Jason McCullough: Research

Research Interests:


Commutative algebra, comptutational algebra, algebraic geometry, coding theory.

My main area of research is commutative algebra. I am especially interested in graded free resolutions and homological questions that relate geometry, topology and combinatorics. Fix a field k. Graded free resolutions are useful objects to study as virtually all invariants of a projective variety (with fixed embedding) in projective space over k can be derived from them. In algebraic language, one begins with a homogeneous ideal I in a standard graded polynomial ring S = k[x_1...x_n] (or the quotient S/I), maps a graded free S-module onto a minimal set of generators; then one maps a graded free S-module onto a minimal set of generators of the kernel of the previous map, and so on. Hilbert's Syzygy Theorem guarantees that this process stops after at most n steps. The minimality of the construction implies that the resulting resolution is unique up to isomorphism of complexes. Many coarser invariants, like Castelnuovo-Mumford regularity and projective dimension, are of great interest.

Papers


  1. 1. "On the Maximal Graded Shifts of Ideals and Modules." submitted. [pdf]

  2. The inspiration for this paper was the paper "The regularity of Tor and graded Betti numbers" by Eisenbud-Huneke-Ulrich in which the authors prove several bounds on the maximal graded shifts of modules of dimension at most one with some assumption about their annihilator. For example, they show that a the degrees of a homogeneous complete intersection contained in the annihilator govern the graded shifts at the end of the resolution. In this paper we extend these to the case when there is no assumption on the dimension of the module at the expense of slightly weaker bounds.


    2. "Regularity of Prime Ideals."(with Giulio Caviglia, Irena Peeva, Matteo Varbaro) submitted. [pdf]
    In this paper we answer a number of natural questions arising from the counterexamples to the Eisenbud-Goto Conjecture (see below), constructed by Peeva and myself; namely, we show (1) There are counterexamples to the Eisenbud-Goto conjecture coming from Rees algebras, (2) there are counterexamples that do not rely on the Mayr-Meyer ideals, (3) there is a bound on the regularity (and projective dimension and graded Betti numbers, etc.) of nondegenerate prime ideals purely in terms of their multiplicity, and (4) there exist nondegenerate prime ideals with bounded generating degree and arbitrarily large regularity.


  3. 3. "Counterexamples to the Eisenbud-Goto regularity conjecture." (with Irena Peeva) to appear in Journal of the American Mathematical Society. [pdf]

  4. There is a general doubly exponential upper bound on the Castelnuovo-Mumford regularity of homogeneous ideals in a standard graded polynomial ring. This bound is nearly sharp since the Mayr-Meyer construction leads to examples of families of ideals attaining doubly exponential regularity. It was expected that much better bounds hold for the defining ideals of geometrically nice projective varieties over an algebraically closed field. The longstanding Regularity Conjecture, by Eisenbud-Goto (1984), predicts the following elegant linear bound in terms of the degree:
                reg(P)  ≤   deg(S/P) -  codim (P) + 1
    for every homogeneous non-degenerate prime ideal P in a standard graded polynomial ring S over an algebraically closed field k.

    The Regularity Conjecture holds if S/P is Cohen-Macaulay by a result of Eisenbud-Goto (1984). It is proved for curves by Gruson-Lazarsfeld-Peskine (1983), completing classical work of Castelnuovo. It is also proved for smooth surfaces by Lazarsfeld (1987) and Pinkham (1986), and for most smooth 3-folds by Ran (1990). In the smooth case, Kwak (1998) gave bounds for regularity in dimensions 3 and 4 that are only slightly worse than the optimal ones in the conjecture. Many other special cases and related results have been proved as well.

    Irena Peeva and I constructed counterexamples to the Regularity Conjecture. Our main theorem is much stronger and shows that the regularity of non-degenerate homogeneous prime ideals is not bounded by any polynomial function of the degree. We provide a family of prime ideals Pr, depending on a parameter r, whose degree is singly exponential in r and whose regularity is doubly exponential in r. For this purpose, we introduce an approach which, starting from a homogeneous ideal I, produces a prime ideal P whose projective dimension, regularity, degree, dimension, depth, and codimension are expressed in terms of numerical invariants of I. The construction involves two new concepts: (1) Rees-like algebras which, unlike the standard Rees algebras, have well-behaved defining equations and free resolutions, and (2) a step-by-step homogenization technique which, unlike classical homogenization, preserves graded Betti numbers.



