An arXiv preprint describes a way to compute free resolutions over the exterior algebra, using a refined Schreyer workflow for an anti-commutative setting. The workflow computes syzygies, meaning relations among algebraic relations, and reports a Gröbner basis for the exterior-algebra syzygy module.
The paper asks whether this algorithm can be adapted to compute free resolutions over the exterior algebra and to illustrate cohomology calculations through the BGG correspondence. Here, BGG is the link the paper draws between exterior-algebra resolutions and information about sheaf cohomology. The study domain is algebraic: modules and coherent-sheaf cohomology computations connected through free resolutions.
To do that, the workflow first computes syzygies of a corresponding lifted module in the anti-commutative setting and uses relative Gröbner bases. The stated result is a Gröbner basis for the exterior-algebra syzygy module. For general readers, the important point is that the method organizes relations through a lifted calculation before producing the basis needed for the original module.
The paper gives a formal correctness statement for three numbered procedures. It says Algorithms 4 and 5 terminate and produce the required liftings, while Algorithm 6 returns a Gröbner basis of the syzygy module. That is a mathematical statement about what the construction returns, not an empirical finding about runtime.
From algebra to cohomology
The proposed use goes beyond the resolution itself. Through the BGG correspondence, the paper presents exterior-algebra resolutions as an application route for obtaining sheaf-cohomology information. The scope is illustrative: the paper does not evaluate that route across a broad benchmark set.
One worked example gives concrete output. In the non-minimized resolution, the displayed sizes at degrees 0 through 4 are 6, 6, 10, 15 and 21. After minimization, the displayed sizes are 6, 3, 1, 1 and 3 across degrees 0 through 4. The figures illustrate the calculation; they are not a performance comparison.
A parallel route, still on paper
Parallel execution is presented as a possible extension. The authors identify independent lifting of terms and graph traversal as sources of possible parallelism. They also reformulate the recursive subtree-lifting procedure as an iterative algorithm for parallel computation.
But that proposal is not yet a speed result. No speedup or scalability measurement is reported, and the current implementation is described as a demonstrator while a Singular-kernel implementation is under development.
The practical test is still ahead
The paper also sets a boundary on what a computation can promise. Exterior-algebra free resolutions may be infinite, so a computation has to specify the homological degree it is meant to reach. That is a scope condition for running the method, not evidence that every resolution terminates finitely.
The algorithm is reported as implemented in sresext.lib, a library of the Singular computer algebra system. The paper says the library is included in the latest Singular release and in its latest GitHub version.
Taken together, the preprint offers a mathematically specified route from lifted syzygy computations to exterior-algebra bases, with an illustrative BGG application and a proposed path to parallel execution. It does not yet answer how the method scales on large resolutions or whether the iterative formulation produces measurable speedups. Those questions remain open while the Singular-kernel implementation is developed.
Paper data and sources
Original title: Schreyer Resolutions over the Exterior Algebra
Authors: Janko Boehm, Lakshmi Ramesh
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-09-29
DOI: Not available
Original paper · Full text