{
  "abstract": "We study super-matrices $M_{m|n}(R)$ over the rank-two exterior algebra $R=K[θ_1,θ_2]/(θ_1^2,θ_2^2,θ_1θ_2+θ_2θ_1)$, the smallest supercommutative ring on which the Berezinian's odd-by-odd correction term $BD^{-1}C$ need not vanish. We recall that the supertrace satisfies graded cyclicity, $\\mathrm{str}(XY)=(-1)^{|X||Y|}\\mathrm{str}(YX)$, so $\\mathrm{str}([A,B])=0$ for even $A,B$, and that an explicit $1|1$ matrix over $R$ has $BD^{-1}C\\neq0$ surviving to second order in the odd generators. We then prove the extension anticipated in the $1|1$ note: a closed-form Berezinian formula for arbitrary even $X\\in M_{m|n}(R)$, $m,n\\geq1$ (Theorem 1), given through a single $m\\times m$ matrix pairing $Δ$ built from the odd blocks of $X$; and a fully general multiplicativity theorem $\\mathrm{Ber}(XY)=\\mathrm{Ber}(X)\\mathrm{Ber}(Y)$ for even invertible $X,Y\\in M_{m|n}(R)$ (Theorem 2), proved by an explicit trace-cyclicity cancellation. Both results specialize exactly to the known $1|1$ formulas when $m=n=1$, and we verify them on a worked $m=2,n=1$ example. We extend the framework from two to an arbitrary number $r\\geq2$ of odd generators, replacing the scalar pairing $Δ$ by a $Λ^2(V)$-valued pairing on the $r$-dimensional space $V$ of odd generators, and identify this pairing, in the scalar case, with a Plücker coordinate of the odd off-diagonal data. We close with a discussion of related work of Khudaverdian--Voronov, and a remark identifying $Δ$ as a matrix-valued contraction against the natural alternating pairing on the space of odd generators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05148",
  "author_total": 1,
  "authors": [
    {
      "name": "Devichandrika V"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/42sa3xhcxpws1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05148",
  "fulltext_url": null,
  "id": "42sa3xhcxpws1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05148",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7211876048",
  "pdf_url": "https://arxiv.org/pdf/2609.05148",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/42sa3xhcxpws1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.05148",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05148"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Explicit Super-Linear Algebra over K[θ1, θ2]: A General Berezinian Correction Formula and Multiplicativity Theorem for Arbitrary Block Size",
  "title_text": "Explicit Super-Linear Algebra over K[θ1, θ2]: A General Berezinian Correction Formula and Multiplicativity Theorem for Arbitrary Block Size",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/42sa3xhcxpws1",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.3842/sigma.2025.***",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}