  5. 4. "A finite classification of (x,y)-primary ideals of low multiplicity." (with Paolo Mantero) to appear in Collectanea Mathematica. [pdf]

  6. Let I be a homogeneous ideal which is primary to a linear prime (x,y) in a standard graded polynomial ring S. Engheta previously showed that there are essentially two forms that I can take when the Hilbert multiplicity of S/I is 2. In previous work with Huneke and Seceleanu, we showed that for multiplicity 3 and higher there are infinitely many such ideals and their projective dimension can be chosen to be arbitrarily high. However, in this paper we give a finite classification in multiplicities 3 and 4 of the linear, quadric and cubic generators of all such ideals, showing that there are essentially 8 cases in multiplicity 3 and 22 cases in multiplicity 4. We apply our classification theorems to improve bounds on the projective dimension of 3 cubics - a special case of Stillman's Question. [Supplemental M2 File] [Output from Supplemental M2 File]


  7. 5. "A tight bound on the projective dimension of four quadrics." (with Craig Huneke, Paolo Mantero and Alexandra Seceleanu) to appear in Journal of Pure and Applied Algebra. [arXiv]

  8. In this paper, we prove an optimal upper bound on the projective dimension of any homogeneous ideal generated by 4 quadrics in a polynomial ring over a field. In this specific case, this result vastly improves the more general upper bound for ideals generated by any number of quadrics given by Ananyan-Hochster. [Supplemental M2 File] [Output from Supplemental M2 File]


  9. 6. "Three themes on syzygies." (with Gunnar Fløystad and Irena Peeva) Bulletin of the American Mathematical Society 53 no. 3, July 2016, 415-435 [pdf] [link]

  10. This survey paper covers three recent developments in commutative algebra: resolutions over complete intersections, Stillman's Question and Boij-Soderberg theory. It is intended for a general audience.


  11. 7. "Multiple structures with arbitrarily large projective dimension on linear subspaces." (with Craig Huneke, Paolo Mantero and Alexandra Seceleanu) Journal of Algebra 447 (2016), 183–205. [arXiv] [link]

  12. In Engheta's work showing that the projective dimension of any ideal generated by 3 cubic forms is bounded, he gave a homological structure theorem for unmixed height-2, multiplicity-2 ideals. The notable case was the primary one, in which he showed that the only two possibilities were (x,y^2) and (x^2,xy,y^2,ax+by), where x,y are independent linear forms and ht(x,y,a,b) = 4 in the latter case. In this paper we show that this kind of result cannot by extended by finding examples of primary ideals with arbitrarily large projective dimension in every larger height and multiplicity case.


  13. 8. "A multiplicity bound for graded rings and a criterion for the Cohen-Macaulay property." (with Craig Huneke, Paolo Mantero and Alexandra Seceleanu) Proceedings of the American Mathematical Society 143 no. 6 (2015), 2365-2377. [arXiv] [link]

  14. This work extends previous work of Engheta who gave an upper bound on the Hilbert-Samuel Multiplicity of a homogeneous almost complete intersection. Our result gives an upper bound on the multiplicity of any ideal of the form I + (f), where I is a Cohen-Macaulay ideal and f is not in I. Moreover, we show that ideals that meet this upper bound must be Cohen-Macaulay themselves. Dual results for almost Gorenstein ideals are also given as well as several examples of ideals that meet this upper bound.


  15. 9. "Infinite graded free resolutions." (with Irena Peeva) Commutative Algebra and Noncommutative Algebraic Geometry Volume 1: Expository Articles (Eisenbud, Iyengar, Singh, Stafford, Van den Bergh eds.), Math. Sci. Res. Inst. Publ. No. 67, Cambridge University Press, New York (2015), 215–258. [pdf]
    (The version linked here is updated from the print version and includes references to recent developments in the theory of Golod rings.)

  16. This survey paper is based on Irena Peeva's series of lectures at the Joint Introductry workshop in Commutative and Cluster Algebras in Fall 2013 at MSRI. We briefly summarize the construction of graded free resolutions. The paper covers topics from Koszul rings, Golod rings, Complete intersections and matrix factorizations, slope, rate and regularity.


  17. 10. "Hypergraphs and regularity of square-free monomial ideals." (with Kuei-Nuan Lin) International Journal of Algebra and Computation 23 no. 7 (2013), 1573–1590. [arXiv] [link]

  18. We define a combinatorial structure which we call a labeled hypergraph associated to any square-free monomial ideal. We then prove a number of special case exact formulas and general upper bounds on the regularity of I in terms of the combinatorial data in its labeled hypergraph. Our hypergraph definition is based on one of Kimura-Terai-Yoshida who studied the arithmetical rank of certain monomial ideals.


