Related papers: A Real Nullstellensatz for Free Modules

In this article, we show a new general linear independence criterion related to values of $G$-functions, including the linear independence of values at algebraic points of contiguous hypergeometric functions, which is not known before. Let…

The simplest version of Bertini's irreducibility theorem states that the generic fiber of a non-composite polynomial function is an irreducible hypersurface. The main result of this paper is its analog for a free algebra: if $f$ is a…

For any two complex numbers $a$ and $b$, $\mathcal{V} ir(a,b)$ is a central extension of $\mathcal{W}(a,b)$ which is universal in the case $(a,b)\neq (0,1)$, where $\mathcal{W}(a,b)$ is the Lie algebra with basis $\{L_n,W_n\mid n\in\Z\}$…

Let $R$ be a commutative Noetherian ring and let $n$ be a non-negative integer. In this article, by using the theory of Gorenstein dimensions, it is shown that whenever $R$ is a homomorphic image of a Noetherian Gorenstein ring, then the…

Let $\fa$ be an ideal of a local ring $(R,\fm)$ and $X$ a $d$-dimensional homologically bounded complex of $R$-modules whose all homology modules are finitely generated. We show that $H^d_{\fa}(X)=0$ if and only if $\dim \hat{R}/\fa…

We prove in ZFC that if $G$ is a (right) $R$-module such that the groups $\Hom_R(\prod_{i\in I}G_i,G)$ and $\prod_{i\in I}\Hom_R(G_i,G)$ are naturally isomorphic for all families of $R$-modules $(G_i)_{i\in I}$ then G=0. The result is valid…

We consider local non-Gorenstein rings of the form $(S_i,\mathfrak{n}_i)=k[X, Y_1, \ldots ,Y_i]/\left(X^2, (Y_1, \ldots, Y_i)^2\right), $ where $i\geq 2.$ We show that every totally reflexive $S_i$-module has a presentation matrix of the…

We prove that for any connected open set $\Omega\subset \R^n$ and for any set of matrices $K=\{A_1,A_2,A_3\}\subset M^{m\times n}$, with $m\ge n$ and rank$(A_i-A_j)=n$ for $i\neq j$, there is no non-constant solution $B\in…

Let $(R, \m)$ be a commutative Noetherian local ring with $\m^3 =(0)$. We give a condition for $R$ to have a non-free module of G-dimension zero. We shall also construct a family of non-isomorphic indecomposable modules of G-dimension zero…

Let $\mathbf X$ be a generic alternating matrix, $\mathbf t$ be a generic row vector, and $J$ be the ideal $\operatorname{Pf}_4({\mathbf X})+I_1({\mathbf {t X}})$. We prove that $J$ is a perfect Gorenstein ideal of grade equal to the grade…

Let $R$ be a commutative Noetherian ring, and let $\mathfrak a$ be a proper ideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with the finite projective dimension $p$. Also, let $N$ be a non-zero finitely generated…

Let A be a connected left artinian ring with radical square zero and with n simple modules. If A is not self-injective, then we show that any module M with Ext^i(M,A) = 0 for 1 \le i \le n + 1 is projective. We also determine the structure…

In modern form, Hilbert's Theorem 90 tells us that R^1f_*(G_m)=0, where f is the canonical map between the etale site and the Zariski site of a scheme X. I construct examples showing that the corresponding statement for algebraic spaces…

Let $S$ be the polynomial ring over a field $K$ in a finite set of variables, and let $ \mathfrak{m}$ be the graded maximal ideal of $S$. It is known that for a finitely generated graded $S$-module $M$ and all integers $k\gg 0$, the module…

Positivstellensatz is a fundamental result in real algebraic geometry providing algebraic certificates for positivity of polynomials on semialgebraic sets. In this article Positivstellens\"atze for trace polynomials positive on…

In this work we prove that an entire function $f(z)$ has only negative zeros if and only if its order is strictly less $1$, its root sequence is real-part dominating and there exists an nonnegative integer $m$ the real function…

We study the category of modules admitting compatible actions of the Lie algebra $\mathcal{V}$ of vector fields on an affine space and the algebra $\mathcal{A}$ of polynomial functions. We show that modules in this category which are…

Let $\fa$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\fd_R X$, $\Gfd_R X$ and $\GCfd_RX$ by $\T(X)$. Let $M$ be an $R$-module such that…

Let $K\left\langle X \right\rangle$ denote the free associative algebra generated by a set $X = \{x_1, \dots, x_n\}$ over a field $K$ of characteristic $0$. Let $I_p$, for $p \geq 2$, denote the two-sided ideal in $K\left\langle X…

We prove the following;Theorem:Let R be a prime noetherian ring with k.dimR = n, n a finite non-negative integer. We refer the reader to the definitions (1.1) of this paper.For a fixed non-negative integer m, m<n let Xm be the full set of…