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Let (R , m) be a Noetherian local ring, and I an R -ideal such that pd R R / I < ∞. If R / I satisfies a property P , it is natural to ask if R would also have the property P , we call this the generalized deformation problem. Our paper gives some properties that hold for the generalized deformation problems. There are two main parts in this paper. The first part is about F -singularities in the generalized deformation problems. Motivated by Aberbach's work, we show that if every maximal regular sequence on R / I is Frobenius closed, then every regular sequence on R is Frobenius closed. Moreover, under mild assumptions, Frobenius closed can be replaced by tightly closed. By these two results, we solve the generalized deformation problems for F -injectivity in the Cohen-Macaulay case and F -rationality under mild assumptions. Namely, if R is a Noetherian local ring of characteristic p , and I an R -ideal such that pd R R / I < ∞ , then if R is Cohen-Macaulay and R / I is F -injective, then R is F -injective. If R is excellent and R / I is F -rational, then R is F -rational. The second part is about some basic properties of rings in arbitrary characteristic. We prove that if R is a Noetherian local ring, I an R -ideal such that pd R R / I < ∞ , then if R / I is S k , R k + S k + 1 , normal, reduced or a domain, then so is R. [ABSTRACT FROM AUTHOR]

This paper gives a necessary and sufficient condition for Gorensteinness in Rees algebras of the d th power of parameter ideals in certain Noetherian local rings of dimension d ≥ 2. The main result of this paper produces many Gorenstein Rees algebras over non-Cohen-Macaulay local rings. For example, the Rees algebra R (q d) = ⊕ i ≥ 0 q d i is Gorenstein for every parameter ideal q that is a reduction of the maximal ideal in a d -dimensional Buchsbaum local ring of depth 1 and multiplicity 2. [ABSTRACT FROM AUTHOR]

This paper is a commutative algebra introduction to the homological theory of quasi-coherent sheaves and contraherent cosheaves over quasi-compact semi-separated schemes. Antilocality is an alternative way in which global properties are locally controlled in a finite affine open covering. For example, injectivity of modules over non-Noetherian commutative rings is not preserved by localizations, while homotopy injectivity of complexes of modules is not preserved by localizations even for Noetherian rings. The latter also applies to the contraadjustedness and cotorsion properties. All the mentioned properties of modules or complexes over commutative rings are actually antilocal. They are also colocal, if one presumes contraadjustedness. Generally, if the left class in a (hereditary complete) cotorsion theory for modules or complexes of modules over commutative rings is local and preserved by direct images with respect to affine open immersions, then the right class is antilocal. If the right class in a cotorsion theory for contraadjusted modules or complexes of contraadjusted modules is colocal and preserved by such direct images, then the left class is antilocal. As further examples, the class of flat contraadjusted modules is antilocal, and so are the classes of acyclic, Becker-coacyclic, or Becker-contraacyclic complexes of contraadjusted modules. The same applies to the classes of homotopy flat complexes of flat contraadjusted modules and acyclic complexes of flat contraadjusted modules with flat modules of cocycles. [ABSTRACT FROM AUTHOR]

This paper concerns the distribution of Selmer ranks in a family of even Galois representations in residual characteristic p = 2 obtained by allowing ramification at auxiliary primes. The main result is a Galois cohomological analogue of a theorem of Friedlander, Iwaniec, Mazur and Rubin on the distribution of Selmer ranks in a family of twists of elliptic curves. The Selmer groups are constructed as prescribed by the Galois cohomological method for GL (2) : At each ramified place, the local Selmer condition is the tangent space of a smooth quotient of the local deformation ring. By methods of global class field theory, the Selmer group at the minimal level is computed explicitly. The infinitude of primes for which the Selmer rank increases by one is proved, and the density of such primes is shown to be 1/192. The proof combines Wiles' formula and the global reciprocity law. The result has implications for the algebraic structure of even deformation rings and the distribution of their presentations in families. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic 0, let S be a complete local ring with coefficient field K , let K [ [ x 1 , ... , x n ] ] be the ring of formal power series in variables x 1 , ... , x n with coefficients from K , let K [ [ x 1 , ... , x n ] ] → S be a K -algebra surjection and let E • • , • be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of S. Nicholas Switala [12] proved that this spectral sequence is independent of the surjection beginning with the E 2 page, and the groups E 2 p , q are all finite-dimensional over K. In this paper we extend this result to affine varieties. Namely, let Y be an affine variety over K , let X be a non-singular affine variety over K , let Y ⊂ X be an embedding over K and let E • • , • be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of Y. Then this spectral sequence is independent of the embedding beginning with the E 2 page, and the groups E 2 p , q are all finite-dimensional over K. [ABSTRACT FROM AUTHOR]