  19. 11. "Projective dimension of codimension two algebras presented by quadrics." (with Craig Huneke, Paolo Mantero and Alexandra Seceleanu) Journal of Algebra 393 (2013), 170-186. [arXiv] [link]

  20. Stillman's Question asks for a bound on the projective dimension of a homogeneous ideal in a polynomial ring purely in terms of the degrees of the minimal generators of the ideal. We give an optimal affirmative answer to this question in the case of height 2 ideals generated by quadrics.


  21. 12. "Bounding projective dimension." (with Alexandra Seceleanu ) Commutative Algebra. Ex- pository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday (I. Peeva, Ed.), Springer-Verlag, London (2013), 551-576. [pdf]

  22. This is a survey paper written in 2013 on the status of Stillman's Question. Stillman asked whether the projective dimension of a homogeneous ideal can be bounded in terms of the degrees of the minimal generators only. Our paper summarizes Caviglia's proof that Stillman's question is equivalent to the analogous question for regularity. We also discuss Ananyan-Hochster's positive answer for Stillman's Question in the case of ideals generated by quadrics along with the construction of ideals of large projective dimension given below in Beder et. al. Much research has occured since this was written. Many of the questions we raised have been answered in the above papers.


  23. 13. "A polynomial bound on the regularity of an ideal in terms of half of the syzygies." Math. Res. Ltrs. 19 (2012), no 3, 555-565.[arXiv] [link]

  24. A famous example of Mayr-Meyer and a modified version of Bayer-Stillman show that the regularity of an ideal may grow doubly exponentially with respect to the degrees of the minimal generators. In his thesis, Engheta asked whether a polynomial upper bound on regularity would be possible if not just the degrees of the generators but the degrees of some of the first several syzygies were used as well. Here I showed that the answer to Engheta's question is yes if the degrees of the first half of the syzygies are used. I also prove a related subadditivity upper bound. Both bounds are proved via the Boij-Soederberg decomposition of the Betti table of the free resolution of R/I.


  25. 14. "Ideals with larger projective dimension and regularity." (with Jesse Beder, Luis Nunez-Betancourt, Alexandra Seceleanu, Bart Snapp and Branden Stone) J. Symbolic Comput. 46 (2011), no. 10, 1105-1113. [arXiv] [link]

  26. Generalizing the construction below of ideals with large projective dimension, we construct a family of ideals associated to tuples of integers that have large projective dimension and regularity relative to the degrees of the generators. We notably give an example of a family of 3-generated ideals whose projective dimension grows exponentially with respect to the degrees of the generators. The generators can be computed easily with the help of the Macaulay2 package BigIdeal.m2. (See below.)


  27. 15. "A family of ideals with few generators in low degree and large projective dimension." Proceedings of the AMS. Volume 139, No. 6. pp. 2017-2023. [arXiv] [link]

  28. This is my first paper regarding Stillman's Question in which I produced a family ideals with large projective dimension relative to the degrees of the generators. My example showed that a question of Z. Yi relating to the vanishing of local cohomology modules over a polynomial ring was false. It also showed bound that any positive answer to Stillman's Question must necessarily be very large.


  29. 16. "A note on the strong direct summand conjecture." Proceedings of the AMS. Volume 127, No. 9. 2009. pp. 2857-2864. [pdf] [link]

  30. Hochster's Direct Summand Conjecture (DSC) and the related Homological Conjectures have many astonishing consequences in commutative algebra and algebraic geometry. While most are known to be true, most are open questions in the mixed characteristic case. (i.e. where (R,m) is a local ring, char(R) = 0 and char(R/m) = p) N. Ranganathan gave a new interpretation of one of the homological conjectures, the Vanishing of Maps of Tor (VMTC), in language similar to that of the Direct Summand Conjecture. Her Strong Direct Summand Conjecture (SDSC) is equivalent to the VMTC and implies the DSC. In this paper, which contains some of the results from my thesis, I prove several special cases of the SDSC.


