Building on work of Briggs, Grifo and Pollitz arXiv:2506.10827, we give an example of two cohomological support varieties of monomial ideals which are not unions of linear subspaces. We provide a procedure for the computation of the cohomological support varieties of certain other monomial ideals - including those with homogeneous generators - with improved computational efficiency, leading to a computer-assisted verification of the existence of a third support variety of a monomial ideal which is not a union of linear subspaces and a computer-assisted proof of a classification of cohomological support varieties of homogeneous monomial ideals over $\mathbb{Q}$ with 6 generators.
Recent work [BGP24,BGP25] has considered the realizability of cohomological support varieties in various levels of generality. Let Q be a regular local ring with residue field k or polynomial ring over k, R = Q/I be a quotient by I = (f ) = (f 1 , . . . , f n ) and M be an finitely generated R-module. The cohomological support variety (CSV) V R (M ), born of the Hom complex Ext • E (M, k) over a certain DG algebra E, is a valuable homological invariant which is intimately related to a number of properties of both M and R [Avr98, AB00b,Pol19,BGP24].
The set of possible values of V R (M ), the subject of the realizability problem for CSVs, has been classified when R is a complete intersection [Ber07] or Golod [BGP24]. Work in [BGP25] restricts attention to the case M = R = Q/(f ) and classifies V R (R) for monomial ideals where f has at most 5 generators, provided that either R = Q/I where Q is regular local and I is in the square of the maximal ideal of Q or Q is a graded polynomial ring and I is generated by forms of degree at least two, which is to say that R has a minimal regular presentation: Theorem 1.1 ([BGP25, Theorem 6.14, Theorem 6.16]). Let Q/I be a minimal regular presentation of R and let f be a minimal generating set of I consisting of n ≤ 5 monomials. Then V R (R) is either a coordinate subspace of A 5 k or a union of two hyperplanes. Using a Macaulay2 calculation, they find a variety which is not a union of linear subspaces which is realizable when n = 6 [BGP25, Example 6.17]. In doing so, they employ a procedure which can be used to compute the CSV of an arbitrary monomial ideal. However, this procedure involves the computation of the kernel of one square matrix of size 2 n+1 and the image of another, making it computationally expensive. For certain monomial ideals, including those with homogeneous generators, we offer a more efficient procedure, which consequently allows us to theoretically verify their computational discovery, and find a second such variety: Theorem A. Every CSV of a ring with a minimal regular presentation given by a homogeneous monomial ideal with n generators is expressible as the homology of a chain complex of vector spaces with total dimension 2 n . Theorem B (cf. [BGP25, Example 6.17]). If R = k[x 1 , . . . , x 6 ]/(x 1 x 2 , x 2 x 3 , x 3 x 4 , x 4 x 5 , x 5 x 6 , x 6 x 1 ), then V R (R) = V(a 1 a 3 a 5 + a 2 a 4 a 6 ) ⊆ A 6 k . Furthermore, if char(k) ̸ ∈ {2, 5} and R = k[x 1 , . . . , x 10 ]/(x 1 x 2 , x 2 x 3 , . . . , x 10 x 1 ), then V R (R) = V(a 1 a 3 a 5 a 7 a 9 + a 2 a 4 a 6 a 8 a 10 ) ⊆ A 10 k .
It also allows us to write a method in Macaulay2 which computes cohomological support varieties of monomial ideals over Q. This allows us to make two additional statements, albeit verified only with computer assistance. First, we are able to give another example of a previously-unknown realizable variety, given by the edge ideal of a 14-cycle over Q:
Example C (Example 5.1). The edge ideal of a 14-cycle over Q has support variety
The second is a partial generalization of Theorem 2.5 to the case n = 6, under the condition that we only consider homogeneous monomials and only over Q:
Theorem D (Theorem 5.2). The CSVs of rings over Q with minimal regular presentations given by ideals with minimal generating sets with 6 elements are all one of the following up to order:
• a linear subspace,
• a union of two hyperplanes,
• V(a 135 + a 246 ).
In Section 2, we provide a synopsis of the relevant background information on cohomological support varieties and their calculation in the general and monomial settings, introducing also for the sake of resolving monomial ideals the Taylor resolution. In Section 3, we will extrapolate on an existing construction of the CSV in the monomial setting, leveraging the fact that our resulting resolution is over k and exploiting the relationship between the Taylor resolution and cellular cochain complexes of certain simplicial complexes, whence Theorem A is derived. In Section 4 we use this information to manually verify Theorem B. In Section 5 we describe our computational verification of Example C and Theorem D. We conclude with Section 6, a short note on future work.
We provide a synopsis of the construction of cohomological support varieties, including the relevant topics surrounding DG algebra and module resolutions, Koszul resolutions, and universal resolutions. This section draws predominantly from [Pol19,BGP25]. The interested reader is encouraged to consult [Avr98] for more background on DG algebras and resolutions and [Pol21] for Koszul resolutions, universal resolutions, and the resulting constructions of the cohomological support variety.
2.1. Preliminary notions. Let Q be a commutative ring and let R = Q/I where I = (f 1 , . . . , f n ). E is the DG algebra
that is, it is the Koszul complex of (f ) endowed with an exterior algebra structure. In particular, note that these e i are anti-commutative.
Let A be a DG Q-alg
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