On almost complete intersections

Let R be a local ring such that R=S/I where S is a regular local ring and I is a prime ideal of height r. In this paper it is shown that if I is minimally generated by r+1 elements, then there exists an R-homomorphism : K_RR^r+1 such that is an injection and R^r+1/(K_R)I/I² where K_R:=Ext _S ^r (R,S) the canonical module of R. Moreover, in case where S is a locality over a perfect field k, it is also shown that if R is Cohen-Macaulay and I² is a primary ideal, then the homological dimension of the differential module _R/k is infinite.The author wishes to thank his colleague Mr.Y.Aoyama for valuable discussions in connection with this subject.     

Existence of a weak solution for a parabolic initial-boundary-value problem in an unbounded domain in the class of rapidly increasing functions

In an unbounded domain Rⁿ) one considers the initial—boundary-value problem, under the condition that. It is proved that in the domain Q_T= × (0, T), T C^–1, there exists a unique weak solution of this problem, whose energy norm in _r × (0, T) is estimated by .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 127, pp. 152–157, 1983.     

A remark on almost complete intersections

Let k be a perfect field and S the quotient ring of a polynomial ring k[X₁,...,X_t] with respect to a prime ideal. Let I be a prime ideal of S such that R=S/I is an almost complete intersection. Then, in his paper [2], Matsuoka proves that the homological dimension of the differential module _R/Kis infinite under the assumption that R is Cohen-Macaulay and I² is a primary idea]. In this paper we prove that the result is valid without the above assumption.     

A Theorem on Fine Connectedness

We prove that E is a p-fine domain whenever R ⁿ is a p-fine domain, E R ⁿ is p-polar, and 1 < p n. By a p-fine domain we understand an open connected set in the p-fine topology, i.e. in the coarsest topology making all p-superharmonic functions continuous. As an application of our main result, we establish a general version of minimum principle.     

Submodule lattice quasivarieties and exact embedding functors for rings with prime power characteristic

Given ringsR with prime power characteristicp ^k , quasivarieties (R) of lattices generated by lattices of submodules ofR-modules are studied. An algebra of expressionsd not dependent onR is developed, such that each suchd uniquely determines a two-sides ideald _R ofR. The main technical result is that (R) (S) makes all implications of the formd _s =S d_R=R true, for any such expressiond. The proof makes use of the known equivalence between (R) (S) and existence of an exact embedding functorR-Mod S -Mod. Fork 2, the ordered setW(p ^k ) of all lattice quasivarieties (R),R having characteristic p _K , is shown to be large and complicated, with ascending and descending chains and antichains having continuously many elements. More precisely,W(p ^k ) has a subset which is order isomorphic to the Boolean algebra of all subsets of a denumerably infinite set. Also, given any prime powerp ^k ,k 2, a ringR can be constructed so that (R) and (R ^op) for the opposite ringR ^op are distinct elements ofW(p ^k ).Presented by R. Freese.Research partially supported by Hungarian National Foundation for Scientific Research grant no. 1903.     

On connectedness and indecomposibility of local cohomology modules

Let I denote an ideal of a local Gorenstein ring . Then we show that the local cohomology module , c = height I, is indecomposable if and only if V(I _d ) is connected in codimension one. Here I _d denotes the intersection of the highest dimensional primary components of I. This is a partial extension of a result shown by Hochster and Huneke in the case I the maximal ideal. Moreover there is an analysis of connectedness properties in relation to various aspects of local cohomology. Among others we show that the endomorphism ring of is a local Noetherian ring if dim R/I  =  1.     

On the grade of some ideals

An additivity formula is obtained for the grade of an ideal in a residue ring R/I, where I is a perfect ideal. This result is then applied to compute the grade of ideals of linear (inhomogeneous) polynomials. Further results on the homological rigidity of the conormal module I/I² are pointed out. Finally, an elementary proof is given of a result of Buchsbaum concerning the grade of ideals of minors of a matrix.Partially supported a CNPq grantPartially supported by NSF and CNPq grants     

Local existence for parabolic problems with fully nonlinear boundary condition; anL p -approach

Summary We study the problem (a_i=a_i (x, t, u, p)) (which is of variational form and arises in heat conduction) in the Sobolev Space W _p ² ( × × (0, T)), (Rⁿ, p>n+2. Existence of a local (in time) solution is proved under a natural compatibility condition between the data (Theorem 2.1). This solution is also globally unique. An outlook to similar problems for parabolic systems is given (section 4). Our method also applies to quasilinear equations with conormal b.c. (cf. (P), end of section 2).     

Shirshov finiteness over rings of constants

Let L be a finite-dimensional restricted differential Lie C-algebra of R-continuous derivations of a prime ring R of characteristic p>0, with generalized centroid C. We prove that if the associative inner part of L is quasi-Frobenius then R contains a nonzero element a and elements v₁,…,v_n, such that for any x∈R we have the expansion , where are homorphisms of right R^L-modules . This gives rise to a certain relation on a ring over some subring, known as Shirshov local finiteness. The structure of (R, R^L)-subbimodules in a left Martindale ring of quotients is elucidated. Supported by RFFR grant No. 95-01-01356a, and by the CONACYT of Mexico, Catedra Patrimonial 940411. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 219–238, March–April, 1997.     

