Related papers: Cancellation problem for projective modules over a…

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…

In this article, we prove that if $R$ is an affine algebra of dimension $d\geq 4$ over $\overline{\mathbb{F}}_{p}$ and $1/(d-1)! \in R,$ then any unimodular row over $R$ of length $d$ can be mapped to a factorial row by elementary…

Let $A$ denote an affine algebra over an algebraically closed field $k$, with $\dim A=d\geq 3$. In the light of availability of cancellation theorems for stably free modules $P$ with $rank(P)=d-1$ (corank one), we try to implement the…

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 prove a descent result for finite projective modules, motivated by a question in perfectoid geometry. Given a commutative ring $A$, we formulate a descent problem for descending a finite projective module over the Novikov ring with…

If $n \equiv 0,1~mod~4$, we prove a sum formula $V_{\theta_{0}} (a_{0},a_{R}^{n}) = n \cdot V_{\theta_{0}} (a_{0},a_{R})$ for the generalized Vaserstein symbol whenever $R$ is a smooth affine algebra over a perfect field $k$ with $char(k)…

We investigate the "theta-deformed spheres" C(S^{3}_{theta}) and C(S^{4}_{theta}), where theta is any real number. We show that all finitely-generated projective modules over C(S^{3}_{theta}) are free, and that C(S^{4}_{theta}) has the…

We have proved the following Problem:{\it Let $R$ be a $\mathbb{C}$-affine domain, let $T$ be an element in $R \setminus \mathbb{C}$ and let $i : \mathbb{C}[T] \hookrightarrow R$ be the inclusion. Assume that $R/TR \cong_{\mathbb{C}}…

The themes of cancellation, internal cancellation, substitution have led to a lot of interesting research in the theory of modules over commutative and noncommutative rings. In this paper, we introduce and study cancellation problem in the…

Let R be a local ring of dimension d. Buchweitz asks if the rank of the d-th syzygy of a module of finite lengh is greater than or equal to the rank of the d-th syzygy of the residue field, unless the module has finite projective dimension.…

The long-standing Auslander and Reiten Conjecture states that a finitely generated module over a finite-dimensional algebra is projective if certain Ext-groups vanish. Several authors, including Avramov, Buchweitz, Iyengar, Jorgensen,…

It's well known that the functional Hilbert space over the unit ball in $B_{d} \in C^d$, with kernel function $K(z,w)=\frac{1}{1-z_{1}w_{1}-... -z_{d}w_{d}}$, admits a natural $A(B_{d})$-module structure. We show the rank of a nonzero…

Given a commutative Noetherian graded domain $R = \bigoplus_{i\ge 0} R_i$ of dimension $d\geq 2$ with $\dim(R_0) \geq 1$, we prove that any unimodular row of length $d+1$ in $R$ can be completed to the first row of an invertible matrix…

Let $R$ be a commutative Noetherian ring of dimension $d$, $M$ a commutative cancellative torsion-free monoid of rank $r$ and $P$ a finitely generated projective $R[M]$-module of rank $t$. $(1)$ Assume $M$ is $\Phi$-simplicial seminormal.…

Let $k$ be a field and let $A$ be a finitely generated $k$-algebra. The algebra $A$ is said to be cancellative if whenever $B$ is another $k$-algebra with the property that $A[x]\cong B[x]$ then we necessarily have $A\cong B$. An important…

We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. We investigate how various properties of a C-module are affected when considered in the module category of B. We give a complete…

Let B be a unital C*-algebra, let A be a unital subalgebra, and let E be a conditional expectation from B to A with index-finite type and a quasi-basis of n elements. Then the topological stable rank satisfies \tsr (B) \leq \tsr (A) + n -…

We call a finite-dimensional K-algebra A geometrically irreducible if for all d, all connected components of the affine scheme of d-dimensional A-modules are irreducible. We show that the geometrically irreducible algebras without loops…

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 expand the existing arsenal of methods for exploring the irreducible components of the varieties $Rep(A,\bold d)$ which parametrize the representations with dimension vector $\bold d$ of a finite dimensional algebra $A$. To do so, we…