Recently, N. Epstein and J. Shapiro introduced and studied the perinormal domains: those domains $A$ whose overrings satisfying going down over $A$ are flat $A$-modules. We show that every Prufer v-multiplication domain is perinormal and has no proper lying over overrings. Conversely, we show that a w-treed perinormal domain is a Prufer v-multiplication domain. We give two pull-back constructions that produce perinormal/non-perinormal domains.

A special case of the Dolgachev-Weisfeiler conjecture asserts that residual coordinates of the polynomial algebra $A =\mathbb{C} [x]^{[n]}$, $n\geq 3$, are coordinates. It is well known that such polynomials are stable coordinates; however, all the examples constructed thus far are actually 1-stable coordinates. In this paper, we show that all residual coordinates of $A $ are 1-stable coordinates.

Let $R$ be a commutative Noetherian local ring with residue field $k$. We show that, if a finite direct sum of syzygy modules of $k$ maps onto `a semidualizing module' or `a non-zero maximal Cohen-Macaulay module of finite injective dimension,' then $R$ is regular. We also prove that $R$ is regular if and only if some syzygy module of $k$ has a non-zero direct summand of finite injective dimension.

We extend a result of Caviglia and Sbarra to a polynomial ring with base field of any characteristic. Given a homogeneous ideal containing both a piecewise lex ideal and an ideal generated by powers of the variables, we find a lex ideal with the following property: the ideal in the polynomial ring generated by the piecewise lex ideal, the ideal of powers and the lex ideal has the same Hilbert function and Betti numbers at least as large as those of the original ideal.

If $U$ is a UFD, then there is a PID $P$ containing $U$ that has the same unit group as $U$. Moreover, $P$ can be taken so that its field of fractions is a pure transcendental extension of the field of fractions of $U$ with trans-cendence degree at most $|U|$.

(1) Let $R$ be a commutative Noetherian ring of dimension $n$ and $P$ a projective $R[X_1,\ldots ,X_m]$-module of rank $n$. In this paper, we associate an obstruction for $P$ to split off a free summand of rank one. (2) Let $R$ be a local ring and $R[X]\subset A\subset R[X,X^{-1}]$. Let $P$ and $Q$ be two projective $A$-modules with $\text {rank}(Q)\lt \text {rank}(P)$. If $Q_f$ is a direct summand of $P_f$ for some special monic polynomial $f\in R[X]$, then $Q$ is also a direct summand of $P$.

Let $G$ be a semisimple algebraic group defined over an algebraically closed field of characteristic 0 and $P$ a parabolic subgroup of $G$. Let $M$ be a $P$-module and $V$ a $P$-stable closed subvariety of $M$. We show in this paper that, if the varieties $V$ and $G\cdot M$ have rational singularities, and the induction functor $R^i ind _P^G(-)$ satisfies certain vanishing conditions, then the variety $G\cdot V$ has rational singularities. This generalizes a result of Kempf on the collapsing of homogeneous bundles. As an application, we prove the property of having rational singularities for nilpotent commuting varieties over $3\times 3$ matrices.

Let $(R,\mathfrak{m} )$ be an unmixed Noetherian local ring, $Q$ a parameter ideal and $K$ an $\mathfrak{m} $-primary ideal of $R$ containing $Q$. We give a necessary and sufficient condition for $R$ to be Cohen-Macaulay in terms of $g_0(Q)$ and $g_1(Q)$, the Hilbert coefficients of $Q$ with respect to $K$. As a consequence, we obtain a result of Ghezzi, et al., which settles the negativity conjecture of Vasconcelos {vanishing-conjecture} in unmixed local rings.

Let $A$ be a commutative Noetherian ring, and let $I$ be an ideal of $A[T]$ containing a quasi-monic polynomial. Assuming that $I/I^2$ is generated by $n$ elements, where $n\geq \dim (A[T]/I)+2$, then, it is proven that any given set of $n$ generators of $I/I^2$ can be lifted to a set of $n$ generators of $I$. It is also shown that various types of Horrocks' type results (previously proven for monic polynomials) can be generalized to the setting of quasi-monic polynomials.

For any commutative ring $R$ of Krull dimension zero and for any discrete Hodge algebra $D$ over $R$, it is proven that, if $n\geq 3$, the group $E_n(D)$ of $n\times n$ elementary matrices acts transitively on $Um_n(D)$, the set of unimodular rows of length $n$ over $D$.