Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.26345

On the notion of $\mathbb{A}^1$-coverings

Marco Giustetto·Thor Wittich
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce the notion of $\mathbb{A}^1$-$n$-coverings and begin the development of their theory over an arbitrary base scheme $S$, in analogy to that of higher coverings in an $\infty$-topos. We show that for $n=0$, our notion recovers Morel's $\mathbb{A}^1$-coverings between simplicial sheaves. Under suitable assumptions, we then provide a Galois-theoretic classification of $\mathbb{A}^1$-$n$-coverings. Finally, we answer a question of Morel concerning a geometric criterion for $\mathbb{A}^1$-coverings in the negative.

Explore related subjects

math.ATmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Marco Giustetto, Thor Wittich. 2026-10-19. On the notion of $\mathbb{A}^1$-coverings. https://arxiv.org/abs/2610.26345

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Type IIA String Theory and tmf with Level Structure

We look at a new string$^h$ tangential structure first introduced by Devalapurkar and relate it to the $W_7=0$ condition of Diaconescu-Moore-Witten for type IIA string theory and M-theory. We show that a string$^h$ structure on the target space automatically satisfies the $W_7=0$ condition and we also explain when the $W_7=0$ condition gives rise to a string$^h$ structure. Devalapurkar initially constructed $MString^h$ in such a way that it orients $tmf_1(3)$; we extend Devalapurkar's result, showing that $MString^h$ orients $tmf_1(n)$. We compute the homotopy groups of $MString^h$ in the dimensions relevant for physical applications, and apply them to anomaly cancellation applications for certain compactifications of type IIA string theory.

math.AT↗
arXiv2026

Persistent (Co)Homology in Matrix Multiplication Time

Most algorithms for computing persistent homology do so by tracking cycles that represent homology classes. There are many choices of such cycles, and specific choices have found different uses in applications. Although it is known that persistence diagrams can be computed in matrix multiplication time [8] for the more general case of zigzag persistent homology, it is not clear how to extract cycle representatives, especially if specific representatives are desired. In this paper, we provide the same matrix multiplication bound for computing representatives for the two choices common in applications in the case of ordinary persistent (co)homology. We first provide a fast version of the reduction algorithm, which is simpler than the algorithm in [8], but returns a different set of representatives than the standard algorithm [6] We then give a fast version of a different variant called the row algorithm [4], which returns the same representatives as the standard algorithm.

math.AT↗
arXiv2026

Adams filtrations and secondary coherence for fibers of Steenrod operations

Let $f:Y\to\prod_λΣ^{d_λ}H\mathbb F_2$ be a locally finite family of positive-degree stable operations, where $Y=H\mathbb F_2$ or $H\mathbb Z$; in the integral case assume $d_λ\geq2$. We determine the Adams filtration of its fiber from the minimal generators of $\operatorname{im}(f^*)$ as a left Steenrod module. An explicit target automorphism separates a minimal subfamily from the redundant coordinates and gives a noncanonical stable splitting. Combining this reduction with the Bruner--Rognes two-extension theorem and an acyclic cancellation gives $E_3=E_\infty$. In positive homotopy degrees, the filtration-one subgroup is the image of the dual of the induced map on minimal-generator quotients, and the redundant coordinates account for filtration zero. We also specify the zero-stem filtration and its possible cyclic extension of order four. A terminating Adem recursion realizes the splitting for the full positive even-square family and its finite truncations. Separately, for a crossed bimodule and fixed lifts of the differentials of a free primary complex, we construct a class in the third cohomology of an explicit Hom complex. Its vanishing is equivalent to the existence of coherent null-composite tracks with those fixed differential lifts.

math.AT↗