## Algebra & Number Theory

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

#### Abstract

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 $Gr(k,ℂn)$. 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

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

Dates
Revised: 4 December 2017
Accepted: 3 January 2018
First available in Project Euclid: 23 May 2018

https://projecteuclid.org/euclid.ant/1527040845

Digital Object Identifier
doi:10.2140/ant.2018.12.285

Mathematical Reviews number (MathSciNet)
MR3803704

Zentralblatt MATH identifier
06880889

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

#### Citation

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. https://projecteuclid.org/euclid.ant/1527040845

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