Modules and Spectral Spaces

We establish conditions for Spec(M) to be Noetherian and spectral space, w.r.t. different topologies. We used rings with Noetherian spectrum to produce plentiful examples of modules with Noetherian spectrum that have not appeared in the literature previously. In particular, we show that every ?-module has Noetherian spectrum. Another main subject of this article is presenting the conditions under which a module is top. In particular, we show that every distributive module is top, every content weak multiplication R-module M is also top, and moreover, if R has Noetherian spectrum, then Spec(M) is a spectral space.     

Local structure theorems for smooth maps of formal schemes

We continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in [L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes, Comm. Alg. 35 (2007) 1341-1367]. In this paper, we focus on some properties which arise specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes. Among our main results, we obtain the characterization of completion morphisms as pseudo-closed immersions that are flat. Also, the local structure of smooth and étale morphisms between locally noetherian formal schemes is described: the former factors locally as a completion morphism followed by a smooth adic morphism and the latter as a completion morphism followed by an étale adic morphism.     

Differential Criterion of Complete Regular Local Rings

Abstract

In the paper by Furuya and Niitsuma [Furuya, M., Niitsuma, H. (2002a). On m -adic higher differentials and regularities of Noetherian complete local rings. J. Math. Kyoto Univ. 42(1):33–40], we gave a regularity criterion of complete Noetherian local rings in terms of the concept of m -adic higher differentials with some assumptions of separability on the residue fields of the local rings. The aim of this paper is to try to weaken “the separability assumptions” in a premise of the theorems in Furuya and Niitsuma (2002a).     

Endomorphism Rings

In this article we construct an affine prime PI ring with a cyclic module whose endomorphism ring is not Jacobson. Then, using similar techniques, we construct an affine prime Noetherian PI ring with a cyclic module whose endomorphism ring is not Noetherian.     

On the Countability of Noetherian Dimension of Modules

Abstract

It is shown that a module M has countable Noetherian dimension if and only if the lengths of ascending chains of submodules of M has a countable upper bound. This shows in particular that every submodule of a module with countable Noetherian dimension is countably generated. It is proved that modules with Noetherian dimension over locally Noetherian rings have countable Noetherian dimension. We also observe that ω^ω is a universal upper bound for the lengths of all chains in Artinian modules over commutative rings.     

Lattice Modules over Principal Element Domains

Abstract

This paper obtains a structure theorem for a finitely generated lattice module 𝔐 over a Noetherian principal element domain 𝔏,with a slightly stronger theorem if the lattice module satisfies a hypothesis valid over principal ideal domains. Additionally,we obtain a new characterization of Dedekind lattice domains as multiplicative lattice domains over which there exists a non torsion,principally generated,Noetherian join-principal-element lattice module.     

Cotangent functors of curve singularities

Let R be a complete analytic k-algebra of dimension one, where k is a field of characteristic 0. For a Noetherian normalization , the cotangent functors T_i(R/A, R) are of finite length. It is shown that if R is smoothable, then for all j, where denotes the complementary module. This result is used to give estimates for the length of the torsion of the module of differentials _R/k.     

Flatness of Noetherian Hopf algebras over coideal subalgebras

It is proved in the paper that a Noetherian residually finite-dimensional Hopf algebra H is a flat module over any right Noetherian right coideal subalgebra A. In the case when A is a Hopf subalgebra we get faithful flatness. These results are obtained by verifying the existence of classical quotient rings of A and H. It is also proved that the antipode of either right or left Noetherian residually finite-dimensional Hopf algebra is bijective. As a consequence, such a Hopf algebra is right and left Noetherian simultaneously.