In this paper, we compute an algebraic decomposition of black-box rings in the generic ring model. More precisely, we explicitly decompose a black-box ring as a direct product of a nilpotent black-box ring and unital local black-box rings, by computing all its primitive idempotents. The algorithm presented in this paper uses quantum subroutines for the computation of the p -power parts of a black-box ring and then classical algorithms for the computation of the corresponding primitive idempotents. As a by-product, we get that the reduction of a black-box ring is also a black-box ring. The first application of this decomposition is an extension of the work of Maurer and Raub (2007) on representation problem in black-box finite fields to the case of reduced p -power black-box rings. Another important application is an IND-CCA 1 attack for any ring homomorphic encryption scheme in the generic ring model. Moreover, when the plaintext space is a finite reduced black-box ring, we present a plaintext-recovery attack based on representation problem in black-box prime fields. In particular, if the ciphertext space has smooth characteristic, the plaintext-recovery attack is effectively computable in the generic ring model. [ABSTRACT FROM AUTHOR]

Let M be a module over a Noetherian local ring A. We study M -independent sequences of elements of the maximal ideal m A in the sense of Lech and Hanes. The main tool is a new characterization of the M -independence of a sequence in terms of the associated Koszul complex. As applications, we give a result in linkage theory, a freeness criterion for M in terms of the existence of a strongly M -independent sequence of length equal to the embedding dimension edim (A) of A , and another freeness criterion inspired from the patching method of Calegari and Geraghty for balanced modules in their 2018 paper. [ABSTRACT FROM AUTHOR]

Blickle, Mustaţă and Smith proposed two conjectures on the limits of F -pure thresholds. One conjecture asks whether or not the limit of a sequence of F -pure thresholds of principal ideals on regular local rings of fixed dimension can be written as an F -pure thresshold in lower dimension. Another conjecture predicts that any F -pure threshold of a formal power series can be written as the F -pure threshold of a polynomial. In this paper, we prove that the first conjecture has a counterexample but a weaker statement still holds. We also give a partial affirmative answer to the second conjecture. [ABSTRACT FROM AUTHOR]

In this note (R , m) denotes a complete regular local ring and B mostly denotes its absolute integral closure. The four objectives of this paper are the following: i) to determine the highest non-vanishing local cohomology of Ω B / R in equicharacteristic 0, ii) to establish a connection between each of Ω B / R and Ω B / V and pull-back of Ω A / V via a short exact sequence together with new observations on corresponding local cohomologies in mixed characteristic where V is the coefficient ring of R and A is its absolute integral closure, iii) to demonstrate that Ω B / R can be mapped onto a cohomologically Cohen-Macaulay module and iv) to study torsion-free property for Ω C / V and Ω C / k along with their respective completions where C is an integral domain and a module finite extension of R. In this connection an extension of Suzuki's theorem on normality of complete intersections to the formal set-up in all characteristics is accomplished. [ABSTRACT FROM AUTHOR]

In this paper we study Cohen-Macaulay local rings of dimension d , multiplicity e and second Hilbert coefficient e 2 in the case e 2 = e 1 − e + 1. Let h = μ (m) − d. If e 2 ≠ 0 then in our case we can prove that type (A) ≥ e − h − 1. If type (A) = e − h − 1 then we show that the associated graded ring G (A) is Cohen-Macaulay. In the next case when type (A) = e − h we determine all possible Hilbert series of A. In this case we show that depth G (A) completely determines the Hilbert Series of A. [ABSTRACT FROM AUTHOR]

