The weakly Arf $(S_2)$-ification of a commutative Noetherian ring $R$ is considered to be a birational extension which is good next to the normalization. The weakly Arf property (WAP for short) of $R$ was introduced in 1971 by J. Lipman (and recently rediscovered), being closely explored with further developments. The present paper aims at constructing, for a given Noetherian ring $R$ which satisfies certain mild conditions, the smallest module-finite birational extension of $R$ which satisfies WAP and the condition $(S_2)$ of Serre. We call this extension the weakly Arf $(S_2)$-ification, and develop the basic theory, including some existence theorems.

We extend the results of the standard monomial theory to the cotangent variety of the Grassmannian, $T^{\ast }\text{Grass}(k,n)$. We exhibit a standard basis in terms of triples of tableaux. We are also able to show that $T^{\ast }\text{Grass}(k,n)$ is arithmetically Cohen–Macaulay and normal in all characteristics. Results on Cohen–Macaulayness and normality are shown for other algebraic groups and cominuscule parabolic subgroups, but only in characteristic zero.

This study investigates the structure of Arf rings. From the perspective of ring extensions, a decomposition of integrally closed ideals is given. Using this, we present a kind of their prime ideal decomposition in Arf rings, and determine their structure in the case where both $R$ and the integral closure $\overline{R}$ are local rings.

This work concerns the Koszul complex $K$ of a commutative noetherian local ring $R$, with its natural structure as a differential graded $R$-algebra. It is proved that under diverse conditions, involving the multiplicative structure of $H(K)$, any $\text{dg }R$-algebra automorphism of $K$ induces the identity map on $H(K)$. In such cases, it is possible to define an action of the automorphism group of $R$ on $H(K)$. On the other hand, numerous rings are described for which $K$ has automorphisms that do not induce the identity on $H(K)$. For any $R$, it is shown that the group of automorphisms of $H(K)$ induced by automorphisms of $K$ is abelian.

We study conditions on a polynomial such that the ideal generated by its orbit under the symmetric group action is a monomial ideal or has a monomial radical. If the polynomial is homogeneous, we expect the ideal to have a monomial radical if the coefficients are sufficiently general with respect to the support of the polynomial. We prove this in the case of a symmetric support set in sufficiently many variables over characteristic zero. If in addition the polynomial has only square-free terms and its coefficients do not sum to zero, then in a larger polynomial ring the ideal itself is square-free monomial. This has implications also for symmetric ideals of the infinite polynomial ring.

We consider rational double point singularities (RDPs) that are nontaut, which means that the isomorphism class is not uniquely determined from the dual graph of the minimal resolution. Such RDPs exist in characteristic $2,3$, and $5$. We compute the actions of Frobenius and other inseparable morphisms on $W_n$-valued local cohomology groups of RDPs. Then we consider RDP K3 surfaces admitting nontaut RDPs. We show that the height of the K3 surface, which is also defined in terms of the Frobenius action on $W_n$-valued cohomology groups, is related to the isomorphism class of the RDP.

Let $R$ be a commutative Noetherian ring, $a$ an element of $R$, and $M$ a nonzero finitely generated $R$-module. We study the nonvanishing of ${\text{Ext }}_R^1(R∕Ra,M)$ and we give some conditions which guarantee that this is the case. In particular, we show that over a Noetherian local ring $(R,\mathfrak{𝔪})$, for an ideal $\mathfrak{𝔟}$ of $R$, if $\mu (\mathfrak{𝔪}∕\mathfrak{𝔟})\le \text{dim}(R∕\mathfrak{𝔟})+1$, then for each $a\in \mathfrak{𝔪}\setminus {\bigcup }_{\mathfrak{𝔭}\in {\text{mAss}}_R(R∕\mathfrak{𝔟})}\mathfrak{𝔭}$, we have ${\text{Ext}}_R^1(R∕Ra,R∕\mathfrak{𝔟})\ne 0$.

We prove that if a local ring admits a pd-test module of finite complete intersection dimension, then it is a complete intersection ring. This answers, positively, a question proposed by Celikbas, Dao and Takahashi. To this aim, we first investigate another question raised by Celikbas and Sather-Wagstaff concerning ascent properties of pd-test complexes under weakly regular homomorphisms.