Related papers: A local ring such that the map between Grothendiec…

The concept of Faltings' local-global principle for the in dimension $< n$ of local cohomology modules over a Noetherian ring $R$ is introduced, and it is shown that this principle holds at levels 1, 2. We also establish the same principle…

Let $\mathfrak{q}$ be an ideal of a Noetherian local ring $(A,\mathfrak{m})$ and $M$ a non-zero finitely generated $A$-module. We present a criterion of Cohen-Macaulayness of the form module $G_M(\mathfrak{q})$ in terms of (non-)vanishing…

Let R be a connected noetherian commutative ring, and let G be a simply connected reductive group over R of isotropic rank ge 2. The elementary subgroup E(R) of G(R) is the subgroup generated by the R-points U_P^+(R) and U_P^-(R) of the…

Given a Noetherian ring $A$, the collection of all integrally closed ideals in $A$ which contain a nonzerodivisor, denoted $ic(A)$, forms a cancellative monoid under the operation $I*J=\overline{IJ}$, the integral closure of the product.…

Let R be a regular local ring, containing an infinite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R.

Finitely generated reflexive modules over commutative Noetherian rings form a key component of Auslander and Bridger's stable module theory and are likewise essential in the study of Cohen--Macaulay representations. Recently, H. Dao…

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $\widehat{R}$ its $\mathfrak{m}$-adic completion. We study the problem of determining when a finitely generated $\widehat{R}$-module arises from an $R$-module, i.e., when it is…

A semidualizing module is a generalization of Grothendieck's dualizing module. For a local Cohen-Macaulay ring $R$, the ring itself and its canonical module are always realized as (trivial) semidualizing modules. Reasonably, one might…

We study Artin algebras $A$ and commutative Noetherian complete local rings $R$ in connection with the following decomposition property of Gorenstein-projective modules: $(*)$ any Gorenstein-projective module is a direct sum of finitely…

Let $(Q, \mathfrak{n})$ be a regular local ring and let $f_1, \ldots, f_c \in \mathfrak{n}^2$ be a $Q$-regular sequence. Set $(A, \mathfrak{m}) = (Q/(\mathbf{f}), \mathfrak{n}/(\mathbf{f}))$. Further assume that the initial forms $f_1^*,…

We develop a theory of G-dimension for modules over local homomorphisms which encompasses the classical theory of G-dimension for finite modules over local rings. As an application, we prove that a local ring R of characteristic p is…

This paper considers the following conjecture: If $R$ is an unmixed, equidimensional local ring that is a homomorphic image of a Cohen-Macaulay local ring, then for any ideal $J$ generated by a system of parameters, the Chern coefficient…

Let $(R,\mathfrak{m},k)$ denote a local ring. For $I$ and $J$ ideals of $R$, for all integer $i$, let $H^i_{I,J}(-)$ denote the $i$-th local cohomology functor with respect to $(I,J)$. Here we give a generalized version of Local Duality…

Applying the authors' preceding work, we construct a version of the moduli space of $G$-torsors over the formal punctured disk for a finite group $G$. To do so, we introduce two Grothendieck topologies, the sur (surjective) and luin…

We define a Grothendieck ring for basic real semialgebraic formulas, that is for systems of real algebraic equations and inequalities. In this ring the class of a formula takes into consideration the algebraic nature of the set of points…

This paper introduces two new Burnside rings for a finite group $G$, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of $G$-sets, and of Galois morphisms of…

We prove that the Farrell-Jones assembly map for connective algebraic K-theory is rationally injective, under mild homological finiteness conditions on the group and assuming that a weak version of the Leopoldt-Schneider conjecture holds…

Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space…

In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring (pure spectrum) are obtained. Especially, Grothendieck type theorem is obtained which states that there is a canonical correspondence between…

Let $R$ be a Gorenstein local ring with maximal ideal $\mathfrak{m}$ satisfying $\mathfrak{m}^3=0\ne\mathfrak{m}^2$. Set $k=R/\mathfrak{m}$ and $e=\text{rank}_{k}(\mathfrak{m}/\mathfrak{m}^2)$. If $e>2$ and $M$, $N$ are finitely generated…