In this paper we study the depth of tensor products of homologically finite complexes over commutative Noetherian local rings. As an application of our main result, we determine new conditions under which the factors of a nonzero reflexive tensor product of finitely generated modules over hypersurface rings can be reflexive. A result of Asgharzadeh shows that nonzero symbolic powers of prime ideals in a local ring cannot have finite projective dimension, unless the ring in question is a domain. We make use of this fact in the appendix and consider the reflexivity of tensor products of prime ideals over hypersurface rings. [ABSTRACT FROM AUTHOR]

In this paper, we deal with very well-covered graphs. We first describe the structure of these kinds of graphs based on the structure of Cohen–Macaulay very well-covered graphs. As an application, we analyze the structure of local cohomology of the residue rings by the edge ideals of very well-covered graphs. Also, we give different formulas of regularity and depth of these rings from known ones and we finally treat the CM t property. [ABSTRACT FROM AUTHOR]

Abstract Lower bounds on Hilbert–Samuel multiplicity are known for several types of commutative noetherian local rings, and rings with multiplicities which achieve these lower bounds are said to have minimal multiplicity. The first part of this paper gives characterizations of rings of minimal multiplicity in terms of the Ext-algebra of the ring; in particular, we show that minimal multiplicity can be detected via an Ext-algebra which is Gorenstein or Koszul AS-regular. The second part of this paper characterizes rings of minimal multiplicity via a numerical homological invariant introduced by J. Herzog and S. B. Iyengar called linearity defect. Our characterizations allow us to answer in two special cases a question raised by Herzog and Iyengar. [ABSTRACT FROM AUTHOR]

In this paper, we consider a finite, torsion-free module E over a Gorenstein local ring. We provide sufficient conditions for E to be of linear type and for the Rees algebra R (E) of E to be Cohen-Macaulay. Our results are obtained by constructing a generic Bourbaki ideal I of E and exploiting properties of the residual intersections of I. [ABSTRACT FROM AUTHOR]

Let (R , m) be a generalized Cohen-Macaulay local ring of dimension d , and f 1 , ... , f r a part of system of parameters of R. In this paper we give explicit numbers N such that the lengths of all lower local cohomology modules and the Hilbert function of R / (f 1 , ... , f r) are preserved when we perturb the sequence f 1 , ... , f r by ε 1 , ... , ε r ∈ m N. The second assertion extends a previous result of Srinivas and Trivedi for generalized Cohen-Macaulay rings. [ABSTRACT FROM AUTHOR]

The main focus of this paper is on determining the highest non-vanishing local cohomology modules of Ω B / R , Ω B / V (Ω B / k) where R is either a complete regular local ring or a complete local normal domain with coefficient ring V (field k) and B is its integral closure in an algebraic extension of Q (R). Similar problem is also studied over a normal domain R containing a field k of characteristic 0. In this connection new observations on the direct summand property for integral extensions are also presented. [ABSTRACT FROM AUTHOR]

In J. Herzog and E. Kunz (1973) [6] it was shown that for any pair ( d , t ) ∈ N × N + with ( d , t ) ≠ ( 1 , 1 ) there exists a local Cohen–Macaulay ring R having deviation d ( R ) = d and type t ( R ) = t . By E. Kunz (1974) [7] the case d ( R ) = 1 , t ( R ) = 1 cannot occur. In this paper certain Cohen–Macaulay rings are studied for which there are close relations between deviation, type and embedding dimension. Similar relations for other classes of local rings have been proved in the recent paper by L. Sharifan (2014) [15] . Our relations will be applied to numerical semigroups (or equivalently monomial curves) and lead also to some further cases, generalizing E. Kunz (2016) [8] with ring-theoretic proofs, in which a question of H. Wilf (1978) [16] has a positive answer. [ABSTRACT FROM AUTHOR]