     

Noether模的Artin根

本文定义了Ｎｏｃｔｈｅｒ模的Ａｒｔｉｎ根，给出了Ｎｏｃｔｈｅｒ半局部环上有限生成模的Ａｒｔｉｎ根的刻划．最后，作为Ａｒｔｉｎ根理论的应用，给出了一个关于Ｃｏｈｅｎ－Ｍａｃａｕｌａｙ环的定理及一个模的用ｄｅｐｔｈ和ｋｒｕｌｌ维数表示的公式。     

The separable transfer

A separable transfer map is defined for finite surjective morphisms from integral schemes to a Noetherian integral normal base, for both chain complexes of sheaves and presheaves of spectra. The spectrum level transfer is a map from a direct image paired with the classifying space of a translation groupoid, and taking values in the original presheaf. More traditional transfer maps are recovered from global sections of the stack associated to this groupoid. These constructions can be made in a big site model for the motivic stable category.     

Universal homogeneous derivations of graded ϵ-commutative algebras

Let K be a commutative ring, let ? be an abelian group, and let ?:?x?→K be a commutation factor over ?.A ? graded K-algebra is said to be ?-commutative if its ?-bracket is identically zero, (K,?) derivations from a given ?-commutative ?-graded K-algebra A into bimodules are studied. It is proved that for each λ?? there exists a universal initial (k,?)-derivation of degree λ of A. For each λ?? a natural module of (K, ?, λ)-differentials of A along with a differential map is constructed. It is proved that each derivation of A canonically equipps this module with a structure of differential module. Applications and examples are given. It is shown that the first order exterior differentials which are known from the theory of smooth graded manifolds are universal initial homogeneous derivations of the sort considered hereby.     

同调光滑连通上链DG代数的一个注记

本文给出了有关同调光滑连通上链微分分次(简称DG)代数的两个重要结论.具体地说,当A是同调光滑连通上链DG代数且其同调分次代数H(A)是诺特分次代数时,证明D_(fg)(A)中的任意Koszul DG A-模都是紧致的.另外,当A是Kozul连通上链DG代数且其同调分次代数H(A)是有平衡对偶复形的诺特分次代数时,证明A的同调光滑性质等价于D_(fg)(A)=D~c(A).     

On arithmetic Macaulayfication of Noetherian rings

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.

     

Zur endlichen Homologischen Dimension von Differentialmoduln

It will be shown that for a reduced separable analytic k-algebra A, with k being a field, a necessary condition for the finite differential module of A to have finite homological dimension is that the canonical module of A is free. This generalizes a result of T. Matsuoka and Y. Aoyama concerning almost complete intersections. In case that A is a normal domain and smooth in codimension 1 an analogous statement can be made for the module of derivations and the module of Zariski differentials of A.     

A Grothendieck module with applications to Poincaré rationality

In this paper, we define a Grothendieck module associated to a Noetherian ring A. This structure is designed to encode relations between A-modules which can be responsible for the relations among Betti numbers and therefore rationality of the Poincaré series. We will define the Grothendieck module, demonstrate that the condition of being torsion in the Grothendieck module implies rationality of the Poincaré series, and provide examples. The paper concludes with an example which demonstrates that the condition of being torsion in the Grothendieck module is strictly stronger than having rational Poincaré series.     

S-TORSION FREE COVERS OF MODULES

Over Pruüfer domains every module has a torsion free cover with a flat kernel. In this paper we show that the commutative Noetherian rings that have this property are the Gorenstein rings with Krull dimension at most one. Here we use the torsion theory defined by the nonzero-divisors of the commutative ring R.     

Base independent algebraic cobordism

The purpose of this article is to show that the bivariant algebraic A-cobordism groups considered previously by the author are independent of the chosen base ring A. This result is proven by analyzing the bivariant ideal generated by the so called snc relations, and, while the alternative characterization we obtain for this ideal is interesting by itself because of its simplicity, perhaps more importantly it allows us to easily extend the definition of bivariant algebraic cobordism to divisorial Noetherian derived schemes of finite Krull dimension. As an interesting corollary, we define the corresponding homology theory called algebraic bordism. We also generalize projective bundle formula, the theory of Chern classes, the Conner–Floyd theorem and the Grothendieck–Riemann–Roch theorem to this setting. The general definitions of bivariant cobordism are based on the careful study of ample line bundles and quasi-projective morphisms of Noetherian derived schemes, also undertaken in this work.