Related papers: Decomposable extensions between rank $1$ modules i…

As a sequel to [Duan-Shu-Yao], we introduce here a category $\mathscr{LC}$ arising from the BGG category $\mathcal{O}$ defined in [Duan-Shu-Yao] for Lie algebras of polynomial vector fields. The objects of $\mathscr{LC}$ are so-called…

Let A be a commutative noetherian ring. Let H(A) be the quotient of the Grothendieck group of finitely generated A-modules by the subgroup generated by pseudo-zero modules. Suppose that the real vector space H(A)_R = H(A) \otimes_Z R has…

For a Galois extension $K/F$ with $\text{char}(K)\neq 2$ and $\text{Gal}(K/F) \simeq \mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$, we determine the $\mathbb{F}_2[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times 2}$. Although…

For every imprimitive complex reflection group of rank 2, we construct a semi-orthogonal decomposition of the derived category of the associated global quotient stack which categorifies the usual decomposition of the orbifold cohomology…

We solve a long standing open problem concerning the structure of finite cycles in the category mod A of finitely generated modules over an arbitrary artin algebra A, that is, the chains of homomorphisms $M_0 \stackrel{f_1}{\rightarrow} M_1…

In this paper we study the derived categories of coherent sheaves on Grassmannians $\operatorname{Gr}(k,n),$ defined over the ring of integers. We prove that the category $D^b(\operatorname{Gr}(k,n))$ has a semi-orthogonal decomposition,…

We study varieties of complexes of projective modules with fixed ranks, and relate these varieties to the varieties of their homologies. We show that for an algebra of global dimension at most two, these two varieties are related by a pair…

We describe new irreducible components of the moduli space of rank $2$ semistable torsion free sheaves on the three-dimensional projective space whose generic point corresponds to non-locally free sheaves whose singular locus is either…

We show that a version of Martin-L\"of type theory with an extensional identity type former I, a unit type N1 , Sigma-types, Pi-types, and a base type is a free category with families (supporting these type formers) both in a 1- and a…

For each Baumslag-Solitar group BS(m,n) (m,n nonzero integers), a totally disconnected, locally compact group, G_{m,n}, is constructed so that BS(m,n) is identified with a dense subgroup of G_{m,n}. The scale function on G_{m,n}, a…

The purpose of this paper is to describe explicitly the modules of (Siegel-)Jacobi forms of degree two of index one of any scalar valued weight with respect to some congruence subgroups of small levels $N\leq 4$. Such a structure for the…

We prove a version of Shelah's Categoricity Conjecture for arbitrary deconstructible classes of modules. Moreover, we show that if $\mathcal{A}$ is a deconstructible class of modules that fits in an abstract elementary class…

We determine, up to isomorphism, the indecomposable maximal Cohen-Macaulay modules over certain complete one-dimensional local rings of finite Cohen-Macaulay type. We then investigate the direct sum relations of maximal Cohen-Macaulay…

A Lefschetz module is a module over a graded algebra $A$ that satisfies analogues of Poincar\'{e} duality, the Hard Lefschetz property, and the Hodge--Riemann relations with respect to an open convex cone $\mathscr{K}$ in the degree one…

In the present paper, we prove that any finite non-trivial irreducible module over a rank two Lie conformal algebra $\mathcal{H}$ is of rank one. We also describe the actions of $\mathcal{H}$ on its finite irreducible modules explicitly.…

Let $\Lambda$ be a finite-dimensional $k$-algebra with $k$ algebraically closed. Bongartz has recently shown that the existence of an indecomposable $\Lambda$-module of length $n > 1$ implies that also indecomposable $\Lambda$-modules of…

Let $G$ be a finite group and $(K,\mathcal{O},k)$ be a $p$-modular system "large enough". Let $R=\mathcal{O}$ or $k$. There is a bijection between the blocks of the group algebra $RG$ and the central primitive idempotents (the blocks) of…

Let R be a ring and G a group. An R-module A is said to be artinian-by-(finite rank) if TorR(A) is artinian and A/TorR(A) has finite R-rank. The authors study ZG-modules A such that A/CA(H) is artinian-by-(finite rank) (as a Z-module) for…

We study the problem of indecomposability of translations of simple modules in the principal block of BGG category $\mathcal{O}$ for $\mathfrak{sl}_n$, as conjectured in \cite{KiM}. We describe some general techniques and prove a few…

Despite being a vast generalization of Garside groups, right $\ell$-groups with noetherian lattice structure and strong order unit share a lot of the properties of Garside groups. In the present work, we prove that every modular noetherian…