An arXiv preprint reports a sharp change in the second cohomology of the truncated Novikov algebra V = k[x]/(x^p). When the parameter λ lies outside the prime field F_p, the cohomology is zero. When λ lies in F_p and p is odd, it has dimension three; when p=2 and λ lies in F_2, it has dimension four. In this calculation, the cohomology classes correspond exactly to equivalence classes of abelian extensions, so the zero case means that every associated extension splits.
The study examines V over a field k of characteristic p > 0, without assuming that k is algebraically closed, and lets λ range over k. Its stated aim is to compute the second cohomology H²(V,M(λ)) for all λ and p and describe the associated abelian extensions.
A parameter controls the cases
The calculation first establishes when two members of the coefficient family represent the same V-bimodule. M(λ) and M(μ) are isomorphic exactly when λ - μ belongs to F_p. In particular, every parameter in F_p gives a module isomorphic to V itself.
The authors organize the computation through a cyclic Z/p-graded presentation of V, which is used to handle truncation and simplify the identities required of cocycles. They also give an explicit cochain complex: cocycles satisfy the two linearized Novikov identities, while coboundaries are the images under the differential δ of one-cochains.
For odd p with λ in F_p, the three dimensions are represented by the basis ψ∗Ψ1, ψ∗Ψ2 and ψ∗Ψ3. The characteristic-two case has a different structure: its four-dimensional cohomology has one-dimensional nonzero graded pieces in degrees -1, 0, 1 and 2, represented by Θ1, Θ2, Θ3 and Θ4.
What the classes mean
The extension result gives the calculation a direct algebraic meaning. Sending a cocycle to the extension it constructs produces a bijection between second-cohomology classes and equivalence classes of abelian extensions. The zero class is the split extension, and zero cohomology is equivalent to every associated extension splitting.
For odd p, the three basis classes also have explicit extension models. The Ψ1 and Ψ2 classes produce non-split extensions. The Ψ3 class produces the truncated polynomial algebra k[x]/(x^{2p}), together with the short exact sequence specified in the construction.
The preprint also reports an exact deformation result for V with adjoint coefficients. Every class in its second cohomology is described as unobstructed and extendable to an exact one-parameter family of Novikov algebras satisfying both defining identities.
The characteristic-zero comparison
For the characteristic-zero comparison, the algebra is P = k[t]. There, the parameter family is a P-bimodule only at λ=0, when it coincides with P. The adjoint second cohomology of P is zero, and the paper describes P as infinitesimally rigid and formally rigid.
The authors keep the interpretation narrow. Within this one-generator family, they say that non-vanishing tracks truncation rather than positive characteristic per se. They do not make a general claim about Novikov algebras.
The manuscript is arXiv:2608.25372v1, listed in math.RA and dated 26 August 2026. It acknowledges P. Kolesnikov for discussions and useful comments, but identifies no funding source.
Paper data and sources
Original title: Cohomology and extensions of Novikov algebras of truncated polynomials
Authors: Hassan Alhussein
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text