This paper has two related parts. The first generalizes Hochster’s formula on resolutions of Stanley–Reisner rings to a colorful version, applicable to any proper vertex-coloring of a simplicial complex. The second part examines a universal system of parameters for Stanley–Reisner rings of simplicial complexes, and more generally, face rings of simplicial posets. These parameters have good properties, including being fixed under symmetries, and detecting depth of the face ring. Moreover, when resolving the face ring over these parameters, the shape is predicted, conjecturally, by the colorful Hochster formula.

We introduce and develop the theory of weakly Arf rings, generalizing the Arf rings originally defined by J. Lipman in 1971. We provide characterizations of the weakly Arf rings and study their relation with the Arf rings and the strictly closed rings. Furthermore, we give various examples of weakly Arf rings that come from idealizations, fiber products, determinantal rings, and invariant subrings.

We set $n\ge 2$ and $d\ge 2$. Let $X$ be the Fermat hypersurface of degree $d$ in the projective space ${\mathbb{P}}^n$. It is known that the automorphism group of $X$ contains a subgroup which is isomorphic to $\mathbb{Z}∕d{\mathbb{Z}}^{\oplus n}$. Let $G$ be a subgroup of the projective linear group $\text{PGL}(n+1,ℂ)$ such that $G$ is isomorphic to $\mathbb{Z}∕d{\mathbb{Z}}^{\oplus n}$. We show that $G$ can be lifted into a subgroup of the general linear group $\text{GL}(n+1,ℂ)$ except for the cases where $(n,d)=(2,3)$ and $(3,2)$. As a result, we show that for a smooth hypersurface $Y$ in ${\mathbb{P}}^n$ of degree $d$, if $\mathbb{Z}∕d{\mathbb{Z}}^{\oplus n}$ acts faithfully on $Y$, then $Y$ is the Fermat hypersurface unless $(n,d)=(2,3)$ and $(3,4)$.

For a valued field $(K,v)$ of arbitrary rank and an extension $w$ of $v$ to $K(X)$, we give a connection between complete sets of abstract key polynomails for $w$ and Mac Lane–Vaquié chains of $w$.

The Tjurina ideal of a germ of a holomorphic function $f$ is the ideal of ${\mathcal{𝒪}}_{{ℂ}^n,0}$ — the ring of those germs at $0\in {ℂ}^n$ — generated by $f$ itself and by its first-order partial derivatives. Here it is denoted by $T(f)$. The ideal $T(f)$ gives the structure of closed subscheme of $({ℂ}^n,0)$ to the singular set of the hypersurface defined by $f$, being an object of central interest in singularity theory. In this note we introduce $T$-fullness and $T$-dependence, two easily verifiable properties for arbitrary ideals of germs of holomorphic functions. These two properties allow us to give necessary and sufficient conditions on an ideal $I\subset {\mathcal{𝒪}}_{{ℂ}^n,0}$ for the equation $I=T(f)$ to admit a solution $f$. As a result we characterize closed subschemes of $({ℂ}^n,0)$ arising as singularities of germs of hypersurfaces.

Given that symbolic and ordinary powers of an ideal do not always coincide, we look for conditions on the ideal such that the equality holds for every natural number. This paper focuses on studying the equality for Derksen ideals defined by finite groups acting linearly on a polynomial ring.

A commutative ring with unity is called an $N$-ring if each of its ideals is contracted from a Noetherian extension ring. The chief result of this paper is a characterisation of local $N$-rings by their subdirectly irreducible quotients. The results are applied to characterise spectral synthesis on $G$-invariant subspaces of the space of complex valued functions on an abelian group $G$.