Functor Categories, Model Theory, Algebraic Analysis and Constructive Methods

We study the equivalences induced by some special silting objects in the derived category over dg-algebra whose positive cohomologies are all zero.

We introduce the notions of pre-morphism and pre-derivation for arbitrary non-associative algebras over a commutative ring k with identity. These notions are applied to the study of pre-Lie k-algebras and, more generally, Lie-admissible k-algebras. Associating with any algebra \((A,\cdot )\) its sub-adjacent anticommutative algebra \((A,[-,-])\) is a functor from the category of k-algebras with pre-morphisms to the category of anticommutative k-algebras. We describe the commutator of two ideals of a pre-Lie algebra, showing that the condition (Huq=Smith) holds for pre-Lie algebras. This allows to make use of all the notions concerning multiplicative lattices in the study of the multiplicative lattice of ideals of a pre-Lie algebra. We study idempotent endomorphisms of a pre-Lie algebra L, i.e., semidirect-product decompositions of L and bimodules over L.

Where can Ore extensions and homological algebra be applied? One possibility is in algebraic analysis: algebraic analysis studies functional linear systems (FLSs) of (ordinary or partial) differential equations, difference equations, differential equations with delay, etc., by means of techniques of homological algebra which use finitely presented modules over noncommutative rings of polynomial type, such as rings of differential operators, rings of polynomials with delay, iterated skew polynomial rings, Ore extensions, or more generally, domains. The structural properties of the FLSs can be described by homological-matrix constructive methods, and their study is one of the central goals of the present work. We have divided the content into four sections. In the first one, we provide enough examples of Ore extensions [28]. We also define a closely related class of rings that generalizes a quite wide class of Ore extensions, the so-called Poincaré-Birkhoff-Witt skew extensions. In the second section,we review some basic topics of homological algebra and define FLSs. The third section studies the decomposition and factorization of FLSs. We will see the relationship between homological properties of modules and triangular decompositions of FLSs, as well as the relationship between idempotent matrices and diagonal block decompositions. In the last section, we present the main ingredients of the noncommutative theory of Gröbner bases for skew PBW extensions and their application in some of the calculations involved in the results of Sects. 2. We illustrate with concrete examples of Ore algebras some of the results by means of Gröbner bases and computational tools.

This survey article on relative homological algebra in bivariant K-thoery is mainly intended for readers with a background knowledge in triangulated categories. We briefly recall the general theory of relative homological algebra in triangulated categories and latter specialize it to the non-equivariant and the equivariant bivariant K-thoery, where the actions on C*-algebras is by a finite cyclic group. We conclude by the explicit computation of the universal abelian invariant (which is a Mackey functor) for separable C*-algebras with the action of \(\mathbb {Z}/4\) by automorphisms.

We describe an abelian category \(\textbf{ab}(M)\) in which the solution sets of finitely many linear equations over an arbitrary ring R with values in an arbitrary left R-module M reside as objects. Such solution sets are also called behaviors in algebraic systems theory. We both characterize \(\textbf{ab}(M)\) by a universal property and give a construction of \(\textbf{ab}(M)\) as a Serre quotient of the free abelian category generated by R. We discuss features of \(\textbf{ab}(M)\) relevant in the context of algebraic systems theory: if R is left coherent and M is an fp-injective fp-cogenerator, then \(\textbf{ab}(M)\) is antiequivalent to the category of finitely presented left R-modules. This provides an alternative point of view to the important module-behavior duality in algebraic systems theory. We also obtain a dual statement: if R is right coherent and M is fp-faithfully flat, then \(\textbf{ab}(M)\) is equivalent to the category of finitely presented right R-modules. As an example application, we discuss delay-differential systems with constant coefficients and a polynomial signal space. Moreover, we propose definitions of controllability and observability in our setup.

Definable categories are axiomatisable additive categories. They appear as definable subcategories of module categories, equivalently as the categories of exact functors on some small abelian category. We give an exposition of their structure and their model theory from an essentially intrinsic point of view. We recall the anti-equivalence between definable categories and small abelian categories and we describe a monoidal version of this due to Wagstaffe.

Assume that R is a non-right perfect ring. Then there is a proper class of classes of (right R-) modules closed under transfinite extensions lying between the classes \(\mathcal P _0\) of projective modules, and \(\mathcal F _0\) of flat modules. These classes can be defined as variants of the class \(\mathcal F \mathcal M\) of absolute flat Mittag-Leffler modules: either as their restricted versions (lying between \(\mathcal P _0\) and \(\mathcal F \mathcal M\)), or their relative versions (between \(\mathcal F \mathcal M\) and \(\mathcal F _0\)). In this survey, we will deal with applications of these classes in relative homological algebra and algebraic geometry. The classes \(\mathcal P _0\) and \(\mathcal F _0\) are known to provide for approximations, and minimal approximations, respectively. We will show that the classes of restricted flat Mittag-Leffler modules, and flat relative Mittag-Leffler modules, have rather different approximation properties: the former classes always provide for approximations, but the latter do not, except for the boundary case of \(\mathcal F _0\). The notion of an (infinite dimensional) vector bundle is known to be Zariski local for all schemes, the key point of the proof being that projectivity ascends and descends along flat and faithfully flat ring homomorphisms, respectively. We will see that the same holds for the properties of being a \(\kappa \)-restricted flat Mittag-Leffler module for each cardinal \(\kappa \ge \aleph _0\), and also a flat \(\mathcal Q\)-Mittag-Leffler module whenever \(\mathcal Q\) is a definable class of finite type. Thus, as in the model case of vector bundles, Zariski locality holds for flat quasi-coherent sheaves induced by each of these classes of modules. Moreover, we will see that the notion of a locally n-tilting quasi-coherent sheaf is Zariski local for all \(n \ge 0\).