We construct a number of different examples of non-Noetherian graded rings that are injective as modules over themselves (or have some related but weaker properties). We discuss how these are related to the theory of triangulated categories, and to Freyd’s generating hypothesis in stable homotopy theory.

Let $p$ be a rational prime, $k$ be a perfect field of characteristic $p$ and $K$ be a finite totally ramified extension of the fraction field of the Witt ring of $k$. Let $\mathcal{G}$ be a finite flat commutative group scheme over ${\mathcal{O}}_K$ killed by some $p$-power. In this paper, we prove a description of ramification subgroups of $\mathcal{G}$ via the Breuil–Kisin classification, generalizing the author’s previous result on the case where $\mathcal{G}$ is killed by $p\ge 3$. As an application, we also prove that the higher canonical subgroup of a level $n$ truncated Barsotti–Tate group $\mathcal{G}$ over ${\mathcal{O}}_K$ coincides with lower ramification subgroups of $\mathcal{G}$ if the Hodge height of $\mathcal{G}$ is less than $\left(p-1\right)∕p^n$, and the existence of a family of higher canonical subgroups improving a previous result of the author.

Let $K$ be a complete discrete valuation field with ring of integers ${\mathcal{O}}_K$ and algebraically closed residue field $k$ of characteristic $p>0$. Let $X∕K$ be a smooth proper geometrically connected curve of genus $g>0$ with $X\left(K\right)\ne \varnothing$ if $g=1$. Assume that $X∕K$ does not have good reduction and that it obtains good reduction over a Galois extension $L∕K$ of degree $p$. Let $\mathcal{Y}∕{\mathcal{O}}_L$ be the smooth model of $X_L∕L$. Let $H:=Gal\left(L∕K\right)$.

In this article, we provide information on the regular model of $X∕K$ obtained by desingularizing the wild quotient singularities of the quotient $\mathcal{Y}∕H$. The most precise information on the resolution of these quotient singularities is obtained when the special fiber ${\mathcal{Y}}_k∕k$ is ordinary. As a corollary, we are able to produce for each odd prime $p$ an infinite class of wild quotient singularities having pairwise distinct resolution graphs. The information on the regular model of $X∕K$ also allows us to gather insight into the $p$-part of the component group of the Néron model of the Jacobian of $X$.

Let $P_d$ be a convex polygon with $d$ vertices. The associated Wachspress surface $W_d$ is a fundamental object in approximation theory, defined as the image of the rational map

$${\mathbb{P}}^2\underset{}{\overset{w_d}{\to }}{\mathbb{P}}^{d-1},$$

determined by the Wachspress barycentric coordinates for $P_d$. We show $w_d$ is a regular map on a blowup $X_d$ of ${\mathbb{P}}^2$ and, if $d>4$, is given by a very ample divisor on $X_d$ so has a smooth image $W_d$. We determine generators for the ideal of $W_d$ and prove that, in graded lex order, the initial ideal of $I_{W_d}$ is given by a Stanley–Reisner ideal. As a consequence, we show that the associated surface is arithmetically Cohen–Macaulay and of Castelnuovo–Mumford regularity $2$ and determine all the graded Betti numbers of $I_{W_d}$.

Let $p$ be a prime. We characterize those finite groups which have precisely one irreducible character of degree divisible by $p$.

Krause studied the homotopy category $K\left(Inj\mathcal{A}\right)$ of complexes of injectives in a locally noetherian Grothendieck abelian category $\mathcal{A}$. Because $\mathcal{A}$ is assumed locally noetherian, we know that arbitrary direct sums of injectives are injective, and hence, the category $K\left(Inj\mathcal{A}\right)$ has coproducts. It turns out that $K\left(Inj\mathcal{A}\right)$ is compactly generated, and Krause studies the relation between the compact objects in $K\left(Inj\mathcal{A}\right)$, the derived category $D\left(\mathcal{A}\right)$, and the category $K_{ac}\left(Inj\mathcal{A}\right)$ of acyclic objects in $K\left(Inj\mathcal{A}\right)$.

We wish to understand what happens in the nonnoetherian case, and this paper begins the study. We prove that, for an arbitrary Grothendieck abelian category $\mathcal{A}$, the category $K\left(Inj\mathcal{A}\right)$ has coproducts and is $\mu$-compactly generated for some sufficiently large $\mu$.

The existence of coproducts follows easily from a result of Krause: the point is that the natural inclusion of $K\left(Inj\mathcal{A}\right)$ into $K\left(\mathcal{A}\right)$ has a left adjoint and the existence of coproducts is a formal corollary. But in order to prove anything about these coproducts, for example the $\mu$-compact generation, we need to have a handle on this adjoint.

Also interesting is the counterexample at the end of the article: we produce a locally noetherian Grothendieck abelian category in which products of acyclic complexes need not be acyclic. It follows that $D\left(\mathcal{A}\right)$ is not compactly generated. I believe this is the first known example of such a thing.

We conclude the computation of the essential dimension of split spinor groups, and an application to algebraic theory of quadratic forms is given. We also compute essential dimension of the split even Clifford group or, equivalently, of the class of quadratic forms with trivial discriminant and Clifford invariant.

We prove a universal property of Deligne’s category ${\underset{¯}{Re}p}^{ab}\left(S_d\right)$. Along the way, we classify tensor ideals in the category $\underset{¯}{Re}p\left(S_d\right)$.

Following and generalizing a construction by Kontsevich, we associate a zeta function to any matrix with entries in a ring of noncommutative Laurent polynomials with integer coefficients. We show that such a zeta function is an algebraic function.