We describe a step toward quantizing deformation theory. The L∞ operad is encoded in a Hochschild cocyle ◦1 in a simple universal algebra (P, ◦0). This Hochschild cocyle can be extended naturally to a star product ⋆ = ◦0+~◦1+~ 2 ◦2+ · · · . The algebraic structure encoded in ⋆ is the properad Ω(coFrob) which, conjecturally, controls a quantization of deformation theory—a theory for which Froben...

The first section of this paper defines and studies a graded ring K . F associated to any field F. By definition, K~F is the target group of the universal n-linear function from F ~ x ... • F ~ to an additive group, satisfying the condition that al • " ' x a, should map to zero whenever a i -q-a i + ~ = 1. Here F ~ denotes the multiplicative group F 0 . Section 2 constructs a homomorphism ~: K,...

Quillen [7] defines an algebraic A -̂functor from the category of associative rings to that of positively graded abelian groups, with Kf(R) = ÏI^BGLR ") for i > 1. Kx and K2 correspond respectively to the 'classical' Bass and Milnor definitions. The /^-images of finite fields and their algebraic closures were computed by Quillen in [8]. Since then, there has been only a handful of complete calcu...

Let D be an integral domain with quotient field K. The Nagata ring D(X) and the Kronecker function ring Kr(D) are both subrings of the field of rational functions K(X) containing as a subring the ring D[X] of polynomials in the variable X. Both of these function rings have been extensively studied and generalized. The principal interest in these two extensions ofD lies in the reflection of vari...

In the title compound, C(20)H(15)F(6)N(3)O(2), the quinoline ring system is almost coplanar with the benzene ring; the dihedral angle between the two planes is 2.31 (8)°. The crystal structure displays an inter-molecular C-H⋯F hydrogen bond. In addition, a weak π-π inter-action is observed between the unfused benzene ring and the benzene ring of quinoline, with a centroid-centroid distance of 3...

In an earlier paper a construction was given for an infinite-dimensional uniserial module over Q for SL(2,Z) whose composition factors are all isomorphic to the standard (two-dimensional) module. In this note we consider generalizations of this construction to other composition factors and to other rings of algebraic integers.

In this course, we will cover the basics of what is called algebraic number theory. Just as number theory is often described as the study of the integers, algebraic number theory may be loosely described as the study of certain subrings of fields K with [K : Q] < ∞; these rings tend to act as natural generalizations of the integers. However, although algebraic number theory has evolved into a s...

Generalizing the concepts of Stanley–Reisner and affine monoid algebras, one can associate to a rational pointed fan Σ in Rd the Zd-graded toric face ring K[Σ]. Assuming that K[Σ] is Cohen–Macaulay, the main result of this paper is to characterize the situation when its canonical module is isomorphic to a Zd-graded ideal of K[Σ]. From this result several algebraic and combinatorial consequences...

We prove a differential analog of a theorem of Chevalley on extending homomorphisms for rings with commuting derivations, generalizing a theorem of Kac. As a corollary, we establish that, under suitable hypotheses, the image of a differential scheme under a finite morphism is a constructible set. We also obtain a new algebraic characterization of differentially closed fields. We show that simil...

In this work we introduce generalized projective geometries which are a natural generalization of projective geometries over a field or ring K but also of other important geometries such as Grassmannian, Lagrangian or conformal geometry (see [3]). We also introduce the corresponding generalized polar geometries and associate to such a geometry a symmetric space over K. In the finite-dimensional...