If the free algebra $F$ on one generator in a variety $V$ of algebras (in the sense of universal algebra) has a subalgebra free on two generators, must it also have a subalgebra free on three generators? In general, no; but yes if $F$ generates the variety $V$.

Generalizing the argument, it is shown that if we are given an algebra and subalgebras, $A_0\supseteq \cdots \supseteq A_n$, in a prevariety ($\mathbb{S}\mathbb{P}$-closed class of algebras) $P$ such that $A_n$ generates $P$, and also subalgebras $B_i\subseteq A_{i-1}$ $\left(0<i\le n\right)$ such that for each $i>0$ the subalgebra of $A_{i-1}$ generated by $A_i$ and $B_i$ is their coproduct in $P$, then the subalgebra of $A$ generated by $B_1,\dots ,B_n$ is the coproduct in $P$ of these algebras.

Some further results on coproducts are noted:

If $P$ satisfies the amalgamation property, then one has the stronger “transitivity” statement, that if $A$ has a finite family of subalgebras ${\left(B_i\right)}_{i\in I}$ such that the subalgebra of $A$ generated by the $B_i$ is their coproduct, and each $B_i$ has a finite family of subalgebras ${\left(C_{ij}\right)}_{j\in J_i}$ with the same property, then the subalgebra of $A$ generated by all the $C_{ij}$ is their coproduct.

For $P$ a residually small prevariety or an arbitrary quasivariety, relationships are proved between the least number of algebras needed to generate $P$ as a prevariety or quasivariety, and behavior of the coproduct operation in $P$.

It is shown by example that for $B$ a subgroup of the group $S=Sym\left(\Omega \right)$ of all permutations of an infinite set $\Omega$, the group $S$ need not have a subgroup isomorphic over $B$ to the coproduct with amalgamation $S{\coprod }_{B}S$. But under various additional hypotheses on $B$, the question remains open.

Our previous theorems on exponential sums often did not apply or did not give sharp results when certain powers of a variable appearing in the polynomial were divisible by $p$. We remedy that defect in this paper by systematically applying $p$-power reduction, making it possible to strengthen and extend our earlier results.

In this paper we study singularities defined by the action of Frobenius in characteristic $p>0$. We prove results analogous to inversion of adjunction along a center of log canonicity. For example, we show that if $X$ is a Gorenstein normal variety then to every normal center of sharp $F$-purity $W\subseteq X$ such that $X$ is $F$-pure at the generic point of $W$, there exists a canonically defined $\mathbb{Q}$-divisor ${\Delta }_W$ on $W$ satisfying $\left(K_X\right){|}_W{\sim }_{\mathbb{Q}}{K}_W+{\Delta }_W$. Furthermore, the singularities of $X$ near $W$ are “the same” as the singularities of $\left(W,{\Delta }_W\right)$. As an application, we show that there are finitely many subschemes of a quasiprojective variety that are compatibly split by a given Frobenius splitting. We also reinterpret Fedder’s criterion in this context, which has some surprising implications.

Following Shokurov’s ideas, we give a short proof of the following klt version of his result: termination of terminal log flips in dimension $d$ implies that any klt pair of dimension $d$ has a log minimal model or a Mori fibre space. Thus, in particular, any klt pair of dimension $4$ has a log minimal model or a Mori fibre space.

Let $\mathcal{C}$ be a fusion category faithfully graded by a finite group $G$ and let $\mathcal{D}$ be the trivial component of this grading. The center $\mathcal{Z}\left(\mathcal{C}\right)$ of $\mathcal{C}$ is shown to be canonically equivalent to a $G$-equivariantization of the relative center ${\mathcal{Z}}_{\mathcal{D}}\left(\mathcal{C}\right)$. We use this result to obtain a criterion for $\mathcal{C}$ to be group-theoretical and apply it to Tambara–Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara–Yamagami categories. Finally, we prove a general result about the existence of zeroes in $S$-matrices of weakly integral modular categories.