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Mappings of degree 5, part I

M. Maciejewski, A. Prószyński (2009)

Colloquium Mathematicae

The class of linear (resp. quadratic) mappings over a commutative ring is determined by a set of equation-type relations. For the class of homogeneous polynomial mappings of degree m ≥ 3 it is so over a field, and over a ring there exists a smallest equationally definable class of mappings containing the preceding one. It is proved that generating relations determining that class can be chosen to be strong relations (that is, of the same form over all commutative rings) if{f} m ≤ 5. These relations...

Matlis dual of local cohomology modules

Batoul Naal, Kazem Khashyarmanesh (2020)

Czechoslovak Mathematical Journal

Let ( R , 𝔪 ) be a commutative Noetherian local ring, 𝔞 be an ideal of R and M a finitely generated R -module such that 𝔞 M M and cd ( 𝔞 , M ) - grade ( 𝔞 , M ) 1 , where cd ( 𝔞 , M ) is the cohomological dimension of M with respect to 𝔞 and grade ( 𝔞 , M ) is the M -grade of 𝔞 . Let D ( - ) : = Hom R ( - , E ) be the Matlis dual functor, where E : = E ( R / 𝔪 ) is the injective hull of the residue field R / 𝔪 . We show that there exists the following long exact sequence 0 H 𝔞 n - 2 ( D ( H 𝔞 n - 1 ( M ) ) ) H 𝔞 n ( D ( H 𝔞 n ( M ) ) ) D ( M ) H 𝔞 n - 1 ( D ( H 𝔞 n - 1 ( M ) ) ) H 𝔞 n + 1 ( D ( H 𝔞 n ( M ) ) ) H 𝔞 n ( D ( H ( x 1 , ... , x n - 1 ) n - 1 ( M ) ) ) H 𝔞 n ( D ( H ( n - 1 M ) ) ) ... , where n : = cd ( 𝔞 , M ) is a non-negative integer, x 1 , ... , x n - 1 is a regular sequence in 𝔞 on M and, for an R -module L , H 𝔞 i ( L ) is the i th local cohomology module of L with respect...

Matlis reflexive and generalized local cohomology modules

Amir Mafi (2009)

Czechoslovak Mathematical Journal

Let ( R , 𝔪 ) be a complete local ring, 𝔞 an ideal of R and N and L two Matlis reflexive R -modules with Supp ( L ) V ( 𝔞 ) . We prove that if M is a finitely generated R -module, then Ext R i ( L , H 𝔞 j ( M , N ) ) is Matlis reflexive for all i and j in the following cases: (a) dim R / 𝔞 = 1 ; (b) cd ( 𝔞 ) = 1 ; where cd is the cohomological dimension of 𝔞 in R ; (c) dim R 2 . In these cases we also prove that the Bass numbers of H 𝔞 j ( M , N ) are finite.

Matrix factorizations and singularity categories for stacks

Alexander Polishchuk, Arkady Vaintrob (2011)

Annales de l’institut Fourier

We study matrix factorizations of a potential W which is a section of a line bundle on an algebraic stack. We relate the corresponding derived category (the category of D-branes of type B in the Landau-Ginzburg model with potential W) with the singularity category of the zero locus of W generalizing a theorem of Orlov. We use this result to construct push-forward functors for matrix factorizations with relatively proper support.

Matrix factorizations for domestic triangle singularities

Dawid Edmund Kędzierski, Helmut Lenzing, Hagen Meltzer (2015)

Colloquium Mathematicae

Working over an algebraically closed field k of any characteristic, we determine the matrix factorizations for the-suitably graded-triangle singularities f = x a + y b + z c of domestic type, that is, we assume that (a,b,c) are integers at least two satisfying 1/a + 1/b + 1/c > 1. Using work by Kussin-Lenzing-Meltzer, this is achieved by determining projective covers in the Frobenius category of vector bundles on the weighted projective line of weight type (a,b,c). Equivalently, in a representation-theoretic context,...

Matrix of ℤ-module1

Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2015)

Formalized Mathematics

In this article, we formalize a matrix of ℤ-module and its properties. Specially, we formalize a matrix of a linear transformation of ℤ-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free ℤ-module V, determinant of its Gramian matrix is constant regardless of selection of its basis. ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattices [22]...

Matroids over a ring

Alex Fink, Luca Moci (2016)

Journal of the European Mathematical Society

We introduce the notion of a matroid M over a commutative ring R , assigning to every subset of the ground set an R -module according to some axioms. When R is a field, we recover matroids. When R = , and when R is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids, i.e. tropical linear spaces, respectively. More generally, whenever R is a Dedekind domain, we extend all the usual properties and operations holding for matroids (e.g., duality), and...

Maximal compatible splitting and diagonals of Kempf varieties

Niels Lauritzen, Jesper Funch Thomsen (2011)

Annales de l’institut Fourier

Lakshmibai, Mehta and Parameswaran (LMP) introduced the notion of maximal multiplicity vanishing in Frobenius splitting. In this paper we define the algebraic analogue of this concept and construct a Frobenius splitting vanishing with maximal multiplicity on the diagonal of the full flag variety. Our splitting induces a diagonal Frobenius splitting of maximal multiplicity for a special class of smooth Schubert varieties first considered by Kempf. Consequences are Frobenius splitting of tangent bundles,...

Maximal non valuation domains in an integral domain

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

Let R be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring R of an integral domain S is called a maximal non valuation domain in S if R is not a valuation subring of S , and for any ring T such that R T S , T is a valuation subring of S . For a local domain S , the equivalence of an integrally closed maximal non VD in S and a maximal non local subring of S is established. The relation between dim ( R , S ) and the number...

Maximal non λ -subrings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

Let R be a commutative ring with unity. The notion of maximal non λ -subrings is introduced and studied. A ring R is called a maximal non λ -subring of a ring T if R T is not a λ -extension, and for any ring S such that R S T , S T is a λ -extension. We show that a maximal non λ -subring R of a field has at most two maximal ideals, and exactly two if R is integrally closed in the given field. A determination of when the classical D + M construction is a maximal non λ -domain is given. A necessary condition is given...

Maximal non-Jaffard subrings of a field.

Mabrouk Ben Nasr, Noôman Jarboui (2000)

Publicacions Matemàtiques

A domain R is called a maximal non-Jaffard subring of a field L if R ⊂ L, R is not a Jaffard domain and each domain T such that R ⊂ T ⊆ L is Jaffard. We show that maximal non-Jaffard subrings R of a field L are the integrally closed pseudo-valuation domains satisfying dimv R = dim R + 1. Further characterizations are given. Maximal non-universally catenarian subrings of their quotient fields are also studied. It is proved that this class of domains coincides with the previous class when R is integrally...

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