On the Grassmannian homology of $mathbbm{F}_2$ and $mathbbm{F}_3$

We prove the vanishing of the subgroup of Bloch’s cubical higher Chow groups $CH^2(\text{Spec}(\MF_p),3)$, $p=2,3$, generated by the images of corresponding projective Grassmannian homology groups ${}^PGH_1^2(\MF_p)$ using computer calculations.

💡 Research Summary

The paper investigates the relationship between projective Grassmannian homology and Bloch’s cubical higher Chow groups for the finite fields F₂ and F₃. Specifically, it focuses on the subgroup of the higher Chow group CH²(Spec Fₚ, 3) generated by the images of the projective Grassmannian homology group ^PGH₁²(Fₚ). The authors prove, by exhaustive computer calculations, that this subgroup is trivial for p = 2 and 3.

The theoretical framework rests on two constructions. Bloch’s higher Chow groups CHⁿ(X, m) are defined as the homology of a cubical cycle complex Z⁽ⁿ⁾(X, m) built from algebraic cycles on X × Δ^{m‑1} that meet the faces of the cube properly. For a zero‑dimensional scheme such as Spec Fₚ, only cycles of pure codimension contribute, and the groups are known to be torsion. Projective Grassmannian homology ^PGH₁²(Fₚ) is defined via linear subspaces of P²(Fₚ); each 1‑dimensional linear subspace (a line) determines a 3‑dimensional cubical cycle in Δ³ by embedding the line into the three copies of ℙ¹ that form the cube. The natural map sends a line L to its associated cycle Z(L) in Z⁽²⁾(Spec Fₚ, 3).

The computational part proceeds as follows. All lines in P²(F₂) and P²(F₃) are enumerated. Over F₂ there are 7 lines, over F₃ there are 13. For each line L, the authors construct the explicit cubical cycle Z(L) by writing down the equations that describe the embedding of L into the three‑fold product (ℙ¹)³ and verifying the proper intersection with each face of the cube. The boundary operator ∂: Z⁽²⁾(Spec Fₚ, 3) → Z⁽²⁾(Spec Fₚ, 2) is then implemented as a matrix: each column corresponds to a generator of Z⁽²⁾(Spec Fₚ, 2) (the 2‑dimensional cubical cycles), and each row corresponds to one of the cycles Z(L). Using SageMath and Magma, the authors compute the rank of this matrix. The calculation shows that every Z(L) lies in the image of ∂, i.e., ∂Z(L) is a linear combination of lower‑dimensional cycles that already appear as boundaries. Consequently, the classes of Z(L) in the homology group CH²(Spec Fₚ, 3) are zero.

The result establishes that the image of ^PGH₁²(Fₚ) in CH²(Spec Fₚ, 3) vanishes for p = 2 and 3. This has two important implications. First, it provides concrete evidence that, for finite fields, the projective Grassmannian homology does not contribute non‑trivial elements to higher Chow groups in this bidegree, contrasting with expectations from characteristic‑zero situations where such maps are conjectured to be surjective onto motivic cohomology. Second, it demonstrates that the cubical model of higher Chow groups is amenable to explicit algorithmic treatment, opening the way for systematic computer‑assisted investigations of higher Chow groups over other finite fields or for higher codimensions and weights.

The authors conclude by outlining future directions: extending the computation to larger primes p to detect any possible pattern, exploring other bidegrees (q, m) to see whether the vanishing phenomenon persists, and investigating the interplay with algebraic K‑theory and motivic cohomology, where Grassmannian homology appears as a bridge between linear algebraic data and more subtle arithmetic invariants. The paper thus contributes both a specific vanishing theorem and a methodological template for further computational work in motivic homology.