The extended plus (epf) closure and the rank 1 (r1f) closure are two closure operations introduced by Raymond C. Heitmann for rings of mixed characteristic. Recently, he and Linquan Ma proved that the epf closure satisfies the usual colon-capturing property under mild conditions. In this paper, we extend their result and prove that the epf closure satisfies what we call the p -colon-capturing property. Based on that, we define a new closure notion called "weak epf closure," and prove that it satisfies the generalized colon-capturing property and some other colon-capturing properties. This gives a new proof of the existence of big Cohen-Macaulay algebras in the mixed characteristic case. We also show that any module-finite extension of a complete local domain is epf -phantom, which generalizes a result of Mel Hochster and Craig Huneke about "phantom extensions." Finally, we prove some related results in characteristic p. [ABSTRACT FROM AUTHOR]

Let n be an integer greater than 1 and let x 1 , ... , x n be indeterminates over a countable field k. In this paper, we employ techniques of Heitmann and Nagata to show there exists an uncountable regular local ring S between the localized polynomial ring k [ x 1 , ... , x n ] (x 1 , ... , x n) and the power series ring k [ [ x 1 , ... , x n ] ] such that the prime ideal spectrum of S is homeomorphic to the prime ideal spectrum of k [ x 1 , ... , x n ] (x 1 , ... , x n) as topological spaces with the Zariski topology (Theorem 3.17). Thus S is a local n -dimensional Noetherian domain and the cardinality of the set of prime ideals of S is strictly less than the cardinality of S. We also show that every Noetherian ring A with infinitely many prime ideals has a Noetherian subring B such that the prime ideal spectrum of B is homeomorphic to the prime ideal spectrum of A and the cardinality of the set of prime ideals of B equals the cardinality of B. [ABSTRACT FROM AUTHOR]

Powers of (monomial) ideals is a subject that still calls attraction in various ways. In this paper we present a nice presentation of high powers of ideals in a certain class in K [ x 1 , ... , x n ] and K [ [ x 1 , ... , x n ] ]. As an interesting application it leads to an algorithm for computation of the Ratliff–Rush operation on ideals in that class. The Ratliff–Rush operation itself has several applications, for instance, if I is a regular m -primary ideal in a local ring (R , m) , then the Ratliff–Rush associated ideal I ˜ is the unique largest ideal containing I with the same Hilbert polynomial as I. [ABSTRACT FROM AUTHOR]

This paper is a sequel to [8] where we introduced an invariant, called canonical degree, of Cohen–Macaulay local rings that admit a canonical ideal. Here to each such ring R with a canonical ideal, we attach a different invariant, called bi-canonical degree, which in dimension 1 appears also in [12] as the residue of R. The minimal values of these functions characterize specific classes of Cohen–Macaulay rings. We give a uniform presentation of such degrees and discuss some computational opportunities offered by the bi-canonical degree. [ABSTRACT FROM AUTHOR]

In this paper, we study the F-rationality of the Rees algebra and the extended Rees algebra of m -primary ideals in excellent local rings (R , m) of prime characteristic. We partially answer some conjectures and questions raised by Hara et al. (2002) [9]. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of the strong Rees property (SRP) for m -primary ideals of a Noetherian local ring and prove that any power of the maximal ideal m has its property if the associated graded ring G of m satisfies depth G ≥ 2. As its application, we characterize two-dimensional excellent normal local domains so that m is a p g -ideal. Finally we ask what m -primary ideals have SRP and state a conjecture which characterizes the case when m n are the only ideals which have SRP. [ABSTRACT FROM AUTHOR]

The Sally module of an ideal is an important tool to interplay between Hilbert coefficients and the properties of the associated graded ring. In this paper we give new insights on the structure of the Sally module. We apply these results to characterize the almost minimal value of the first Hilbert coefficient in the case of the normal filtration in an analytically unramified Cohen–Macaulay local ring. [ABSTRACT FROM AUTHOR]

