Arbitrary Ground Rings Research Articles

Superfiltered A∞$A_\infty$‐deformations of the exterior algebra, and local mirror symmetry

The exterior algebra $E$ on a finite-rank free module $V$ carries a $\mathbb{Z}/2$-grading and an increasing filtration, and the $\mathbb{Z}/2$-graded filtered deformations of $E$ as an associative algebra are the familiar Clifford algebras, classified by quadratic forms on $V$. We extend this result to $A_\infty$-algebra deformations $\mathcal{A}$, showing that they are classified by formal functions on $V$. The proof translates the problem into the language of matrix factorisations, using the localised mirror functor construction of Cho-Hong-Lau, and works over an arbitrary ground ring. We also compute the Hochschild cohomology algebras of such $\mathcal{A}$. By applying these ideas to a related construction of Cho-Hong-Lau we prove a local form of homological mirror symmetry: the Floer $A_\infty$-algebra of a monotone Lagrangian torus is quasi-isomorphic to the endomorphism algebra of the expected matrix factorisation of its superpotential.

Tilting theory via stable homotopy theory

Abstract We show that certain tilting results for quivers are formal consequences of stability, and as such are part of a formal calculus available in any abstract stable homotopy theory. Thus these results are for example valid over arbitrary ground rings, for quasi-coherent modules on schemes, in the differential-graded context, in stable homotopy theory and also in the equivariant, motivic or parametrized variant thereof. In further work, we will continue developing this calculus and obtain additional abstract tilting results. Here, we also deduce an additional characterization of stability, based on Goodwillie’s strongly (co)cartesian n-cubes. As applications we construct abstract Auslander–Reiten translations and abstract Serre functors for the trivalent source and verify the relative fractionally Calabi–Yau property. This is used to offer a new perspective on May’s axioms for monoidal, triangulated categories.

Tilting theory for trees via stable homotopy theory

We show that variants of the classical reflection functors from quiver representation theory exist in any abstract stable homotopy theory, making them available for example over arbitrary ground rings, for quasi-coherent modules on schemes, in the differential-graded context, in stable homotopy theory as well as in the equivariant, motivic, and parametrized variant thereof. As an application of these equivalences we obtain abstract tilting results for trees valid in all these situations, hence generalizing a result of Happel.The main tools introduced for the construction of these reflection functors are homotopical epimorphisms of small categories and one-point extensions of small categories, both of which are inspired by similar concepts in homological algebra.

Rational Modules for Corings

The so called dense pairings were studied mainly by Radford in his work on coreflexive coalegbras over fields. They were generalized in a joint paper with Gómez-Torricillas and Lobillo to the so called rational pairings over a commutative ground ring R to study the interplay between the comodules of an R-coalgebra C and the modules of an R-algebra A that admits an R-algebra morphism κ : A → C*. Such pairings, satisfying the so called α-condition, were called in the author's dissertation measuring α-pairings and can be considered as the corner stone in his study of duality theorems for Hopf algebras over commutative rings. In this paper we lay the basis of the theory of rational modules of corings extending results on rational modules for coalgebras to the case of arbitrary ground rings. We apply these results mainly to categories of entwined modules (e.g., Doi-Koppinen modules, alternative Doi-Koppinen modules) generalizing results of Doi, Koppinen, Menini et al.

On the cyclic homology of commutative algebras over arbitrary ground rings

On the cyclic homology of commutative algebras over arbitrary ground rings

Singularities in the Nilpotent Scheme of a Classical Group

If (X, x) is a pointed scheme over a ring k, we introduce a (generalized) partition ord(x, X/k). If G is a reductive group scheme over k, the existence of a nilpotent subscheme N(G) of Lie(G) is discussed. We prove that ord(x, N(G)/k) characterizes the orbits in N(G), their codimension and their adjacency structure, provided that G is Gln, or SPn and 1/2 E k. For SOn only partial results are obtained. We give presentations of some singularities of N(G). Tables for its orbit structure are added. Introduction. Let G be a reductive algebraic group over a field of characteristic p. Let g be its Lie-algebra and N(G) the closed subset of the nilpotent elements of g, cf. [19]. The G-orbits in N(G) are characterized by weighted Dynkin diagrams,cf. [20, III]. Consider the following question. Is it possible to classify the orbits in N(G) using only the local structure of the variety N(G)? We prove in (4.3) that the answer is positive if G is Gln or if G is SPn and p * 2. To this end we introduce a local invariant ord for any pointed scheme in ? 1. We develop the theory of N(G) over an arbitrary ground ring k in ?2. In ?3 we restrict our attention to the classical group schemes. Using a cross section we obtain information about the orbit structure of N(G). Our main theorem (4.2) relates ord(x, N(G)/k) to the Jordan normal form of the nilpotent endomorphism induced by x in the classical representation. This paper is a condensed version of [13]. The author wishes to express his gratitude to his thesis adviser, Professor T. A. Springer. Conventions and notations. The cardinality of a set V is denoted by # V Any infinite cardinal is represented by oo. If x is a real number then [x] is thc greatest integer in x. All rings are commutative with 1. Let M be a module over a ring A. If M is free the rank of M is denoted by rgAM. An element r E A is called M-regular if a: M M is injective. Let a = (a,, .. . , ar) be a Received by the editors March 18, 1975. AMS (MOS) subject classifications (1970). Primary 14B05, 14L15; Secondary 05A17, 10C30, 13H15, 20G35.