  31. 17. "SRPT optimally uses faster machines to minimize flow time." ( with Eric Torng ) ACM Transactions on Algorithms. Volume 5, No. 1. November 2008. pp. 1-25. [pdf]

  32. This is research I worked on as an undergraduate with Prof. Eric Torng at MSU. The problem comes from scheduling theory in computer science in which one imagines an idealized setting where m parallel processors must collectively schedule and process jobs of known length and to minimize flow time (or total waiting time). The problem is known to be NP-hard, but we show that given machines running the SRPT (Shortest Remaining Processing Time) algorithm with speed 2-1/m faster processors than an optimal schedule to which we are compared, the flow time is 2-1/m times smaller. This result is sharp in the sense that for any speed-up less than 2-1/m, there are schedules for which optimal speed-1 machines will outperform the faster SRPT machines.


  33. 18. "A Generalized Floor Bound on the Minimum Distance of Geometric Goppa Codes." (with Benjamin Lundell) Journal of Pure and Applied Algebra. Volume 207, Issue 1. September 2006. pp. 155-164. [link] [pdf]

  34. This work grew out of a summer reading course Ben Lundell and I took with Prof. Iwan Duursma. We showed how to generalize a bound on the minimum distance of certain algebraic geometry codes called Goppa codes in a way that improved the known bounds for certain choices of divisors on curves over finite fields. We include specific comparisons of our bound to that of Matthews and Kirfel-Pelikaan for Suzuki and Hermitian curves. Improvements to our bound have since been found by Gunneri-Stichtenoth-Taskin, Duursma-Kirov-Park, and others.


Slides/Video from Conference/Invited Talks


  • "On the Maximal Graded Shifts of Modules over a Polynomial Ring" AMS Sectional Meeting. Special Session on Commutative Algebra. Hunter College. New York City, NY. 2017. [Slides]


  • "A Tight Bound on the Projective Dimension of Four Quadrics" AMS Sectional Meeting. Special Session on Combinatorial and Computational Commutative Algebra. University of Georgia. Athens, GA. 2016. [Slides]

  • "Graded Free Resolutions of Ideals" Department Colloquium. University of Nebraska. Lincoln, NE. 2015. [Slides]

  • "Graded Free Resolutions of Ideals: A UCR Story" Department Colloquium. UC Riverside. Riverside, CA. 2014. [Slides]

  • "Multiple Structures With Arbitrarily Large Projective Dimension Supported on Linear Spaces." AMS Sectional Meeting. Special Session on Recent Advances in Commutative Algebra. University of Louisville. Louisville, KY. 2013. [Slides]

  • "Bounds on the Projective Dimension and Regularity of Ideals." MSRI workshop on "Representation Theory, Homological Algebra, and Free Resolutions" Berkeley, CA. 2013. [Video]

  • "Hypergraphs and Regularity of Square-free Monomial Ideals." Joint AMS-MAA Math Meetings. Special Session on Commutative Algebra and Algebraic Geometry. San Diego, CA. 2013. [Slides]

  • "Syzygy Bounds on the Regularity of Ideals." AMS Sectional Meeting. Special Session on Commutative Algebra. University of Arizona. Tuscon, AZ. 2012. [Slides]

  • "MSRI Tutorial Session on Infinite Free Resolutions." Joint Workshop on Commutative Algebra and Cluster Algebras. Berkeley, CA. 2012 [M2 File of Examples]

  • "Ideals with Large(r) Projective Dimension and Stillman's Question." AMS Sectional Meeting, Special on Commutative Algebra. University of Utah. Salt Lake City, UT. 2011 [Slides]

  • "Ideals with Large(r) Projective Dimension and Regularity." Computational and Commutative Algebra Seminar. Cornell Univeristy. Ithaca, NY. 2011. [Slides] [M2 File]

  • "Lifting Splittings and the Strong Direct Summand Conjecture." Joint Math Meetings, Special Session on Local Commutative Algebra. New Orleans, LA, 2011. [Slides]

  • "On the Strong Direct Summand Conjecture." AMS Spring Central Sectional Meeting, Special Session on Local and Homological Methods in Commutative Algebra, Urbana, IL, 2009. [Slides]

  • (with Ben Lundell) "A Generalized Floor Bound on the Minimum Distance of Geometric Goppa Codes." Joint Mathematics Meetings. Special Session on Algebraic-Geometry Codes. Atlanta, GA, 2005.

Macaulay 2 Packages



  • BigIdeal.m2 - This package generates the ideals defined in "Ideals with Larger Projective Dimension and Regularity" by Beder, McCullough, Nunez, Seceleanu, Snapp and Stone. These ideals have very large projective dimension and regularity relative to the degree and number of generators.

  • PowerSeries.m2 - This package allows for computation of and manipulation of power series in which more series terms may always be computed later on. Support for rational functions and generating functions is built in.