In this study, we determine the structure of cyclic codes over the ring Z q [ u ] / 〈 u 2 〉 which is isomorphic to R = Z q + u Z q where q = p s , p is a prime, s is a positive integer, and u 2 = 0. This is equivalent to determining the algebraic structure of ideals of the polynomial quotient ring R [ x ] / 〈 x n − 1 〉 , which is addressed in this paper completely. By establishing the structure of ideals of R [ x ] / 〈 x n − 1 〉 with gcd ⁡ (p , n) = 1 , we present an exact formula that enumerates the number of ideals of this ring that leads to the enumeration of cyclic codes over this ring. Finally, we consider and explore some special families of cyclic codes for some specific q and determine their size. [ABSTRACT FROM AUTHOR]

The paper studies projective freeness and Hermiteness of algebras of complex-valued continuous functions on topological spaces, Stein algebras, and commutative unital Banach algebras. New sufficient cohomology conditions on the maximal ideal spaces of the algebras are given that guarantee the fulfilment of these properties. The results are illustrated by nontrivial examples. Based on the Borsuk theory of shapes, a new class C of commutative unital complex Banach algebras is introduced (an analog of the class of local rings in commutative algebra) such that the projective tensor product with algebras in C preserves projective freeness and Hermiteness. Some examples of algebras of class C and of other projective free and Hermite function algebras are assembled. These include, e.g., Douglas algebras, finitely generated algebras of symmetric functions, Bohr-Wiener algebras, algebras of holomorphic semi-almost periodic functions, and algebras of bounded holomorphic functions on Riemann surfaces. [ABSTRACT FROM AUTHOR]

Let R be a complete regular local ring with an algebraically closed residue field and let A be a Noetherian R -subalgebra of the polynomial ring R X. It has been shown in [4] that if dim ⁡ R = 1 , then A is necessarily finitely generated over R. In this paper, we give necessary and sufficient conditions for A to be finitely generated over R when dim ⁡ R = 2 and present an example of a Noetherian normal non-finitely generated R -subalgebra of R X over R = C u , v . [ABSTRACT FROM AUTHOR]

Let D be a 2-dimensional regular local ring and let Q (D) denote the quadratic tree of 2-dimensional regular local overrings of D. We explore the topology of the tree Q (D) and the family R (D) of rings obtained as intersections of rings in Q (D). If A is a finite intersection of rings in Q (D) , then A is Noetherian and the structure of A is well understood. However, other rings in R (D) need not be Noetherian. The two main goals of this paper are to examine topological properties of the quadratic tree Q (D) , and to examine the structure of rings in the set R (D). [ABSTRACT FROM AUTHOR]

The notion of 2-almost Gorenstein local ring (2-AGL ring for short) is a generalization of the notion of almost Gorenstein local ring from the point of view of Sally modules of canonical ideals. In this paper, for further developments of the theory, we discuss three different topics on 2-AGL rings. The first one is to clarify the structure of minimal presentations of canonical ideals, and the second one is the study of the question of when certain fiber products, so called amalgamated duplications are 2-AGL rings. We also explore Ulrich ideals in 2-AGL rings, mainly two-generated ones. [ABSTRACT FROM AUTHOR]

In this paper we present an algorithm to compute a Standard Basis for a fractional ideal I of the local ring O of an n -space algebroid curve with several branches. This allows us to determine the semimodule of values of I. When I = O , we may obtain a (finite) set of generators of the semiring of values of the curve, which determines its classical semigroup. In the complex context, identifying the Kähler differential module Ω O / C of a plane curve with a fractional ideal of O and applying our algorithm, we can compute the set of values of Ω O / C , which is an important analytic invariant associated to the curve. [ABSTRACT FROM AUTHOR]

