arXiv subjects

Martin J. Taylor

Publications and source records attributed to Martin J. Taylor.

5 recordsLinked to original sources

arXiv2026

The Nil K-groups of finite groups

We show that for every finite group $G$ and every integer $n$, the Nil group $\mathrm{NK}_n(\mathbb{Z}[G])$ of the integral group ring of $G$ has finite exponent, and give a bound depending on n and the order $|G|$. This bound is explicit when every prime divisor of $|G|$ is at least $3+n/2$. More generally, our finite exponent result applies to $\mathrm{NK}_n(R[G])$ whenever $R$ is a regular, torsion-free, Noetherian commutative ring such that $R/p$ is regular for every prime $p$ that divides $|G|$. In the Appendix, M. Morrow provides an alternative approach and also shows that the Nil group $\mathrm{NK}_n(X)$ of an excellent Noetherian scheme $X$, with $X[1/p]$ regular, is annihilated by a finite power of $p$, provided $X$ admits a suitable resolution of singularities. By extending his argument to certain noncommutative rings, we also show that, for $R$ as above and $α$ any automorphism of $G$, the Farrell Nil groups $\mathrm{NK}_n(R[G],α)$ have finite exponent. As a consequence, for every virtually cyclic group $Γ$, the group $\mathrm{K}_n(\mathbb{Z}[Γ])$ is a direct sum of a finitely generated abelian group and an infinite countable direct sum of copies of a finite abelian group.

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arXiv2025

On the Quadratic Structure of Torsors over Affine Group Schemes

Let $\mathcal{G}=\mathrm{Spec}(A)$ be a finite and flat group scheme over the ring of algebraic integers $R$ of a number field $K$ and suppose that the generic fiber of $\mathcal{G}$ is the constant group scheme over $K$ for a finite group $G$. Then the $R$-dual $A^D$of $A$ identifies as a Hopf $R$-order in the group algebra $K[G]$. If $B$ is a principal homogeneous space for $A$, then it is known that $B$ is a locally free $A^D$-module. By multiplying the trace form of $B_K/K$ by a certain scalar we obtain a $G$-invariant form $Tr'_B$ which provides a non-degenerate $R$-form on $B$. If $G$ has odd order, we show that the $G$-forms $(B, Tr'_B)$ and $(A, Tr'_A)$ are locally isomorphic and we study the question of when they are globally isomorphic. Suppose now that $K$ is a finite extension of $\mathbb Q_p$ with valuation ring $R$. In the course of our study we are led to consider the extension of scalars map $φ_K: G_0(A^D)\rightarrow G_0(A^D_K)=G_0(K[G])$. When $A^D$ is the group ring $R[G]$, Swan showed that $φ_K$ is an isomorphism. Jensen and Larson proved that $φ_K$ is also an isomorphism for any Hopf $R$-order $A^D$ of $K[G]$ when $G$ is abelian and $K$ is large enough. Here we prove that $\ker φ_K$ is at most a finite abelian $p$-group. However, numerous examples lead us to conjecture that Swan's result extends to all Hopf $R$-orders in $K[G]$, i.e. $\ker φ_K$ is always trivial.

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arXiv2018

Cup products in the etale cohomology of number fields

This paper concerns cup product pairings in étale cohomology related to work of M. Kim and of W. McCallum and R. Sharifi. We will show that by considering Ext groups rather than cohomology groups, one arrives at a pairing which combines invariants defined by Kim with a pairing defined by McCallum and Sharifi. We also prove a formula for Kim's invariant in terms of Artin maps in the case of cyclic unramified Kummer extensions. One consequence is that for all $n > 1$, there are infinitely many number fields $F$ over which there are both trivial and non-trivial Kim invariants associated to cyclic groups of order $n$.

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arXiv2017

On the trace forms of Galois algebras

We study the trace form $q_L$ of $G$-Galois algebras $L/K$ when $G$ is a finite group and $K$ is a field of characteristic different from $2$. We introduce in this paper the category of $2$-reduced groups and, when $G$ is such a group, we use a formula of Serre to compute the second Hasse-Witt invariant of $q_L$. By combining this computation with work of Quillen we determine the isometry class of $q_L$ for large families of $G$-Galois algebras over global fields. We also indicate how our results generalize to Galois $G$-covers of schemes.

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arXiv2013

The classifying topos of a group scheme and invariants of symmetric bundles

Let $Y$ be a scheme in which 2 is invertible and let $V$ be a rank $n$ vector bundle on $Y$ endowed with a non-degenerate symmetric bilinear form $q$. The orthogonal group ${\bf O}(q)$ of the form $q$ is a group scheme over $Y$ whose cohomology ring $H^*(B_{{\bf O}(q)},{\bf Z}/2{\bf Z})\simeq A_Y[HW_1(q),..., HW_n(q)]$ is a polynomial algebra over the étale cohomology ring $A_Y:=H^*(Y_{et},{\bf Z}/2{\bf Z})$ of the scheme $Y$. Here the $HW_i(q)$'s are Jardine's universal Hasse-Witt invariants and $B_{{\bf O}(q)}$ is the classifying topos of ${\bf O}(q)$ as defined by Grothendieck and Giraud. The cohomology ring $H^*(B_{{\bf O}(q)},{\bf Z}/2{\bf Z})$ contains canonical classes $\mathrm{det}[q]$ and $[C_q]$ of degree 1 and 2 respectively, which are obtained from the determinant map and the Clifford group of $q$. The classical Hasse-Witt invariants $w_i(q)$ live in the ring $A_Y$. Our main theorem provides a computation of ${det}[q]$ and $[C_{q}]$ as polynomials in $HW_{1}(q)$ and $HW_{2}(q)$ with coefficients in $A_Y$ written in terms of $w_1(q),w_2(q)\in A_Y$. This result is the source of numerous standard comparison formulas for classical Hasses-Witt invariants of quadratic forms. Our proof is based on computations with (abelian and non-abelian) Cech cocycles in the topos $B_{{\bf O}(q)}$. This requires a general study of the cohomology of the classifying topos of a group scheme, which we carry out in the first part of this paper.

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