Noetherian K-algebra Research Articles

The module structure of a group action on a ring

Consider a finite group G acting on a graded Noetherian k-algebra S, for some field k of characteristic p; for example S might be a polynomial ring. Regard S as a kG-module and consider the multiplicity of a particular indecomposable module as a summand in each degree. We show how this can be described in terms of homological algebra and how it is linked to the geometry of the group action on the spectrum of S.

Universal deformation rings of modules for generalized Brauer tree algebras of polynomial growth

Let k be an arbitrary field, be a k-algebra and V be a -module. When it exists, the universal deformation ring of V is a k-algebra whose local homomorphisms to R parametrize the lifts of V up to , where R is any complete, local commutative Noetherian k-algebra with residue field k. Symmetric special biserial algebras, which coincide with Brauer graph algebras, can be viewed as generalizing the blocks of finite type p-modular group algebras. Bleher and Wackwitz classified the universal deformation rings for all modules for symmetric special biserial algebras with finite representation type. In this paper, we begin to address the tame case. Specifically, let be any 1-domestic, symmetric special biserial algebra. By viewing as generalized Brauer tree algebras and making use of a derived equivalence, we classify the universal deformation rings for those -modules V with stable endomorphism ring isomorphic to k. The latter is a natural condition, since it guarantees the existence of the universal deformation ring .

Homological aspects of derivation modules and critical case of the Herzog–Vasconcelos conjecture

Let R be a Noetherian local k-algebra whose derivation module $${\mathrm{Der}}_k(R)$$ is finitely generated. Our main goal in this paper is to investigate the impact of assuming that $${\mathrm{Der}}_k(R)$$ has finite projective dimension (or finite Gorenstein dimension), mainly in connection with freeness, under a suitable hypothesis concerning the vanishing of (co)homology or the depth of a certain tensor product. We then apply some of our results towards the critical case $${\mathrm{depth}}\,R=3$$ of the Herzog–Vasconcelos conjecture and consequently to the strong version of the Zariski–Lipman conjecture.

Computing Quot schemes via marked bases over quasi-stable modules

Let k be a field of arbitrary characteristic, A a Noetherian k-algebra and consider the polynomial ring A[x]=A[x0,…,xn]. We consider graded submodules of A[x]m having a special set of generators, a marked basis over a quasi-stable module. Such a marked basis shares many interesting properties with a Gröbner basis, including the existence of a Noetherian reduction relation. The set of submodules of A[x]m having a marked basis over a given quasi-stable module possesses a natural affine scheme structure which we will exhibit. Furthermore, the syzygies of a module with such a marked basis are generated by a marked basis, too (over a suitable quasi-stable module in ⊕i=1m′A[x](−di)). We apply marked bases and related properties to the investigation of Quot functors (and schemes). More precisely, for a given Hilbert polynomial, we explicitly construct (up to the action of a general linear group) an open cover of the corresponding Quot functor, made up of open subfunctors represented by affine schemes. This provides a new proof that the Quot functor is the functor of points of a scheme. We also exhibit a procedure to obtain the equations defining a given Quot scheme as a subscheme of a suitable Grassmannian. Thanks to the good behaviour of marked bases with respect to Castelnuovo-Mumford regularity, we can adapt our methods in order to study the locus of the Quot scheme given by an upper bound on the regularity of its points.

The Dixmier-Moeglin equivalence, Morita equivalence, and homeomorphism of spectra

Let k be a field and let R be a left noetherian k-algebra. The algebra R satisfies the Dixmier-Moeglin equivalence if the annihilators of irreducible representations are precisely those prime ideals that are locally closed in the Spec(R) and if, moreover, these prime ideals are precisely those whose extended centres are algebraic extensions of the base field. We show that if R and S are two left noetherian k-algebras with dimk(R),dimk(S)<|k| then if R and S have homeomorphic spectra then R satisfies the Dixmier-Moeglin equivalence if and only if S does. In particular, the topology of Spec(R) can detect the Dixmier-Moeglin equivalence in this case. In addition, we show that if k is uncountable and R is affine noetherian and its prime spectrum is a finite disjoint union of locally closed subspaces that are each homeomorphic to the spectrum of an affine commutative ring then R satisfies the Dixmier-Moeglin equivalence. We show that neither of these results need hold if k is countable and R is infinite-dimensional. Finally, we make the remark that satisfying the Dixmier-Moeglin equivalence is a Morita invariant and finally we show that R and S are left noetherian k-algebras that satisfy the Dixmier-Moeglin equivalence then R⊗kS does too, provided it is left noetherian and satisfies the Nullstellensatz; and we show that eRe also satisfies the Dixmier-Moeglin equivalence, where e is a nonzero idempotent of R.

