Gwyneth R. Whieldon. Cornell University Department of Mathematics

Transcription

1 Gwyneth R. Whieldon Cornell University Department of Mathematics Research Statement My research is at the crossroads of commutative algebra, combinatorics and algebraic geometry, focused on the study of rings and modules via their underlying combinatorial structure. The core mission is to understand the geometry of varieties and schemes via properties of their free resolutions, exploration of the generating sets and decompositions of their defining ideals, and analyzing the topological properties of associated spaces. Main Themes of Research Program. (1) To study the algebraic invariants of squarefree monomial ideals in an attempt to better understand extremal behavior of ideals and modules over polynomial rings. (2) To answer questions on the structure of modules via decompositions of their free resolutions, either through initial ideals or in the style of Boij-Soederberg theory. (3) To apply the tools of topological combinatorics to algebra and geometry. Understanding the combinatorial structure of the free resolutions of modules relies on numerous tools from topology, combinatorics, and algebra, and permits a rich interplay between the geometry of these rings and the topology of combinatorial objects. Specific Areas of Research. Within these broader themes of algebraic geometry and computational commutative algebra, my specific problems and results are in the areas of: (1) Studying Castelnuovo-Mumford regularity and projective dimension of modules M and providing both upper and lower bounds on the growth of these invariants in terms of combinatorial data of the ring. (2) Studying the asymptotic behavior of powers of monomial ideals, via polarization, the topology and decompositions of their Stanley-Reisner complexes, and the bigraded structure of their Rees algebras. (3) Classifying Betti diagrams of special classes of monomial ideals in terms of their structure, with a goal of algebraically approaching combinatorial questions such as possible f-vectors or h-vectors of spheres and reconstructibility of graphs. (4) Translating questions about topology of simplicial complexes into algebraic calculations via free resolutions. Most of my approaches to these problems are flavored with topological combinatorics. Discrete Morse theory plays a large role, as do combinatorial expressions of the Betti numbers via Hochster s formula and the lcm-lattice. Investigating the Stanley-Reisner complexes of squarefree monomial ideals permits broad algebraic questions to be turned into topological computations such as homology, k-connectedness, or colorability. More generally, algebraic concepts such as Alexander duality, linkage of ideals, or the primary decomposition and associated primes of an ideal can often be translated into combinatorial conditions. This connection permits rich results on both the topological and algebraic sides. 1. Background (Commutative Algebra and Free Resolutions) A central object in commutative algebra is the free resolution of module M, which can be thought of as a chain complex of modules measuring precisely how far a module is from free. At each stage of a resolution, we approximate the module by taking the module of

2 2 relations on the generators of M, then the module of relations on those relations, etc. The i th module in this list is called the i th syzygy module of M. Free Resolutions of Graded Modules. More formally, we say that the free resolution of a graded module M over a ring R is an exact sequence of free modules of the form: F : R( j) β ϕ i,j i R( j) β ϕ 1,j 1 R( j) β ϕ 0,j 0 M, where ϕ i : R( j) β i,j R( j) β i 1,j are graded degree zero maps with entries in R. Such a resolution is minimal precisely when the entries in the ϕ are in R >0, the maximal ideal of R. The β i,j = β i,j (M), the rank of the graded component in degree j at the i th homological stage of the resolution, are an algebraic invariant of M. Many invariants of modules, including Tor, Ext, Castelnuovo-Mumford regularity, depth, dimension, Hilbert function, etc. can be obtained via these free resolutions. Computer algebra systems such as Macaulay 2 (M2) permit explicit calculation of these resolutions (and other algebraic invariants.) This makes their use in investigating the geometry of these rings invaluable, and a strong component of my research agenda includes continued exploration of geometry of modules via such computer algebra systems. The packages I have written for Macaulay 2 use the explicit computation of Betti numbers via free resolutions to investigate extremal properties of topological and geometric structures. 2. Bounds on Growth of Algebraic Invariants: Known and Conjectured Free resolutions have been studied extensively over the last century, but outside of a few narrow cases, sharp bounds on the growth in degree or length of such in terms of general data of the module remain elusive. The Hilbert Syzygy Theorem provides an upper bound on the length of a free resolution of M for modules over polynomial rings in n variables, but upper bounds on the growth independent of the global dimension of our ring R generally require increasingly specific classes of modules Extremal Examples. In the course of answering this question, I exhibited modules with the highest currently known projective dimension (length of minimal resolution) among ideals generated in degree d with k generators, independent of the number of generators of the ring. Theorem 2.1. [Whi11] Let k be a field, d, k N with k = l + m some fixed partition of k. For N = ( ) m+d 2 d 1, in the polynomial ring with m + l N variables, we have the ideal R = k[x 1, x 2,..., x m, y i,j : 1 i l, 1 j N], N N I d,k,m = (x d 1, x d 2,..., x d m, y 1,j m j,..., y l,j m j ), j=1 with m 1,..., m N all monomials of degree d in the x-variables, has projective dimension equal to the number of variables in the ring, or pd(i d,k,m ) = m + l (m+d 2 ). j=1 d 1

