Quotients of Strongly Proper Forcings and Guessing Models

We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have simple universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the $\omega_1$-approximation property. We prove that the existence of stationarily many $\omega_1$-guessing models in $P_{\omega_2}(H(\theta))$, for sufficiently large cardinals $\theta$, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss

Lattice theory of torsion classes

The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set $\operatorname{\mathsf{tors}} A$ of torsion classes over a finite-dimensional algebra $A$. We show that $\operatorname{\mathsf{tors}} A$ is a complete lattice which enjoys very strong properties, as bialgebraicity and complete semidistributivity. Thus its Hasse quiver carries the important part of its structure, and we introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of $\operatorname{\mathsf{tors}} A$. In particular, we give a representation-theoretical interpretation of the so-called forcing order, and we prove that $\operatorname{\mathsf{tors}} A$ is completely congruence uniform. When $I$ is a two-sided ideal of $A$, $\operatorname{\mathsf{tors}} (A/I)$ is a lattice quotient of $\operatorname{\mathsf{tors}} A$ which is called an algebraic quotient, and the corresponding lattice congruence is called an algebraic congruence. The second part of this paper consists in studying algebraic congruences. We characterize the arrows of the Hasse quiver of $\operatorname{\mathsf{tors}} A$ that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, we study in detail the case of preprojective algebras $\Pi$, for which $\operatorname{\mathsf{tors}} \Pi$ is the Weyl group endowed with the weak order. In particular, we give a new, more representation theoretical proof of the isomorphism between $\operatorname{\mathsf{tors}} k Q$ and the Cambrian lattice when $Q$ is a Dynkin quiver. We also prove that, in type $A$, the algebraic quotients of $\operatorname{\mathsf{tors}} \Pi$ are exactly its Hasse-regular lattice quotients.Comment: 65 pages. Many improvements compared to the first version (in particular, more discussion about complete congruence uniform lattices