Extended Local Cohomology and Local Homology

We present an in-depth exploration of the module structures of local (co)homology modules (moreover, for complexes) over the completion \(\widehat {R}^{\mathcal {a}}\) of a commutative noetherian ring R with respect to a proper ideal \(\mathcal {a}\). In particular, we extend Greenlees-May Duality and MGM Equivalence to track behavior over \(\widehat {R}^{\mathcal {a}}\), not just over R. We apply this to the study of two recent versions of homological finiteness for complexes, and to certain isomorphisms, with a view toward further applications. We also discuss subtleties and simplifications in the computations of these functors.     

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.     

IBN rings and orderings on grothendieck groups

LetR be a ring with an identity element.R∈IBN means thatR ^m⋟Rⁿ impliesm=n, R∈IBN ₁ means thatR ^m⋟Rⁿ⊕K impliesm≥n, andR∈IBN ₂ means thatR ^m⋟R^m⊕K impliesK=0. In this paper we give some characteristic properties ofIBN ₁ andIBN ₂, with orderings on the Grothendieck groups. In addition, we obtain the following results: (1) IfR∈IBN ₁ and all finitely generated projective leftR-modules are stably free, then the Grothendieck groupK ₀(R) is a totally ordered abelian group. (2) If the pre-ordering of the Grothendieck groupK ₀(R) of a ringR is a partial ordering, thenR∈IBN ₁ orK ₀(R)=0. Supported by National Nature Science Foundation of China.     

Rings over which all modules are I 0-modules. II

All right R-modules are I ₀-modules if and only if either R is a right SV-ring or R/I ⁽²⁾ (R) is an Artinian serial ring such that the square of the Jacobson radical of R/I ⁽²⁾ (R) is equal to zero.     

Faserungen,die Aus Reguli mit Einem Gemeinsamen Geradenpaar Zusammengesetzt Sind

Let _pp be a pappian projective 3-space and be a set of lines of _pp; we define:-a lineg of _pp has theproperty R with respect to C, if all lines of meetingg form a regulus- has theproperty E ₂, if there exists a pencil of lines containing only lines having the propertyR with respect to . A spread having the propertyE ₂ is built up of reguli which have two lines in common. In the present paper, we point out a method of constructing a spread of _pp which has the propertyE ₂. This construction is applied to the real 3-space to produce an example of a spread which has the property E₂ and is different from an elliptic linear line congruence; the discussed example represents a 4-dimensional translation plane with a 6-dimensional collineation group. Finally, the propertiesE ₂ andR are used to characterize the elliptic linear line congruences of a pappian 3-space.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

О росте вдоль прямой ц елых функций экспоне нциального типа с заданными нулями

The following result is proved. Let=_n} be a sequence of complex numbers with ¦Re _n¦¦ _n ¦, >0, and letg be an entire function of exponential type with a sequence of zeros which satisfies the same condition. There exists an entire function of exponential typef0 such thatf()=0 and ¦f(iy)¦¦g(iy)¦,yR, if and only if there exists a constantM such that for all numbersr andR, 0rR<>, we have

Approximate radial symmetry for solutions of a class of boundary value problems in ring-shaped domains

This paper deals with some classical boundary value problems defined in the ring-shaped region W\[`(B(R₀))] ì \BbbR^N\Omega \backslash\overline{B(R_0)} \subset \Bbb{R}^N, where B(R ₀) is the N-ball of radius R ₀ centered at the origin and contained in . The authors investigate how the geometry of is affected by some overdertermined data. In particular they show that must be nearly spherical if the overdetermined data are nearly constants.     

Solvability “in the small” of a nonstationary flow problem for an ideal incompressible fluid,I

For the system of Euler equations and the incompressibility equation one considers the following initial-boundary value problem: the field of velocities is prescribed at the initial moment and for all to one gives the following boundary conditions: on the entire boundary of the normal component of the velocity is prescribed and on that part S₁ of the boundary of where inflow occurs one prescribes the value of the velocity =rot |S₂, whose components satisfy a certain necessary equality, derived in the paper. For such a problem one proves its unique solvability on a small interval of time.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 96, pp. 39–56, 1980.     

Problèmes de Sturm-Liouville: bornes pour les valeurs propres et les zéros des fonctions propres

Résumé Pour évaluer ₂, on partage l'intervalleI=[a,b] en deux:I=[a,y] etI=[y,b]. A l'aide de l'inégalité de Barta et du processus d'itération de Schwarz, on détermine, en partant d'une fonction particulière, des bornes inférieuresv _n ^– (y) et _n ^– (y) pour la première valeur propre des nouveaux problèmes définis dansI etI. Auk-ième pas de l'itération, la meilleure borne possible pour ₂ est donnée parv _k ^– (y _k ), oùy _k est la racine de l'équationv _k ^– – _k ^– . De plus,v _k ^– (y _k ) ₂ ety _k tend vers le zéro de la deuxième fonction propre.

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Summary In order to evaluate ₂, we cut the intervalI=[a,b] into two parts:I=[a,y] andI=[y,b]. Using Barta's inequality and Schwarz's iteration procedure and starting from a particular function, we determine lower boundsv _k ^– (y) and _k ^– (y) for the first eigenvalue of the new problems defined inI andI. Afterk iterations, the best possible bound for ₂ isv _k ^– (y _k ), wherey _k is the root of the equationv _k ^– – _k ^– . Moreoverv _k ^– (y _k ) ₂ andy _k tends to the zero of the second eigenfunction.