3 The rapid increase in projective dimension of these ideals relative to the degree and number of the generators hints at the computational difficulty inherent even in resolutions of standardly graded ideals in a polynomial ring Combinatorics of Free Resolutions Stepping from Algebraic Geometry to Combinatorial Algebra. Classifying free resolutions of modules over polynomial rings in general can be quite difficult. Via initial ideals 1 and polarization 2, many algebraic invariants of general resolutions can be reduced to problems in resolutions of squarefree monomial ideals. Pushing these questions into ideals such as these allows the use of tools from simplicial topology. Stanley-Reisner Ideals and Hypergraph Ideals. The set of squarefree ideals in a polynomial ring in n variables can be put in 1:1 correspondence with simplicial complexes on n vertices in two standard ways - via the Stanley-Reisner ideal or the facet (or hypergraph) ideal of a simplicial complex. The Stanley-Reisner ideal is generated by squarefree monomials corresponding to the nonfaces of, while the facet ideal is generated by monomials corresponding to the facets of. Example 3.1. Let be the following simplicial complex. b a c d Then in the ring R = k[a, b, c, d] we have the Stanley-Reisner ideal SR = (ac, bcd) and the facet ideal I = (abd, bc, cd). We let k[ ] := R/SR for ease of notation. The combinatorics of simplicial complexes are connected to algebraic invariants of the Stanley-Reisner ring k[ ] through Hochster s formula, which allows us to rewrite the Betti numbers β i,j (k[ ]) in terms of the homologies of subcomplexes of. Theorem 3.2. (Hochster s Formula [Hoc77]) Let k[ ] be a Stanley-Reisner ring on variables X = {x 1,..., x n }. Then if m is a squarefree monomial with support W = {x i1,..., x ij } X with deg(m) = j, we have β i,m (k[ ]) = dim H j i 1 ( W, k), where W is the induced subcomplex of on vertices in W. Viewing a squarefree monomial ideal I as either a facet ideal (as in [Far02], or as surveyed in [HVT08]) or a Stanley-Reisner ideal (as in Chapter 5 of [BH93]) will showcase different combinatorial aspects of the complex. 1 The initial ideal in(i) is the monomial ideal generated by leading terms of all polynomials in an ideal I. 2 The polarization of a monomial ideal is the monomial ideal obtained by replacing monomial generators with an equivalent squarefree generating set in a larger number of variables.

