We identify the *sl(n*+1) isotypical components of the global Weyl modules *W*(*k**ω*1) with certain natural subspaces of the polynomial ring in *k* variables. We then apply the representation theory of current algebras to classical problems in invariant theory.

We apply Orlov's equivalence to derive formulas for the Betti numbers of maximal Cohen-Macaulay modules over the cone an elliptic curve $(E,x)$ embedded into $\mathbb{P}^{n-1}$, by the full linear system $|\mathcal{O}(nx)|$, for $n>3$. The answers are given in terms of recursive sequences. These results are applied to give a criterion of (Co-)Koszulity.

In the last two sections of the paper we apply our methods to study the cases $n=1,2$\,. Geometrically these correspond to the embedding of an elliptic curve into a weighted projective space. The singularities of the corresponding cones are called minimal elliptic. They were studied by K.Saito \cite{Saito74}, where he introduced the notation $\widetilde{E_8}$ for $n=1$, $\widetilde{E_7}$ for $n=2$ and $\widetilde{E_6}$ for the cone over a smooth cubic, that is, for the case $n=3$\,. For the singularities $\widetilde{E_7}$ and $\widetilde{E_8}$ we obtain formulas for the Betti numbers and the numerical invariants of MCM modules analogous to the case of a plane cubic.

Triangular decomposition is one of the standard ways to represent the radical of a polynomial ideal. A general algorithm for computing such a decomposition was proposed by A. Szántó. In this paper, we give the first complete bounds for the degrees of the polynomials and the number of components in the output of the algorithm, providing explicit formulas for these bounds.

Kronecker's Theorem and Rabin's Theorem are fundamental results about computable fields *F* and the decidability of the set of irreducible polynomials over *F*. We adapt these theorems to the setting of differential fields *K*, with constrained pairs of differential polynomials over *K* assuming the role of the irreducible polynomials. We prove that two of the three basic aspects of Kronecker's Theorem remain true here, and that the reducibility in one direction (but not the other) from Rabin's Theorem also continues to hold.

We prove general results about completeness of cotorsion theories and existence of covers and envelopes in locally presentable abelian categories, extending the well-established theory for module categories and Grothendieck categories. These results are then applied to the categories of contramodules over topological rings, which provide examples and counterexamples.

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of afinitely generated algebra of invariants. As an application, we provide a combinatorial GIT-type construction of categorical quotients for actions of not necessarily reductive groups on, e.g. complete varieties with finitely generated Cox ring via lifting to the characteristic space

A ring R is left exact if, for every finitely generated left submodule S⊂Rn, every left R-linear function from S to R extends to a left R-linear function from R^n to R. The class of exact rings generalizes that of self-injective rings and has been introduced in a recent paper by Wilding, Johnson, and Kambites. In our paper we show that the group ring of a group G over a ring R is left exact if and only if R is left exact and G is locally finite.

We give explicit realizations of irreducible representations of the Yangian of the general linear Lie algebra and of its twisted analogues, corresponding to symplectic and orthogonal Lie algebras. In particular, we develop the fusion procedure for twisted Yangians. For the non-twisted Yangian, this procedure goes back to the works of Cherednik.

This paper is a greatly expanded version of [37, Section 9.11]. A series of definitions and results illustrating the thesis in the title (where quasi-formality means vanishing of a certain kind of Massey multiplications in the cohomology) is presented. In particular, we include a categorical interpretation of the “Koszulity implies K(π,1)” claim, discuss the differences between two versions of Massey operations, and apply the derived nonhomogeneous Koszul duality theory in order to deduce the main theorem. In the end we demonstrate a counterexample providing a negative answer to a question of Hopkins and Wickelgren about formality of the cochain DG-algebras of absolute Galois groups, thus showing that quasi-formality cannot be strengthened to formality in the title assertion.

We call a graded connected algebra R effectively coherent, if for every linear equation over R with homogeneous coefficients of degrees at most d, the degrees of generators of its module of solutions are bounded by some function D(d). For commutative polynomial rings, this property has been established by Hermann in 1926. We establish the same property for several classes of noncommutative algebras, including the most common class of rings in noncommutative projective geometry, that is, strongly Noetherian rings, which includes Noetherian PI algebras and Sklyanin algebras. We extensively study so-called universally coherent algebras, that is, such that the function D(d) is bounded by 2d for d≫0. For example, finitely presented monomial algebras belong to this class, as well as many algebras with finite Groebner basis of relations.

In this paper we investigate the connection between the Mac Lane (co)homology and Wieferich primes in finite localizations of global number rings. Following the ideas of Polishchuk–Positselski [29], we define the Mac Lane (co)homology of the second kind of an associative ring with a central element. We compute these invariants for finite localizations of global number rings with an element w and obtain that the result is closely related to the Wieferich primes to the base w. In particular, for a given non-zero integer w, the infiniteness of Wieferich primes to the base w turns out to be equivalent to the following: for any positive integer n , we have View the MathML source. As an application of our technique, we identify the ring structure on the Mac Lane cohomology of a global number ring and compute the Adams operations (introduced in this case by McCarthy [26]) on its Mac Lane homology.

Given an operad P with a finite Gröbner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dimP(n)} is differential algebraic, and in fact algebraic if P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions of P(n) is bounded by an exponent of n(or a polynomial of n, in the non-symmetric case) then, moreover, the ordinary generating function for the above sequence {dimP(n)} is rational. We give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads.

Let K be a ﬁeld and A be a commutative associative K-algebra which is an integral domain. The Lie algebra DerA of all K-derivations of A is an A-module in a natural way and if R is the quotient ﬁeld of A then RDerA is a vector space over R. It is proved that if L is a nilpotent subalgebra of RDerA of rank k over R (i.e. such that dimR RL = k), then the derived length of L is at most k and L is ﬁnite dimensional over its ﬁeld of constants. In case of solvable Lie algebras over a ﬁeld of characteristic zero their derived length does not exceed 2k. Nilpotent and solvable Lie algebras of rank 1 and 2 (over R) from the Lie algebra RDerA are characterized. As a consequence we obtain the same estimations for nilpotent and solvable Lie algebras of vector ﬁelds with polynomial, rational, or formal coeﬃcients.

We consider three types of rings of supersymmetric poly-nomials: polynomial ones Λm,n, partially polynomial Λ+ym,nand Laurent supersymmetric rings Λ±m,n. For each type of rings we give their descriptions in terms of generators and relations. As a corollary we get for n ≥man isomorphism Λ+ym,n=Λ+ym,m⊗Λ+y0,n−m. It is also true for polynomial rings, but in this case the isomorphism does not preserve the grading. For each type of rings some natural basis consisting of Euler supercharacters is constructed.