Related papers: A note on a sheaf of real regular functions

The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudo-reflection groups over regular domains. More precisely, let $A$ be a regular domain and let $K$ be its field of…

In the first chapter, we will present a computation of the square value of the module of L functions associated to a Dirichlet character. This computation suggests to ask if a certain ring of arithmetic multiplicative functions exists and…

Let $ A$ be a complex unital Banach algebra. An element $a \in A$ is said to be Hermitian, if $ \| \exp (ita) \| =1$ for all $t\in R$. In the case of the algebra of bounded linear operators in a Hilbert space this Hermitian property agrees…

Let $M$ be a left module over a ring $R$ and $I$ an ideal of $R$. $M$ is called an $I$-supplemented module (finitely $I$-supplemented module) if for every submodule (finitely generated submodule) $X$ of $M$, there is a submodule $Y$ of $M$…

The Bott-Thurston cocycle is a $2$-cocycle on the group of orientation-preserving diffeomorphisms of the circle. We introduce and study a formal analog of Bott-Thurston cocycle. The formal Bott-Thurston cocycle is a $2$-cocycle on the group…

Let $M$ be a left module over a ring $R$ and $I$ an ideal of $R$. We call $(P, f)$ a (locally)projective $I$-cover of $M$ if $f$ is an epimorphism from $P$ to $M$, $P$ is (locally)projective, $Kerf\subseteq IP$, and whenever $P=Kerf+X$,…

Let $(A,\mathfrak{m})$ be a complete equicharacteristic Noetherian domain of dimension $d + 1 \geq 2$. Assume $k = A/\mathfrak{m}$ has characteristic zero and that $A$ is not a regular local ring. Let $Sing(A)$ the singular locus of $A$ be…

Let $\cal R$ be either the Grothendieck semiring (semiring with multiplication) of complex algebraic varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class of the complex affine line. We…

We present, in the same vein as in [20] and [21], some results of the so-called "Smooth (or $\mathcal{C}^\infty$) Commutative Algebra", a version of Commutative Algebra of $\mathcal{C}^{\infty}-$rings instead of ordinary commutative unital…

A generic computation of a subset $A$ of $\mathbb{N}$ is a computation which correctly computes most of the bits of $A$, but which potentially does not halt on all inputs. The motivation for this concept is derived from complexity theory,…

Since a Poisson Structure is a smooth bivector field, we use a ring-valued sheaf $\OO_{X}$ on a manifold with corners $X$, we can interpret $\OO_{X}(U)$ as the ring of admissible smooth functions where $U$ is an open subset on $X$, in this…

In [TV], Bertrand To\"en and Michel Vaqui\'e define a scheme theory for a closed monoidal category $(\mathcal{C},\otimes,1)$. One of the key ingredients of this theory is the definition of a Zariski topology on the category of commutative…

The main goal of this work is classifying the singularities of slice regular functions over a real alternative *-algebra A. This function theory has been introduced in 2011 as a higher-dimensional generalization of the classical theory of…

We prove, with no claim to originality, a relative version of the Fujita-Zariski theorem. When the base is a field, this result is due to Fujita (1983) and states that if an invertible sheaf on a proper variety is ample on its base locus,…

A number of spectrum constructions have been devised to extract topological spaces from algebraic data. Prominent examples include the Zariski spectrum of a commutative ring, the Stone spectrum of a bounded distributive lattice, the Gelfand…

Numerical semigroup rings are investigated from the relative viewpoint. It is known that algebraic properties such as singularities of a numerical semigroup ring are properties of a flat numerical semigroup algebra. In this paper, we show…

Let $X\subset{\mathbb R}^n$ be a (global) real analytic surface. Then every positive semidefinite meromorphic function on $X$ is a sum of $10$ squares of meromorphic functions on $X$. As a consequence, we provide a real Nullstellensatz for…

In this paper we study certain sheaves of $p$-adically complete rings of differential operators on semistable models of the projective line over the ring of integers in a finite extension $L$ of ${\mathbb Q}_p$. The global sections of these…

We say that a formal power series $\sum a_n z^n$ with rational coefficients is a 2-function if the numerator of the fraction $a_{n/p}-p^2 a_n$ is divisible by $p^2$ for every prime number $p$. One can prove that 2-functions with rational…

Let $\mathbf{F}_q$ be a finite field of $q$ elements. For multiplicative characters $\chi_1,\dots, \chi_m$ of $\mathbf{F}_q^\times$, we let $J(\chi_1,\dots, \chi_m)$ denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for…