4 4 4. Edge Ideals and Facet Ideals My personal research has been focused on applications of Hochster s formula to computations of Betti numbers on facet ideals of degree 2, called edge ideals in recognition of their corresponding simplicial complexes being graphs. Properties of edge ideals have been explored by Van Tuyl, Tai Ha, Francisco, Villareal, Faridi and others, leading to surprisingly fertile combinatorial structure even in this seemingly simple case. Numerous graph theoretic problems can be turned into questions in algebra, and vice versa. Proposition 4.1. [Vil90] Let G be a simple graph on vertex set [n] = {1, 2,..., n} with edge set E, and let I G = (x i x j : {i, j} E) R = k[x 1,..., x n ] be the edge ideal of G. Then the Stanley-Reisner complex of I G, denoted (I G ), is given by (I G ) = Ĝc, the clique closure [i.e. adding a face σ to whenever σ ] of the complement graph of G in [n]. Linear Quotients of Powers of Edge Ideals. One common graph that arises in combinatorial investigations is the graph of the anticycle, meaning the complement graph of a cycle of length n. In joint work with A. Hoefel, we exhibited a linear quotient ordering for the second power of this ideal, giving us that the Stanley-Reisner complex of the Alexander dual of the polarization of IG 2 is shellable. Theorem 4.2. [Whi11] Let G be the complement graph of the n-cycle. Then I 2 G has a linear resolution with linear quotients. This meshes well with my general research program s goal: To classify invariants of edge ideals (and other squarefree ideals) and the powers of these ideals in terms of combinatorial data. While having linear quotients guarantees a linear resolution of these particular ideals IG 2, a better understanding of the shapes of the Betti diagrams of edge ideals 4.1. Syzygies of Edge Ideals of Graphs with C 4 -free Complement. As an additional result in this direction, I was able to classify the first Betti number that is of degree two higher than the linear strand in terms of the location of the first nonlinear Betti number. Theorem 4.3. [Whi10b] Let G be a graph with the complement graph G c having no induced 4-cycles [a graph with C 4 -free complement.] If the first nonlinear Betti number in the resolution of R/I G occurs at stage i of the resolution, then the earliest a syzygy of higher degree can occur is at stage 2i. The complement graph of the 1-skeleton of the icosahedron is an example of a stronger conjectured bound: Namely, in graphs with the first nonlinear Betti number in the 3 rd stage of the resolution, the soonest the second nonlinear Betti number can occur is at the 9 th stage of the resolution. G total: : 1 1: : : 1

5 Theorem 4.4. [Whi11] Given an edge ideal I G with a complement graph G c with no induced cycles of length 4 and β 2,5 (I G ) 0, all Betti number in the 4-strand are zero until β 3,12 (I G ). The proof of these relies on an examination of the topology of the clique closures of complements graphs a fairly standard theme in edge ideals. Future research plans include further characterization of graph structures which force bounds on projective dimension and regularity (a generalization of maximum degree of generators of our ideal I measuring the complexity of the resolution.) For ideals I G with regularity higher than 4, the degree sequences can have more interesting patterns - but these can be characterized via topological joins of Stanley-Reisner complexes of other ideals, and the Betti numbers of some classes of ideals can be completely characterized [Whi10a]. 5. Asymptotic Behavior of Powers of Equigenerated Monomial Ideals My research also addresses the behavior of higher powers of edge ideals and other facet ideals. The following result was via an examination of the bigraded and multigraded structures of the Rees algebras of equigenerated monomial ideals, refining work in [Cha97]. Stabilization of Betti Diagrams of Monomial Ideals I. Theorem 5.1. Let I = (m 1,..., m k ) R be an equigenerated monomial ideal of degree d, i.e. deg(m i ) = d. Then there exists an N such that for all n > N shape of the Betti diagrams of I n stabilize, e.g. for all i, j N. β,i,dn+j(r/i) 0 β,i,dn+j(r/i) 0 This N marks the power of I that needs to be taken to fix the shape of the Betti table of the resolution of R/I at all subsequent powers, the locations of nonzero entries in the Betti table can be obtained by shifting down by d places all of the nonzero entries in the Betti table of the previous power. This result generalized to non-equigenerated ideals easily, but the translation of the Betti tables are less clear. This stabilization happens at or before the x-regularity of the Rees algebra of I. Precise calculation of this stabilization number in terms of data of I remains elusive, as the bound given by the x-regularity is not sharp. 6. Undergraduate Research Program Connections Between Graph Theory and Algebra. As my research has a deep focus on graph theory with a wide application to algebra many of the problems are amenable to adaption to undergraduate research. Creating a dictionary between graph invariants of G and algebraic properties of I G is a rich and accessible field. Previous REU Experience: Combinatorics of Free Probability. A future direction for research with undergraduates is a continuation of a previous REU I helped run, where we helped students connect the combinatorics of poset structures arising from noncrossing pairings on bit-strings of 0 s and 1 s to asymptotics of random matrices and free probability. This approach was an extension of work in [NS06]. The advantage of approaching these problems combinatorially rather than through heavy analysis is twofold - it permits an easy extension of the free central limit theorem to the multidimensional case, and it gives rise to an interesting interplay between the topology of the poset of noncrossing pairings [and special subposets of it corresponding to particular strings] and moments of free variables 5