Deformations of complexes for finite dimensional algebras

Let k be a field and let Λ be a finite dimensional k-algebra. We prove that every bounded complex V• of finitely generated Λ-modules has a well-defined versal deformation ring R(Λ,V•) which is a complete local commutative Noetherian k-algebra with residue field k. We also prove that nice two-sided tilting complexes between Λ and another finite dimensional k-algebra Γ preserve these versal deformation rings. Additionally, we investigate stable equivalences of Morita type between self-injective algebras in this context. We apply these results to the derived equivalence classes of the members of a particular family of algebras of dihedral type that were introduced by Erdmann and shown by Holm to be not derived equivalent to any block of a group algebra.

On Solutions for Derivations of a Noetherian k-algebra and Local Simplicity

We introduce a general notion of solution for a Noetherian differential k-algebra and study its relationship with simplicity, where k is an algebraically closed field; then we analyze conditions under which such solutions may exist and be unique, with special emphasis in the cases of k-algebras of finite type and formal series rings over k. Using that notion we generalize a criterion for simplicity due to Brumatti–Lequain–Levcovitz and give a geometric characterization of that; as an application we give a new proof of a classification theorem for local simplicity due to Hart and obtain a general result for simplicity of formal series rings over k.

A relative Hilbert–Mumford criterion

We generalize the classical Hilbert-Mumford criteria for GIT (semi-)stability in terms of one parameter subgroups of a linearly reductive group G over a field k, to the relative situation of an equivariant, projective morphism X -> Spec A to a noetherian k-algebra A. We also extend the classical projectivity result for GIT quotients: the induced morphism X^ss/G -> Spec A^G is projective. As an example of applications to moduli problems, we consider degenerations of Hilbert schemes of points.

Universal Extension Classes for GL 2

In this note we give a new existence proof for the universal extension classes for G L 2 previously constructed by Friedlander and Suslin via the theory of strict polynomial functors. The key tool in our approach is a calculation of Parker showing that, for suitable choices of coefficient modules, the Lyndon–Hochschild–Serre spectral sequence for S L 2 relative to its first Frobenius kernel stabilizes at the E 2-page. Consequently, we obtain a new proof that if G is an infinitesimal subgroup scheme of G L 2, then the cohomology ring H∙(G,k) of G is a finitely-generated noetherian k-algebra.

Universal deformation rings of modules over Frobenius algebras

Let k be a field, and let Λ be a finite dimensional k-algebra. We prove that if Λ is a self-injective algebra, then every finitely generated Λ-module V whose stable endomorphism ring is isomorphic to k has a universal deformation ring R(Λ,V) which is a complete local commutative Noetherian k-algebra with residue field k. If Λ is also a Frobenius algebra, we show that R(Λ,V) is stable under taking syzygies. We investigate a particular Frobenius algebra Λ0 of dihedral type, as introduced by Erdmann, and we determine R(Λ0,V) for every finitely generated Λ0-module V whose stable endomorphism ring is isomorphic to k.

A class of noncommutative projective surfaces

Let A = ⊕i⩾0Ai be a connected graded, noetherian k-algebra that is generated in degree one over an algebraically closed field k. Suppose that the graded quotient ring Q(A) has the form Q(A) = k(X)[t, t−1; σ], where σ is an automorphism of the integral projective surface X. Then we prove that A can be written as a naïve blowup algebra of a projective surface 𝕏 birational to X. This enables one to obtain a deep understanding of the structure of these algebras; for example, generically they are not strongly noetherian and their point modules are not parametrized by a projective scheme. This is despite the fact that the simple objects in qgr-A will always be in (1-1) correspondence with the closed points of the scheme 𝕏.

