LOCAL DUALITY FOR THE SINGULARITY CATEGORY OF A FINITE DIMENSIONAL GORENSTEIN ALGEBRA

A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the $\mathfrak{p}$ -local and $\mathfrak{p}$ -torsion subcategories of the derived category, for each homogeneous prime ideal $\mathfrak{p}$ arising from the action of a commutative ring via Hochschild cohomology.

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On Grothendieck's local duality theorem

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100055870 ◽

1979 ◽

Vol 85 (3) ◽

pp. 431-437 ◽

Cited By ~ 3

Author(s):

M. H. Bijan-Zadeh ◽

R. Y. Sharp

Keyword(s):

Algebraic Geometry ◽

Commutative Algebra ◽

Noetherian Ring ◽

Duality Theorem ◽

Great Emphasis ◽

Triangulated Category ◽

Elementary Approach ◽

Commutative Noetherian Ring ◽

Local Duality ◽

Dualizing Complexes

In (11) and (12), a comparatively elementary approach to the use of dualizing complexes in commutative algebra has been developed. Dualizing complexes were introduced by Grothendieck and Hartshorne in (2) for use in algebraic geometry; the approach to dualizing complexes in (11) and (12) differs from that of Grothendieck and Hartshorne in that it avoids use of the concepts of triangulated category, derived category, and localization of categories, and instead places great emphasis on the concept of quasi-isomorphism of complexes of modules over a commutative Noetherian ring.

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AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

International Journal of Algebra and Computation ◽

10.1142/s021819670700372x ◽

2007 ◽

Vol 17 (03) ◽

pp. 527-555 ◽

Cited By ~ 5

Author(s):

YOU'AN CAO ◽

DEZHI JIANG ◽

JUNYING WANG

Keyword(s):

Automorphism Group ◽

Commutative Ring ◽

Lie Algebra ◽

Lie Algebras ◽

Commutative Rings ◽

Nilpotent Lie Algebras ◽

Simple Lie Algebra ◽

Finite Dimensional ◽

Positive Roots ◽

Nilpotent Subalgebra

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

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A generalized ideal-based zero-divisor graph

Journal of Algebra and Its Applications ◽

10.1142/s0219498815500796 ◽

2015 ◽

Vol 14 (06) ◽

pp. 1550079 ◽

Cited By ~ 1

Author(s):

M. J. Nikmehr ◽

S. Khojasteh

Keyword(s):

Prime Ideal ◽

Commutative Ring ◽

Chromatic Number ◽

Zero Divisor ◽

Undirected Graph ◽

Clique Number ◽

Proper Ideal ◽

Graph Structure ◽

Zero Divisor Graph ◽

Vertex Set

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.

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Tate and Tate–Hochschild cohomology for finite dimensional Hopf algebras

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2013.01.008 ◽

2013 ◽

Vol 217 (10) ◽

pp. 1967-1979 ◽

Cited By ~ 2

Author(s):

Van C. Nguyen

Keyword(s):

Hopf Algebras ◽

Hochschild Cohomology ◽

Finite Dimensional

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On the Integral Extensions of Quadratic Forms Over Local Fields

Canadian Journal of Mathematics ◽

10.4153/cjm-1970-037-x ◽

1970 ◽

Vol 22 (2) ◽

pp. 297-307 ◽

Cited By ~ 2

Author(s):

Melvin Band

Keyword(s):

Prime Ideal ◽

Local Field ◽

Quadratic Forms ◽

Sufficient Conditions ◽

Local Fields ◽

Ring Of Integers ◽

Finite Dimensional ◽

Integral Analogue ◽

Necessary And Sufficient ◽

Special Case

Let F be a local field with ring of integers and unique prime ideal (p). Suppose that V a finite-dimensional regular quadratic space over F, W and W′ are two isometric subspaces of V (i.e. τ: W → W′ is an isometry from W to W′). By the well-known Witt's Theorem, τ can always be extended to an isometry σ ∈ O(V).The integral analogue of this theorem has been solved over non-dyadic local fields by James and Rosenzweig [2], over the 2-adic fields by Trojan [4], and partially over the dyadics by Hsia [1], all for the special case that W is a line. In this paper we give necessary and sufficient conditions that two arbitrary dimensional subspaces W and W′ are integrally equivalent over non-dyadic local fields.

