Algebra & Number Theory

Towards Boij–Söderberg theory for Grassmannians: the case of square matrices

Nicolas Ford, Jake Levinson, and Steven V Sam

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We characterize the cone of GL-equivariant Betti tables of Cohen–Macaulay modules of codimension 1, up to rational multiple, over the coordinate ring of square matrices. This result serves as the base case for “Boij–Söderberg theory for Grassmannians,” with the goal of characterizing the cones of GLk-equivariant Betti tables of modules over the coordinate ring of k×n matrices, and, dually, cohomology tables of vector bundles on the Grassmannian Gr(k,n). The proof uses Hall’s theorem on perfect matchings in bipartite graphs to compute the extremal rays of the cone, and constructs the corresponding equivariant free resolutions by applying Weyman’s geometric technique to certain graded pure complexes of Eisenbud–Fløystad–Weyman.

Article information

Algebra Number Theory, Volume 12, Number 2 (2018), 285-303.

Received: 21 August 2016
Revised: 4 December 2017
Accepted: 3 January 2018
First available in Project Euclid: 23 May 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13D02: Syzygies, resolutions, complexes
Secondary: 05E99: None of the above, but in this section

Boij–Söderberg theory Betti table cohomology table Schur functors Grassmannian free resolutions equivariant K-theory


Ford, Nicolas; Levinson, Jake; Sam, Steven V. Towards Boij–Söderberg theory for Grassmannians: the case of square matrices. Algebra Number Theory 12 (2018), no. 2, 285--303. doi:10.2140/ant.2018.12.285.

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  • C. Berkesch, J. Burke, D. Erman, and C. Gibbons, “The cone of Betti diagrams over a hypersurface ring of low embedding dimension”, J. Pure Appl. Algebra 216:10 (2012), 2256–2268.
  • M. Boij and J. Söderberg, “Graded Betti numbers of Cohen–Macaulay modules and the multiplicity conjecture”, J. Lond. Math. Soc. $(2)$ 78:1 (2008), 85–106.
  • M. Boij and J. Söderberg, “Betti numbers of graded modules and the multiplicity conjecture in the non-Cohen–Macaulay case”, Algebra Number Theory 6:3 (2012), 437–454.
  • D. Eisenbud and D. Erman, “Categorified duality in Boij–Söderberg theory and invariants of free complexes”, J. Eur. Math. Soc. 19:9 (2017), 2657–2695.
  • D. Eisenbud and F.-O. Schreyer, “Betti numbers of graded modules and cohomology of vector bundles”, J. Amer. Math. Soc. 22:3 (2009), 859–888.
  • D. Eisenbud, G. Fløystad, and J. Weyman, “The existence of equivariant pure free resolutions”, Ann. Inst. Fourier $($Grenoble$)$ 61:3 (2011), 905–926.
  • D. Erman and S. V. Sam, “Questions about Boij–Söderberg theory”, pp. 285–304 in Surveys on recent developments in algebraic geometry (Salt Lake City, 2015), edited by I. Coskun et al., Proc. Sympos. Pure Math. 95, Amer. Math. Soc., Providence, RI, 2017.
  • G. Fløystad, “Boij–Söderberg theory: introduction and survey”, pp. 1–54 in Progress in commutative algebra, I, edited by C. Francisco et al., de Gruyter, Berlin, 2012.
  • G. Fløystad, J. McCullough, and I. Peeva, “Three themes of syzygies”, Bull. Amer. Math. Soc. $($N.S.$)$ 53:3 (2016), 415–435.
  • N. Ford and J. Levinson, “Foundations of Boij–Söderberg theory for Grassmannians”, preprint, 2016.
  • W. Fulton, Young tableaux: with applications to representation theory and geometry, Lond. Math. Soc. Student Texts 35, Cambridge Univ. Press, 1996.
  • I. Gheorghita and S. V. Sam, “The cone of Betti tables over three non-collinear points in the plane”, J. Commut. Algebra 8:4 (2016), 537–548.
  • D. R. Grayson and M. E. Stillman, “Macaulay2, a software system for research in algebraic geometry”,
  • M. Kummini and S. V. Sam, “The cone of Betti tables over a rational normal curve”, pp. 251–264 in Commutative algebra and noncommutative algebraic geometry, II (Berkeley, CA 2012/2013), edited by D. Eisenbud et al., Math. Sci. Res. Inst. Publ. 68, Cambridge Univ. Press, 2015.
  • L. Lovász and M. D. Plummer, Matching theory, North-Holland Math. Studies 121, North-Holland, Amsterdam, 1986.
  • The Sage Developers, SageMath, the Sage Mathematics Software System, 2016,
  • S. V. Sam and A. Snowden, “Introduction to twisted commutative algebras”, preprint, 2012.
  • S. V. Sam and J. Weyman, “Pieri resolutions for classical groups”, J. Algebra 329 (2011), 222–259.
  • J. Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics 149, Cambridge Univ. Press, 2003.