We construct bar-invariant $\mathbb{Z}\left[q^{\pm 1/2}\right]$-bases of the quantum cluster algebra of the valued quiver $A_2^{\left(2\right)}$, one of which coincides with the quantum analogue of the basis of the corresponding cluster algebra discussed in P. Sherman, A. Zelevinsky: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., 4, 2004, 947–974.

Let $A$ and $B$ be commutative rings with identity. An $A$-$B$-biring is an $A$-algebra $S$ together with a lift of the functor ${Hom}_A(S,-)$ from $A$-algebras to sets to a functor from $A$-algebras to $B$-algebras. An $A$-plethory is a monoid object in the monoidal category, equipped with the composition product, of $A$-$A$-birings. The polynomial ring $A\left[X\right]$ is an initial object in the category of such structures. The $D$-algebra $Int\left(D\right)$ has such a structure if $D=A$ is a domain such that the natural $D$-algebra homomorphism ${\theta }_n:{{⨂}_D}_{i=1}^{n}Int\left(D\right)\to Int\left(D^n\right)$ is an isomorphism for...

Nous montrons dans cet article des bornes pour la régularité de Castelnuovo-Mumford d’un schéma admettant des singularités, en fonction des degrés des équations définissant le schéma, de sa dimension et de la dimension de son lieu singulier. Dans le cas où les singularités sont isolées, nous améliorons la borne fournie par Chardin et Ulrich et dans le cas général, nous établissons une borne doublement exponentielle en la dimension du lieu singulier.

We investigate factorization of elements in overrings of a half-factorial domain $A$ in relation with the behaviour of the boundary map of $A$. It turns out that a condition, called ${\mathcal{C}}^{\star }$, on the extension plays a central role in this study. We finally apply our results to the special case of $A+XB\left[X\right]$ polynomial rings.

In rings ${\Gamma }_M$ of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence $M={\left(M_l\right)}_{l\ge 0}$ (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in ${\Gamma }_M$ such that F is divisible by Φ in the usual ring of all power series. We give first a simple proof of the fact that F/Φ belongs also to ${\Gamma }_M$, provided ${\Gamma }_M$ is stable under derivation. By a further development of the method, we obtain the main result of the paper, stating that...