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Tensor Products and Bimorphisms

Canadian Mathematical Bulletin ◽

10.4153/cmb-1976-060-2 ◽

1976 ◽

Vol 19 (4) ◽

pp. 385-402 ◽

Cited By ~ 48

Author(s):

Bernhard Banaschewski ◽

Evelyn Nelson

Keyword(s):

Tensor Product ◽

Commutative Ring ◽

Tensor Products ◽

Vector Spaces ◽

Dimensional Vector ◽

Bilinear Maps ◽

Special Situation ◽

Dual Spaces ◽

Finite Dimensional

The binary tensor product, for modules over a commutative ring, has two different aspects: its connection with universal bilinear maps and its adjointness to the internal hom-functor. Furthermore, in the special situation of finite-dimensional vector spaces, the tensor product can also be described in terms of dual spaces and the internal hom-functor. The aim of this paper is to investigate these relationships in the setting of arbitrary concrete categories.

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A characterization of minimal prime ideals

Glasgow Mathematical Journal ◽

10.1017/s0017089500032547 ◽

1998 ◽

Vol 40 (2) ◽

pp. 223-236 ◽

Cited By ~ 13

Author(s):

Gary F. Birkenmeier ◽

Jin Yong Kim ◽

Jae Keol Park

Keyword(s):

Positive Integer ◽

Prime Ideal ◽

Commutative Ring ◽

Minimal Prime Ideal ◽

Prime Ideals ◽

Sheaf Representations ◽

Minimal Prime

AbstractLet P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

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On the Prime Ideals in a Commutative Ring

Canadian Mathematical Bulletin ◽

10.4153/cmb-2000-038-7 ◽

2000 ◽

Vol 43 (3) ◽

pp. 312-319 ◽

Cited By ~ 4

Author(s):

David E. Dobbs

Keyword(s):

Prime Ideal ◽

Commutative Ring ◽

Polynomial Ring ◽

Sufficient Conditions ◽

Prime Ideals ◽

Preceding Result ◽

Finite Commutative Ring ◽

The Axiom Of Choice ◽

Necessary And Sufficient ◽

Positive Integers

AbstractIf n and m are positive integers, necessary and sufficient conditions are given for the existence of a finite commutative ring R with exactly n elements and exactly m prime ideals. Next, assuming the Axiom of Choice, it is proved that if R is a commutative ring and T is a commutative R-algebra which is generated by a set I, then each chain of prime ideals of T lying over the same prime ideal of R has at most 2|I| elements. A polynomial ring example shows that the preceding result is best-possible.

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Normality in Elementary Subgroups of Chevalley Groups Over Rings

Canadian Journal of Mathematics ◽

10.4153/cjm-1976-042-2 ◽

1976 ◽

Vol 28 (2) ◽

pp. 420-428 ◽

Cited By ~ 1

Author(s):

James F. Hurley

Keyword(s):

Normal Subgroup ◽

Commutative Ring ◽

Lie Algebra ◽

Chevalley Group ◽

Complex Field ◽

Normal Subgroups ◽

Chevalley Groups ◽

Simple Lie Algebra ◽

Elementary Subgroup ◽

Finite Dimensional

In [6] we have constructed certain normal subgroups G7 of the elementary subgroup GR of the Chevalley group G(L, R) over R corresponding to a finite dimensional simple Lie algebra L over the complex field, where R is a commutative ring with identity. The method employed was to augment somewhat the generators of the elementary subgroup EI of G corresponding to an ideal I of the underlying Chevalley algebra LR;EI is thus the group generated by all xr(t) in G having the property that ter ⊂ I. In [6, § 5] we noted that in general EI actually had to be enlarged for a normal subgroup of GR to be obtained.

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