Characterization Of Regular Local Rings Research Articles

Criteria for prescribed bound on projective dimension

We establish a prescribed upper bound for the projective dimension of a finitely generated module over a Cohen-Macaulay local ring (with canonical module) satisfying certain cohomological conditions. The central hypothesis is the vanishing of finitely many suitable Ext modules. We then derive corollaries which provide freeness criteria and a characterization of regular local rings. Finally, we raise two main questions related to our results; one generalizes a problem due to D. Jorgensen, and the other retrieves the long-standing Auslander-Reiten conjecture in the Gorenstein case.

Characterizations of regular local rings via syzygy modules of the residue field

Let $R$ be a commutative Noetherian local ring with residue field $k$. We show that if a finite direct sum of syzygy modules of $k$ surjects 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.

A DESCENT THEOREM FOR FORMAL SMOOTHNESS

We give a descent result for formal smoothness having interesting applications: we deduce that quasiexcellence descends along flat local homomorphisms of finite type, we greatly improve Kunz’s characterization of regular local rings by means of the Frobenius homomorphisms as well as André and Radu relativization of this result, etc. In the second part of the paper, we study a similar question for the complete intersection property instead of formal smoothness, giving also some applications.

A new proof of Serre’s homological characterization of regular local rings

We give a new proof of Serre’s result that a Noetherian local ring is regular if and only if it has finite global dimension. Our proof avoids the explicit construction of a Koszul complex.

Characterizations of regular local rings in positive characteristics

In this note, we provide several characterizations of regular local rings in positive characteristics, in terms of the Hilbert-Kunz multiplicity and its higher $\mathrm {Tor}$ counterparts ${\mathfrak {i}} t_i={\lim }_{n \to \infty } \ell (\mathrm {Tor}_i(k,{}^{f^n}\!\! R))/p^{nd}$. We also apply the characterizations to improve a recent result by Bridgeland and Iyengar in the characteristic $p$ case. Our proof avoids using the existence of big Cohen-Macaulay modules, which is the major tool in the proof of Bridgeland and Iyengar.

A Characterization of Regular Local Rings

A Characterization of Regular Local Rings

A characterization of regular local rings

For a local ring, ( A , M ) (A,\mathfrak {M}) of positive depth regularity is shown to be equivalent to the symmetric algebra of M \mathfrak {M} being torsion free.

Characterizations of Regular Local Rings of Characteristic p

Characterizations of Regular Local Rings of Characteristic p

Notes on differenntial theoretic characterization of regular local rings

Notes on differenntial theoretic characterization of regular local rings

On the flatness of complete formally projective modules

In the paper [4], the author introduced the notion of m-adic free modules in order to give a differential characterization of regular local rings. Later, more generally, A. Grothendieck introduced the notion of formally projective modules for the similar purpose in his [3]. Comparisons of these two notions in important cases were given in [5]. In the present paper, we intend to study the relationship between the flatness and the formally projectivity. The main purpose is to show the flatness of complete formally projective modules over complete Zariski rings, using results in [5]. As a result of it, we can characterize the formally projectivity of complete modules over complete semilocal rings by the flatness. Meanwhile, we prove the separatedness of finite coefficients extension of complete formally projective modules over complete Zariski rings. Let R be a commutative ring with an identity and m be an ideal. Let M be an R-module. We consider the m-adic topologies both in R and M. By the completeness of R-modules, we always mean the separated completeness. We follow the terminology in [5]. We begin with a supplement to [5]. The following lemma is a part of Proposition 4.1, n04, IV, in [2].

Commutative algebras and cohomology

This paper has three purposes: (1) to develop the machinery of a commutative cohomology theory for commutative algebras, (2) to apply this cohomology theory to give a working theory of ring extensions for commutative algebras, and (3) to employ and test the theory by relating certain natural cohomological conditions to corresponding conditions in algebraic geometry. The first purpose is approached in the first and third sections. Our cohomology has several properties not enjoyed by the Hochschild theory (an explicit relation for the tensor product, a workable sequence for changes from an algebra to a factor algebra, and an explicit formula for going local); however, unlike the Hochschild theory, we have no dimension shifting techniques. For this reason, we restrict consideration to the first, second, and third cohomology modules. The second purpose is considered in the second section. For B and A commutative algebras, we let M(A) denote HomA(A, A), taken modulo the ideal of multiplications by elements in A. Following the prototype theory which Eilenberg and Mac Lane developed for groups, we associate to each algebra homomorphism from B into M(A), an element in a third cohomology module. The homomorphism is called unobstructed if this element is zero. The unobstructed homomorphisms, together with the elements of a certain second cohomology module, are associated in a one-one fashion with the commutative algebra extensions of A by B. The purpose of the fourth and last section is best expressed by considering a point p on an affine variety V over a perfect field k. Let A be the coordinate ring of V, , be the prime ideal associated with p, and Q be the quotient field of A/p. We prove that p is a simple point if and only if the second cohomology module of A with coefficients in Q is zero. In this manner we give a characterization of regular local rings. We also prove that p will be a complete intersection if and only if the difference between the dimensions of the first and second cohomology modules (both of A with coefficients in Q) is the dimension of V. Our third main result is that V will be nonsingular if and only if A has trivial second cohomology with respect to all finitely generated Amodules.