6 6 and free cumulants. The mixed flavor of this project - a melange of combinatorics, probability, topology, and computer algebra system exploration - was rewarding and exciting for the students and I would be very pleased to return to this project. 7. Future Research Program While many of the previously mentioned work is still ongoing and productive, some additional avenues of research I would be interested in exploring include: Classifying shapes of Betti diagrams of edge ideals (or general squarefree monomial ideals) and their powers in terms of combinatorial data of the graphs or hypergraphs. I would like as well to have better combinatorial data determining the power I N required to stabilize the shape of the Betti diagrams of I n. Using topological or algebraic constructions (nerves and inverse nerves of simplicial complexes, discrete Morse theory, Rees algebras, whiskered complexes, etc.) to extend the computational power of these resolutions. Some additional current results of mine connect structures in graphs (and other simplicial complexes ) to the Ext modules of the Stanley Reisner ideals of the nerve of, which potentially provide a direct combinatorial interpretation of Ext modules for squarefree monomial ideals. My goal is to extend these results further than the few current cases known [Whi10b]. To produce large classes of ideals maximizing algebraic invariants with restraints. For example, classes achieving maximal or minimal Betti numbers for a fixed Hilbert function, Gotzmann growth in the general monomial ideal case, maximum regularity or projective dimension with certain forbidden minors of G, etc. To examine decompositions of resolutions of Stanley-Reisner ideals into their pure parts via Boij-Soederberg theory. Some current work of mine has been focused on examining the decompositions of 2-stranded resolutions (quotients of ideals with regularity 3) into their pure parts, in an attempt to give combinatorial significance to coefficients appearing in the decomposition. While the modules resolved to obtain the pure resolutions aren t (necessarily) obtained as quotients of polynomial rings, other recent work [BEKS10] has indicated that there is hope at a better understanding of these module structures. Most generally, to continue to extend the reach of topological combinatorics into calculation or specification of algebraic invariants of resolutions. References [BEKS10] Christine Berkesch, Daniel Erman, Manoj Kummini, and Steven Sam. Poset structures in boijsderberg theory. unpublished, [BH93] Winfried Bruns and Jürgen Herzog. Cohen-Macaulay rings, volume 39 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, [Cha97] Karen A. Chandler. Regularity of the powers of an ideal. Comm. Algebra, 25(12): , [DE09] Anton Dochtermann and Alexander Engström. Algebraic properties of edge ideals via combinatorial topology. Electron. J. Combin., 16(2, Special volume in honor of Anders Bjorner):Research Paper 2, 24, [Far02] Sara Faridi. The facet ideal of a simplicial complex. Manuscripta Math., 109(2): , 2002.

7 [Hoc77] [HVT08] Melvin Hochster. Cohen-Macaulay rings, combinatorics, and simplicial complexes. In Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975), pages Lecture Notes in Pure and Appl. Math., Vol. 26. Dekker, New York, Huy Tài Hà and Adam Van Tuyl. Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers. J. Algebraic Combin., 27(2): , [NS06] Alexandru Nica and Roland Speicher. Lectures on the combinatorics of free probability, volume 335 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, [Vil90] Rafael H. Villarreal. Cohen-Macaulay graphs. Manuscripta Math., 66(3): , [Whi10a] Gwyneth R. Whieldon. Jump sequences of edge ideals. Arxiv, [Whi10b] Gwyneth R. Whieldon. Nerves of simplicial complexes. In preparation, [Whi11] Gwyn Whieldon. Bounding invariants of Stanley-Reisner ideals. Ph.D. Thesis,