相关论文: Stably free modules of rank $2$ over certain real …

We study the cancellation property of projective modules of rank $2$ with a trivial determinant over Noetherian rings of dimension $\leq 4$. If $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $k$ such that…

We prove that stably free modules of rank d-1 over a smooth affine algebra of dimension d over an algebraically closed field k are free, provided (d-1)! is nonzero in k.

We prove that the stably free modules over a smooth affine threefold over an algebraically closed field of characteristic different from 2 are free.

We prove constructively that for any finite-dimensional commu- tative ring R, every stably free module over R[X;X^{1}] of rank > dim R is free, i.e., R[X;X^{-1}] is (dimR)-Hermite.

For all $k \ge 2$, we show that there exists a group $G$ and a non-free stably free $\mathbb{Z} G$-module of rank $k$. We use this to show that, for all $k \ge 2$, there exist homotopically distinct finite $2$-complexes with fundamental…

We study modules over a generalized Weyl algebra $R(\sigma,a)$ which are free when restricted to the base ring $R$. When $R$ is an integral domain, we construct all such finite-rank modules up to isomorphism, leading to new simple modules…

We provide an explicit classification of all simple $\mathfrak{sl}_2$-modules that are torsion free of rank $1$ over the Cartan subalgebra. We also establish a similar result for the first Weyl algebra and for the Lie superalgebra…

Let $F_m$ be the free group on $m$ generators and let $G$ be a finite nilpotent group of non square-free order; we show that for each $m\ge 2$ the integral group ring ${\bf Z}[G\times F_m]$ has infinitely many stably free modules of rank 1.

Let $A$ be a ring of dimension $d$ containing an infinite field $k$, $T_1,\ldots,T_r$ be variables over $A$ and $P$ be a projective $A[T_1,\ldots,T_r]$-module of rank $n$. Assume one of the following conditions hold. (1) $2n\geq d+3$ and…

Constructive proofs of fact that a stably free left $S$-module $M$ with rank$(M)\geq$sr$(S)$ is free, where sr$(S)$ denotes the stable rank of an arbitrary ring $S$, were developed in some articles. Additionally, in such papers, are…

Let $p\neq 2$, and let $R$ be a smooth affine algebra of dimension $3$ over $\overline{F}_p$ and $P, Q$ be projective $R$-modules of rank $2$, each with trivial determinant. We prove: $P$ is isomorphic to $Q$ if and only if there is an…

We study simple $\mathfrak{sl}(2)$-modules over $\mathbb C$ that are free of finite rank as $U(\mathfrak h)$-modules, where $\mathfrak h$ is a Cartan subalgebra of $\mathfrak{sl}(2)$. Our main result is an explicit classification of the…

Let $(R,\mathfrak{m}_R,k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero. R. Berger conjectured that $R$ is regular if and only if the universally finite module of differentials $\Omega_R$ is…

Let $F$ be a non-archimedean local field of odd residue characteristic $p$. Let $G$ be the unramified unitary group $U(2, 1)(E/F)$ in three variables, and $K$ be a maximal compact open subgroup of $G$. For an irreducible smooth…

In a recent paper [El 13], M.E. Kahoui has shown that if $R$ is a polynomial ring over $\mathbb{C}$, $A$ an $\mathbb{A}^3$-fibration over $R$, and $W$ a residual variable of $A$ then $A$ is stably polynomial over $R[W]$. In this article we…

We show that there exists a stably free module over a polynomial ring which is not extended from the ground ring. This provides a counterexample to the Hermite ring conjecture.

Let $R$ be a commutative ring. One may ask when a general $R$-module $P$ that satisfies $P \oplus R \cong R^n$ has a free summand of a given rank. M. Raynaud translated this question into one about sections of certain maps between Stiefel…

Let $(R,\mm,K)$ be a regular local ring containing a field $k$ such that either char $k=0$ or char $k=p$ and tr-deg $K/\BF_p\geq 1$. Let $g_1,\ldots,g_t$ be regular parameters of $R$ which are linearly independent modulo $\mm^2$. Let…

We study two categories of ${U}(\mathfrak h)$-free $\mathfrak{sl}(m|n)$-modules of total rank 2: $\mathcal{M}_{\mathfrak{sl}(m|n)}(2)$, whose objects are free of rank 2 over ${U}(\mathfrak h)$ which are not necessarily $\mathbb Z_2$-graded,…

It is an open problem of Mazari-Armida whether every abstract elementary class of $R$-modules $(\mathbf{K}, \leq_{\mathrm{pure}})$, with $\leq_{\mathrm{pure}}$ the pure submodule relation, is stable. We answer this question in the negative…