Reduced Tangent Cones and Conductor at Multiplanar Isolated Singularities

Let A be a noetherian local k-algebra with residue field k and associated graded ring G(A). Let 𝔟 be the conductor of A in its normalization , which we assume to be regular and finite over A. Assume also that Proj(G(A)) is reduced and . Let r be the embedding dimension of A and be the Hilbert function of . We show that if the Hilbert function of Proj(G(A)) is equal to , then G(A) is reduced. This allows us to compute, in the general case, the conductor of the local ring of singular isolated points on surfaces whose tangent cone is multiplanar; that is, it consists—as a set—of planes. We explicitly construct a class of such surfaces in the rational case.

Noetherian algebras over algebraically closed fields

Let k be an uncountable algebraically closed field and let A be a countably generated left Noetherian k-algebra. Then we show that A ⊗ k K is left Noetherian for any field extension K of k. We conclude that all subfields of the quotient division algebra of a countably generated left Noetherian domain over k are finitely generated extensions of k. We give examples which show that A ⊗ k K need not remain left Noetherian if the hypotheses are weakened.

The rigid dualizing complex of a universal enveloping algebra

Let k be a field and A a noetherian (noncommutative) k-algebra. The rigid dualizing complex of A was introduced by Van den Bergh. When A= U( g) , the enveloping algebra of a finite dimensional Lie algebra g , Van den Bergh conjectured that the rigid dualizing complex is ( U( g)⊗⋀ n g)[n] , where n= dim g . We prove this conjecture, and give a few applications in representation theory and Hochschild cohomology.

Classification of group representations on a local [formula omitted]-algebra

Let O be a local and complete noetherian k-algebra and let φ : Aut .5ptk( O) → Aut .5ptk( G m ̄ O) be the natural morphism between the group of automorphisms of O and the group of automorphisms of its tangent space. The main result of this paper is to show that Ker φ is an inverse limit of unipotent algebraic groups, that is, Ker φ= lim ←K j , j ∈ N . As a consequence: (1) we can describe the structure of the set of classes of representations ρ : G → Aut .5ptk( O) in terms of the groups H ℓ(G,N i) , ℓ=1,2, where N i are the additive factor groups of a natural resolution of K j ; (2) theorems are obtained for classifying representations valid for any dimension, any characteristic and even when O is not regular; (3) we obtain the complete classification of the representations of Z/p on k[[z]] when p=char(k)>0; and (4) an algorithm Σ is obtained which allows us to calculate successive approximations of the set of classes of representations of Z/p on k[[u,v]], p=char(k)>0.

Localisation in rings equipped with a solvable Lie algebra action

Let k be an algebraically closed uncountable field of characteristic 0,g a finite dimensional solvable k-Lie algebraR a noetherian k-algebra on which g acts by k-derivationsU(g) the enveloping algebra of g,A=R*g the crossed product of R by U(g)P a prime ideal of A and Ω(P) the clique of P. Suppose that the prime ideals of the polynomial ring R[x] are completely prime. If R is g-hypernormal, then Ω(P) is classical. Denote by AT the localised ring and let M be a primitive ideal of AT Set Q=P∩R In this note, we show that if R is a strongly (R,g)-admissible integral domain and if QRQ is generated by a regular g-centralising set of elements, then (1)M is generated by a regular g-semi-invariant normalising set of elements of cardinald = dim (RQ 0 + ∣XA (P)∣ (2)d gldim(AT ) = Kdim(AT ) = ht(M) = ht(P).

A note on linearization of actions of finitely semisimple Hopf algebras on local algebras

Let H be a Hopf algebra over a field k and let H ⊗ A → A, h ⊗ a → h. a, be an action of H on a commutative local Noetherian kalgebra ( A, m). We say that this action is linearizable if there exists a minimal system x 1, …, x n of generators of the maximal ideal m such that h. x i ϵ kx 1 + …+ kx n for all h ϵ H and i = 1, …, n. In the paper we prove that the actions from a certain class are linearizable (see Theorem 4), and we indicate some consequences of this fact.