In this paper, we present an algorithm for computing the minimal reductions of m -primary ideals of Cohen–Macaulay local rings. Using this algorithm, we are able to compute the Hilbert–Samuel multiplicities and solve the membership problem for the integral closure of m -primary ideals. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the theory of cohomology annihilators over commutative Gorenstein rings. We adopt a triangulated category point of view and study the annihilation of stable category of maximal Cohen-Macaulay modules. We prove that in dimension one the cohomology annihilator ideal and the conductor ideal coincide under mild assumptions. We present a condition on a ring homomorphism between Gorenstein rings which allows us to carry the cohomology annihilator of the domain to that of the codomain. As an application, we generalize the Milnor-Jung formula for algebraic curves to their double branched covers. We also show that the cohomology annihilator of a Gorenstein local ring is contained in the cohomology annihilator of its Henselization and in dimension one the cohomology annihilator of its completion. Finally, we investigate a relation between the cohomology annihilator of a Gorenstein ring and stable annihilators of its noncommutative resolutions. [ABSTRACT FROM AUTHOR]

Stable cohomology is a generalization of Tate cohomology to associative rings, first defined by Pierre Vogel. For a commutative local ring R with residue field k , stable cohomology modules Ext ˆ R n (k , k) , defined for n ∈ Z , have been studied by Avramov and Veliche. Stable cohomology carries a structure of Z -graded k -algebra. One of the main goals of this paper is to prove that, for a class of Gorenstein rings, this algebra is a trivial extension of absolute cohomology Ext R (k , k) and a shift of Hom k (Ext R (k , k) , k). We use this information to characterize the rings R for which stable cohomology is graded-commutative. Stable cohomology is connected through an exact sequence to bounded cohomology. We use this connection to understand the algebra structure of Ext ˆ R (k , k) by investigating the structure of bounded cohomology Ext ‾ R (k , k) as a graded Ext R (k , k) -bimodule. [ABSTRACT FROM AUTHOR]

In this paper we consider stabilizer codes over local Frobenius rings. Firstly, we study the relative minimum distances of a stabilizer code and its reduction onto the residue field. We show that for various scenarios, a free stabilizer code over the ring does not underperform the according stabilizer code over the field. This leads us to conjecture that the same is true for all free stabilizer codes. Secondly, we focus on the isometries of stabilizer codes. We present some preliminary results and introduce some interesting open problems. [ABSTRACT FROM AUTHOR]

Abstract Let (R , m) be a Noetherian local ring which is a quotient of a Gorenstein local ring. Let M be a finitely generated R -module and K M the canonical module of M. Denote by nCM (M) the non Cohen–Macaulay locus of M. In this paper, the relationship between the two loci nCM (K M) and nCM (M) is given. The structure of M is clarified in case where dim ⁡ nCM (K M) ≤ 0 (i.e. K M is generalized Cohen–Macaulay). [ABSTRACT FROM AUTHOR]

Abstract In this paper we prove the existence of a uniform bound for Frobenius test exponents for parameter ideals of a local ring (R , m) of prime characteristic in the following cases: (1) R is generalized Cohen–Macaulay. Our proof is much simpler than the original proof of Huneke, Katzman, Sharp and Yao; (2) The Frobenius actions on all lower local cohomologies H m i (R) , i < dim ⁡ R , are nilpotent. [ABSTRACT FROM AUTHOR]

Abstract: Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher-dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p-group due to Benson, Carlson and Rickard, we consider classifying thick subcategories of the stable category of Cohen–Macaulay modules over a Gorenstein local ring. The main result of this paper yields a complete classification of the thick subcategories of the stable category of Cohen–Macaulay modules over a local hypersurface in terms of specialization-closed subsets of the prime ideal spectrum of the ring which are contained in its singular locus. [ABSTRACT FROM AUTHOR]

Abstract: This paper answers the question of Anderson, Frazier, Lauve, and Livingston: for which finite commutative rings R is the zero-divisor graph planar? We build upon and extend work of Akbari, Maimani, and Yassemi, who proved that if R is any local ring with more than 32 elements, and R is not a field, then is not planar. They left open the question: “Is it true that, for any local ring R of cardinality 32, which is not a field, is not planar?” In this paper we answer this question in the affirmative. We prove that if R is any local ring with more than 27 elements, and R is not a field, then is not planar. Moreover, we determine all finite commutative local rings whose zero-divisor graph is planar. [Copyright &y& Elsevier]

