balanced Implies VCM (UMN REU) Project 6 July 24, 2019 2 / 20
Stanley-Reisner Theory Definition An abstract simplicial complex ∆ on vertex set X is a collection of subsets of X such that A ∈ ∆ whenever A ⊆ B ∈ ∆ . b X = { a , b , c , d , e , f } ∆ = 2 { a , b , d , e } ∪ 2 { b , c , e , f } facets : { a , b , d , e } , { b , c , e , f } d f dimension : 3 pure ? yes gallery-connected ? no c a e (UMN REU) Project 6 July 24, 2019 3 / 20
Stanley-Reisner Theory b Given a simplicial complex ∆ on X , the Stanley-Reisner ideal of ∆ is the following ideal in k [ X ] : d f � I ∆ = ( x i : x i / ∈ A ) = ( m A : A �∈ ∆) . c a e A ∈ ∆ I ∆ = � c , f � ∩ � a , d � = � ac , af , cd , df � . (UMN REU) Project 6 July 24, 2019 4 / 20
Simplicial Complex in P � n From now on we will be working in the product n = P n 1 × · · · × P n r and we projective space P � use the following notation. b S := k [ x i , j : 1 ≤ i ≤ r , 0 ≤ j ≤ n i ] B := � r i = 1 � x i , 0 , x i , 1 , . . . , x i , n i � is the irrelevant ideal of S . Note that d f V ( B ) = ∅ . n is a A simplicial complex in P � c a simplicial complex on the vertex set n = � r e X � i = 1 { x i , j : 0 ≤ j ≤ n i } . The Stanley-Reisner ring of ∆ is the quotient ring k [∆] := S / I ∆ . (UMN REU) Project 6 July 24, 2019 5 / 20
Free Resolutions & Virtual Resolutions Definition A complex of free S -modules, φ 1 φ 2 φ n F · : 0 ← F 0 ← − F 1 ← − · · · ← − F n , is a free resolution of S / I if � H i ( F · ) = 0 for i ≥ 1 1 � H 0 ( F · ) = F 0 / im φ 1 = S / I 2 It is a virtual resolution of S / I if 1 rad ann H i F · ⊇ B for all i > 0 2 ann H 0 F · : B ∞ = I : B ∞ (UMN REU) Project 6 July 24, 2019 6 / 20
Cohen-Macaulay & Virtually Cohen-Macaulay Definition (Cohen-Macaulay) A simplicial complex ∆ on X is Cohen-Macaulay if there exists a free resolution of k [∆] of length codim I ∆ . Definition (Virtually Cohen-Macaulay) A simplicial complex ∆ on X � n is virtually Cohen-Macaulay if there exists a virtual resolution of k [∆] of length codim I ∆ . (UMN REU) Project 6 July 24, 2019 7 / 20
Resolutions of Ideals with Same Variety Lemma For two ideals I , J ⊂ S with V ( I ) = V ( J ) , then any free resolution r of S / J is a virtual resolution of S / I . Recall that B = � r u be i = 1 � x i , 0 , x i , 1 , . . . , x i , n i � . Let B � � r i = 1 � x i , 0 , x i , 1 , . . . , x i , n i � u i . Since V ( I ∩ B � u ) = V ( I ) ∪ V ( B � u ) = V ( I ) , a free resolution of S / ( I ∩ B � u ) is a virtual resolution of S / I . (UMN REU) Project 6 July 24, 2019 8 / 20
Irrelevant & Relevant Faces Since I ∆ = � A ∈ ∆ ( x i : x i / ∈ A ) , adding a face F to ∆ is equivalent to ∈ F ) . intersecting I ∆ with the ideal I = ( x : x / Definition A face F of a simplicial complex ∆ is relevant if it contains at least one vertex from every color; otherwise it is irrelevant . V ( I ) = ∅ if and only if F is irrelevant. (UMN REU) Project 6 July 24, 2019 9 / 20
Virtually Equivalent Simplical Complexes From the previous observation, we have the following important lemma. Lemma Let ∆ , ∆ ′ be two simplicial complexes in P � n such that ∆ \ ∆ ′ and ∆ ′ \ ∆ contain only irrelevant faces. Then the free resolution of I ∆ ′ is a virtual resolution of I ∆ . We call such ∆ and ∆ ′ virtually equivalent . Figure 1: ∆ , in P 2 × P 2 × P 2 Figure 2: ∆ ′ = ∆ ∪ { Irrelevant Facets } (UMN REU) Project 6 July 24, 2019 10 / 20
The Intersection Method Theorem (Herzog-Takayama-Terai) Let I be a monomial ideal, then if I is Cohen-Macaulay, rad( I ) is also Cohen-Macaulay. Lemma u ∈ { 0 , 1 } r such that I ′ = I ∩ B � u is Cohen-Macaulay, then I If there exists � is virtually Cohen-Macaulay. Then we obtain the following: Proposition Let ∆ be a simplicial complex on the product projective space P � n . If there exists J a monomial ideal with V ( J ) = ∅ such that I ∆ ∩ J is Cohen-Macaulay, then there exists ∆ ′ containing only irrelevant facets such that rad( J ) = I ∆ ′ and I ∆ ∩ I ∆ ′ is Cohen-Macaulay. In particular, this implies ∆ ∪ ∆ ′ is Cohen-Macaulay and ∆ is virtually Cohen-Macaulay. (UMN REU) Project 6 July 24, 2019 11 / 20