Rings of Operators on Modules over Commutative Rings and Their Right Ideals

Suppose that there is an inclusion ofk-algebrasR⊆E⊆EndkMwithRcommutative andEnon-commutative. We introduce and impose conditions under which the finitely generated essential right ideals ofEmay be classified in terms ofk-submodules ofM. This yields a classification of the domains Morita equivalent toEwhenEis a Noetherian domain. For example, a special case of our results is:THEOREM.Let R be a commutative Noetherian k-algebra which is domain. Let E be a simple Ore extension of R of the form R[x,x−1;σ]or R[x;δ] (in the latter case we must also assume R⊃3).Then,for a certain sublatticeLof the lattice of k-submodules of R:(a)Every non-zero right ideal of E is isomorphic to one of the form[formula]for some V∈L.(b)Every domain Morita equivalent to E is isomorphic to[formula]for some V∈L.Conversely,if R is Dedekind,then E(V)is Morita equivalent to E,for V∈L.

Residues and effective Nullstellensatz

Let K be a commutative field; an algorithmic approach to residue symbols defined on a Noetherian K-algebra R has been developed. It is used to prove an effective Nullstellensatz for polynomials defined over infinite factorial rings A equipped with a size. This result extends (and slightly improves) the previous work of the authors in the case A = Z. Let p1, . . . , pM be polynomials in n variables with coefficients in an integral domain A, and respective degrees D1 ≥ D2 ≥ · · · ≥ DM , with no common zeros in an integral closure of the quotient field K of A. It follows from the versions of the Hilbert Nullstellensatz in [B], [CGH], [Ko] that one can find an element r0 ∈ A\{0} and polynomials qj ∈ A[x] such that r0 = M ∑ j=1 qjpj (1) with a priori estimates on the degrees max j deg qj ≤ (3/2)D1 · · ·Dμ, where μ = min{n,M} and ι = max{j : 1 ≤ j < μ − 1, Dj = 2}. When A = Z, using analytic methods, and mainly integral representation formulas and multidimensional residues in C, one can show [BGVY, Section 5] that system (1) can be solved with the estimates  max j deg qj ≤ n(2n+ 1)(3/2) μ ∏ j=1 Dj , maxh(qj) ≤ κ(n)D 1 ( μ ∏ j=1 Dj )8 (h+ logM +D1 logD1), (2) for the Mahler size h(qj) in terms of the maximal Mahler size h of the original polynomials pj . Recall that the Mahler size of p ∈ Z[x1, . . . , xn] is given by h(p) = ∫ [0,1]n log |p(e1 , . . . , en)| dθ . For A = Z, one can also recover the estimate for log |r0|, implicit in (2), from the Arithmetic Bezout Theorem (see [Ph2],[BGS, Theorem 5.4.4]), which shows that Received by the editors April 15, 1996. 1991 Mathematics Subject Classification. Primary 14Q20; Secondary 13F20, 14C17, 32C30.

Standard prime ideals and lying over for finite extensions of Noetherian algebras

Let k be a field, let R be a noetherian k-algebra of finite Gelfand-Kirillov dimension GK(R), and let M be a finitely generated right R-module. A standard prime factor series for M is a finite sequence of submodules 0 = N0 ⊂ N1 ⊂…⊂ Ni−1 ⊂ Ni ⊂.… ⊂ Nn = M, such that for each i the annihilator Pi = rR (Ni/Ni−1) is the unique associated prime of Ni/Ni−1 and GK(R/Pi)≤ GK(R/Pj) whenever i≤ j. The set of prime ideals arising from such a series is an invariant of M, called the set of standard primes St(M) of M. The concept, inspired by the notion of a standard affiliated series introduced by Lenagan and Warfield in [7], has been developed in [5], where it was shown that St(M) coincides with the set of all those prime ideals that are minimal over the annihilator of a nonzero submodule of M.