We study Gorenstein flat objects in the category Rep(Q,R) of representations a left rooted quiver Q with values Mod(R), all R-modules, where R is an arbitrary associative ring. show that representation X if and only for each vertex i canonical homomorphism φiX:⊕a:j→iX(j)→X(i) injective, R-modules X(i) CokerφiX are flat. As application, we obtain model structure on which give explicit descriptio...

One of the main open questions in liaison theory is whether every homogeneous Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the G-liaison class of a complete intersection. We give an affirmative answer to this question for StanleyReisner ideals defined by simplicial complexes that are weakly vertex-decomposable. This class of complexes includes matroid, shifted and Goren...

We generalize the notions of F-regular and F-pure rings to those for a pair (R, a) of a ring R and an ideal a ⊂ R with real exponent t > 0, and investigate those properties. Those “F-singularities of pairs” correspond to singularities of pairs of arbitrary codimension in birational geometry. Via this correspondence, we prove Inversion of Adjunction of arbitrary codimension, which states that fo...

The Auslander-Reiten conjecture is a notorious open problem about the vanishing of Ext modules. In Cohen-Macaulay complete local ring R with parameter ideal Q, holds for if and only it residue R/Q. former part this paper, we study R/Qℓ in connection that R, prove equivalence them case where Gorenstein ℓ≤dim⁡R. latter part, generalize result minimal multiplicity by J. Sally. Due to these two our...

We further the classification of rational surface singularities. Suppose $(S, \mathfrak{n}, \mathcal{k})$ is a strictly Henselian regular local ring mixed characteristic $(0, p > 5)$. classify functions $f$ for which $S/(f)$ has an isolated singularity at maximal ideal $\mathfrak{n}$. The such are used to show that if $(R, \mathfrak{m}, excellent, Henselian, Gorenstein dimension $2$ and 5)$, th...

An ideal / in a commutative noetherian ring R is a Gorenstein ideal of grade g if pdR(R/I) = grade I = g and the canonical module HxtsR(R/I, R) is cyclic. Serre showed that if g = 2 then / is a complete intersection, and Buchsbaum and Eisenbud proved a structure theorem for the case g = 3. We present generic resolutions for a class of Gorenstein ideals of grade 4, and we illustrate the structur...

Introduction. Let A be a noetherian ring. When A is commutative (of finite Krull dimension), A is said to be Gorenstein if its injective dimension is finite. If A has finite global dimension, one says that A is regular. If A is arbitrary, these hypotheses are not sufficient to obtain similar results to those of the commutative case. To remedy this problem, M. Auslander has introduced a suppleme...

The Goto number of a parameter ideal Q in a Noetherian local ring (R, m) is the largest integer q such that Q : m is integral over Q. The Goto numbers of the monomial parameter ideals of R = k[[x1 , x2 , . . . , xν ]] are characterized using the semigroup of R. This helps in computing them for classes of numerical semigroup rings, as well as on a case by case basis. The minimal Goto number of R...

In this paper, we extend the well-known Hilbert’s syzygy theorem to the Gorenstein homological dimensions of rings. Also, we study the Gorenstein homological dimensions of direct products of rings. Our results generate examples of non-Noetherian rings of finite Gorenstein dimensions and infinite classical weak dimension.