The Intersection Method Fact Cohen-Macaulay complexes are pure and gallery-connected. Corollary u ∈ Z r such that I ∆ ∩ B � u is For a simplicial complex ∆ , if there exists � Cohen-Macaulay: u ∈ { 0 , 1 } r , then (supp � Consider supp � u ) i can be 1 only if dim P n i = dim ∆ . ∆ is pure and gallery-connected up to adding irrelevant facets. (UMN REU) Project 6 July 24, 2019 12 / 20
Balanced Complexes Definition Let ∆ be a pure simplicial complex on the product of projective spaces n = P n 1 × · · · × P n r . We say that a facet F ∈ ∆ is balanced if it contains P � exactly one vertex of every component. We say that a simplicial complex is balanced if all of its facets are balanced. Theorem The Stanley-Reisner ring of a pure shellable simplicial complex is Cohen-Macaulay. Strategy : Add all possible irrelevant facets of same dimension and show the new complex is shellable. (UMN REU) Project 6 July 24, 2019 13 / 20
Balanced Complex Definition (Shellability) A shelling of ∆ is an ordered list F 1 , F 2 , . . . , F m of its facets such that for all i = 2 , . . . , m , ( � i − 1 k = 1 F k ) ∩ F i is pure of codimension 1. If a simplicial complex is pure and has a shelling, then it is shellable . Definition Given a vertex set V on the product projective space P � n . Then the irrelevant complex supported on V is defined to be ∆ irr := { σ ∈ 2 V | | σ | = n , | col( σ ) | < n } . Strategy : show that any balanced complex with all the irrelevant facets added in yields a shellable complex. (UMN REU) Project 6 July 24, 2019 14 / 20
Balanced Complex Proposition Let ∆ irr be the irrelevant complex supported on V in the product projective P n . Then there exists a balanced facet R such that ∆ = ∆ irr ∪ { R } is shellable. Observation : Adding more balanced facet still maintains a shelling. (UMN REU) Project 6 July 24, 2019 15 / 20
Balanced Complex Theorem If ∆ is a pure and balanced in the product projective space P � n , then ∆ is virtually Cohen-Macaulay. (UMN REU) Project 6 July 24, 2019 16 / 20
Future work Analogue for Reisner’s criterion for virtual Cohen-Macaulayness? (UMN REU) Project 6 July 24, 2019 17 / 20
Acknowledgements We would like to thank Christine Berkesch, Greg Michel, Vic Reiner, and Jorin Schug for their patient guidance and inspiring ideas throughout this project. (UMN REU) Project 6 July 24, 2019 18 / 20
References Christine Berkesch Zamaere, Daniel Erman, and Gregory G. Smith. “Virtual Resolutions for a Product of Projective Spaces”. In: arXiv e-prints (Mar. 2017). arXiv: 1703.07631 [math.AC] . Anders Björner and ML Wachs. “Shellable nonpure complexes and posets. II”. In: Transactions of the American Mathematical Society 349 (Oct. 1997), pp. 3945–3975. DOI: 10.1090/S0002-9947-97-01838-2 . John A. Eagon and Victor Reiner. “Resolutions of Stanley-Reisner rings and Alexander duality”. In: J. Pure Appl. Algebra 130.3 (1998), pp. 265–275. ISSN: 0022-4049. DOI: 10.1016/S0022-4049(97)00097-2 . URL: https://doi.org/10.1016/S0022-4049(97)00097-2 . Christopher A. Francisco, Jeffrey Mermin, and Jay Schweig. “A survey of Stanley-Reisner theory”. In: Connections between algebra, combinatorics, and geometry . Vol. 76. Springer Proc. Math. Stat. Springer, New York, 2014, pp. 209–234. DOI: 10.1007/978-1-4939-0626-0_5 . URL: https://doi.org/10.1007/978-1-4939-0626-0_5 . Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in algebraic geometry . Available at http://www.math.uiuc.edu/Macaulay2/ . Ezra Miller and Bernd Sturmfels. Combinatorial Commutative Algebra . Springer, 2005. (UMN REU) Project 6 July 24, 2019 19 / 20
Questions? Figure 3: confused mudkip. (UMN REU) Project 6 July 24, 2019 20 / 20
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