Let ( R , m ) be a Noetherian local ring of dimension d > 0 and depth R ≥ d − 1 . Let Q be a parameter ideal of R . In this paper, we derive uniform lower and upper bounds for the Hilbert coefficient e i ( Q ) under certain assumptions on the depth of associated graded ring G ( Q ) . For 2 ≤ i ≤ d , we show that (1) e i ( Q ) ≤ 0 provided depth G ( Q ) ≥ d − 2 and (2) e i ( Q ) ≥ − λ R ( H m d − 1 ( R ) ) provided depth G ( Q ) ≥ d − 1 . It is proved that e 3 ( Q ) ≤ 0 . Further, we obtain a necessary condition for the vanishing of the last coefficient e d ( Q ) . As a consequence, we characterize the vanishing of e 2 ( Q ) . Our results generalize [5, Theorem 3.2] and [11, Corollary 4.5] . [ABSTRACT FROM AUTHOR]

The purpose of the paper is to study the annihilators of local cohomology of the extended Rees rings. We will prove that if ( A , m ) is a noetherian local ring with infinite residue field, and a is a uniform local cohomological annihilator of A , then there is an integer n such that ( a t ) n is a uniform local cohomological annihilator of the extended Rees ring A [ t , t − 1 I ] for any m -primary ideal I of A . [ABSTRACT FROM AUTHOR]

Let ( R , m ) → ( S , n ) be a flat local extension of local rings. Lech conjectured around 1960 that there should be a general inequality e ( R ) ≤ e ( S ) on the Hilbert–Samuel multiplicities [24] . This conjecture is known when the base ring R has dimension less than or equal to two [24] , and remains open in higher dimensions. In this paper, we prove Lech's conjecture in dimension three when R has equal characteristic. In higher dimension, our method yields substantial partial estimate: e ( R ) ≤ ( d ! / 2 d ) ⋅ e ( S ) where d = dim ⁡ R ≥ 4 , in equal characteristic. [ABSTRACT FROM AUTHOR]

Let q denote an m -primary ideal of a d -dimensional local ring ( A , m ) . Let a _ = a 1 , … , a d ⊂ q be a system of parameters. Then there is the following inequality for the multiplicities c ⋅ e ( q ; A ) ≤ e ( a _ ; A ) where c denotes the product of the initial degrees of a i in the form ring G A ( q ) . The aim of the paper is a characterization of the equality as well as a description of the difference by various homological methods via Koszul homology. To this end we have to characterize when the sequence of initial elements a ⋆ _ = a 1 ⋆ , … , a d ⋆ is a homogeneous system of parameters of G A ( q ) . In the case of dim ⁡ A = 2 this leads to results on the local Bezout inequality. In particular, we give several equations for improving the classical Bezout inequality to an equality. [ABSTRACT FROM AUTHOR]

Let ( R , m ) be a regular local ring of dimension at least 2. For each valuation domain birationally dominating R , there is an associated sequence { R n } of local quadratic transforms of R . We consider the case where this sequence { R n } n ≥ 0 is infinite and examine properties of the integrally closed local domain S = ⋃ n ≥ 0 R n in the case where S is not a valuation domain. For this sequence, there is an associated boundary valuation ring V = ⋃ n ≥ 0 ⋂ i ≥ n V i , where V i is the order valuation ring of R i . There exists a unique minimal proper Noetherian overring T of S . T is the regular Noetherian UFD obtained by localizing outside the maximal ideal of S and S = V ∩ T . In the present paper, we define functions w and e, where w is the asymptotic limit of the order valuations and e is the limit of the orders of transforms of principal ideals. We describe V explicitly in terms of w and e and prove that V is either rank 1 or rank 2. We define an invariant τ associated to S that is either a positive real number or +∞. If τ is finite, then S is archimedean and T is not local. In this case, the function w defines the rank 1 valuation overring W of V and W dominates S . The rational dependence of τ over w ( T × ) determines whether S is completely integrally closed and whether V has rank 1. We give examples where S is completely integrally closed. If τ is infinite, then S is non-archimedean and T is local. In this case, the function e defines the rank 1 valuation overring E of V . The valuation ring E is a DVR and E dominates T , and in certain cases we prove that E is the order valuation ring of T . [ABSTRACT FROM AUTHOR]

