arXiv subjects

Ahina Nandy

Publications and source records attributed to Ahina Nandy.

3 recordsLinked to original sources

arXiv2026

On the metalinear algebraic cobordism spectrum

In this paper, we study the metalinear algebraic cobordism spectrum $\mathrm{MML}$ (also sometimes denoted $\mathrm{MSL}^c$), which is built from the structure groups of oriented vector bundles. We establish an interpolation between $\mathrm{MSL}$ and $\mathrm{MML}$ and deduce that the canonical morphism $\mathrm{MSL}\to \mathrm{MML}$ admits a retraction. We parametrize all such retractions in the category of $\mathrm{MSL}$-modules and, after fixing one of them, obtain an equivalence $\mathrm{MML}\cong\mathrm{MSL}\oplus \Sigma^{2,1}\mathrm{MGL}$. As an application of these results, we determine various invariants of the metalinear algebraic cobordism spectrum over a field (after inverting the exponential characteristic). More precisely, we determine the first few Milnor-Witt stems of $\mathrm{MML}$ in terms of the very effective algebraic and hermitian K-theory spectra, and the geometric diagonal of $\mathrm{MML}$ in terms of Stong's complex-spin cobordism ring. We also compute the slices and use them to describe the category of 2-inverted modules over the $\mathbb{E}_\infty$-ring spectrum $\mathrm{MML}$.

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arXiv2026

Slices of the special linear algebraic cobordism spectrum

Let $F$ be a field of exponential characteristic $e$. We compute the slices of $\mathbf{MSL}[e^{-1}]$, where $\mathbf{MSL}$ is the special linear algebraic cobordism spectrum defined by Panin and Walter. The answer is expressed in terms of the second page of the Adams-Novikov spectral sequence for the special unitary cobordism spectrum, which was explicitly determined by Novikov. Its applicability is demonstrated by computations with the slice spectral sequence for $\mathbf{MSL}$, which determine the first few Milnor-Witt stems of its homotopy groups (up to the third) in terms of very effective hermitian $K$-theory. We also establish a decomposition of the rational special linear algebraic cobordism spectrum over an arbitrary qcqs scheme.

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arXiv2024

An interpolation between special linear and general algebraic cobordism $\text{MSL}$ and $\text{MGL}$

Conner and Floyd determined the torsion in the special unitary bordism $\text{MSU}$ back in the late 1960s. One of the ingredients of their work was an interpolation between $\text{MSU}$ and unitary bordism $\text{MU}$. In this work, we prove an exactly similar relation between the special, and general linear algebraic cobordism $\text{MSL}$, and $\text{MGL}$ in the stable motivic homotopy category over any Noetherian base scheme of finite Krull dimension. This gives a filtration of $\text{MGL}$ in terms of $\text{MSL}$, and the skeletal filtration of $\mathbb{P}^{\infty}$. Using that, we compute the first slice of $\text{MSL}$ over characteristic zero fields. We also compute the first line of the stable homotopy groups of $\text{MSL}$ over fields of characteristic zero.

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