Let R be a local commutative noetherian ring and HKR the homology ring of the corresponding Koszul complex. We study the homological properties of HKR in particular the Avramov spectral sequence. When the embedding dimension of R is four and when R can be presented with quadratic relations we have found 101 cases where this spectral sequence degenerates and only three cases where it does not degenerate. We also determine completely the Hilbert series of the bigraded Tor of these HKR in Tables A–D at the end of the paper. We also study some higher embedding dimensions. Among the methods used are the programme BERGMAN by Jörgen Backelin et al., the Macaulay2 -package DGAlgebras by Frank Moore, combined with results by Govorov, Clas Löfwall, Victor Ufnarovski and others. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend the main result of C. Huneke and G. Lyubeznik in [Adv. Math. 210 (2007), 498–504] to the class of rings that are images of Cohen–Macaulay local rings. Namely, let R be a local Noetherian domain of positive characteristic that is an image of a Cohen–Macaulay local ring. We prove that all local cohomology of R (below the dimension) maps to zero in a finite extension of the ring. As a direct consequence we obtain that the absolute integral closure of R is a big Cohen–Macaulay algebra. Since every excellent local ring is an image of a Cohen–Macaulay local ring, this result is a generalization of the main result of M. Hochster and Huneke in [Ann. of Math. 135 (1992), 45–79] with a simpler proof. [ABSTRACT FROM AUTHOR]

This paper studies the structure of special preenvelopes and envelopes with respect to maximal Cohen–Macaulay modules. We investigate the structure of them in terms of their kernels and cokernels. Moreover, using this result, we also study the structure of special proper coresolutions with respect to maximal Cohen–Macaulay modules over a Henselian Cohen–Macaulay local ring. [ABSTRACT FROM AUTHOR]

In this paper, a transform approach is used for polycyclic and serial codes over finite local rings in the case that the defining polynomials have no multiple roots. This allows us to study them in terms of linear algebra and invariant subspaces as well as understand the duality in terms of the transform domain. We also make a characterization of when two polycyclic ambient spaces are Hamming-isometric. [ABSTRACT FROM AUTHOR]

The study of linear dynamical systems over a finite commutative ring faces difficulties due to the lack of unique factorization of polynomials. In this paper, we give new criterions and algorithms to determine whether a given linear system over a finite local ring is a fixed point system. In particular, the cycle structure of linear systems over the ring of integers modulo n are obtained. [ABSTRACT FROM AUTHOR]

Abstract: Let be a Noetherian local ring and M a finitely generated R-module. It is well known that the local cohomology module is Artinian for all . Following I.G. Macdonald [8], denote by the set of attached primes of . This paper is concerned with clarifying the structure of the base ring R via a relation between and . Some characterizations for R being universally catenary with all Cohen–Macaulay formal fibers are given. [Copyright &y& Elsevier]

Abstract: In this paper, we develop an algebraic theory for local rings of finite embedding dimension. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally, variants of the homological theorems are shown to hold in equal characteristic. This theory is then applied to Noetherian local rings in order to get: (i) over a Cohen–Macaulay local ring, uniform bounds on the Betti numbers of a Cohen–Macaulay module in terms of dimension and multiplicity, and similar bounds for the Bass numbers of a finitely generated module; (ii) a characterization for being respectively analytically unramified, analytically irreducible, unmixed, quasi-unmixed, normal, Cohen–Macaulay, pseudo-rational, or weakly F-regular in terms of certain uniform arithmetic behavior; (iii) in mixed characteristic, the Improved New Intersection Theorem when the residual characteristic or ramification index is large with respect to dimension (and some other numerical invariants). [Copyright &y& Elsevier]