Invariant Subring Research Articles

Normal subalgebras of a polynomial ring

Let A be a finitely generated subalgebra of a polynomial ring k[x,y,z] over the complex field k. Assuming that A is normal, we clarify the structure of A under additional assumptions if dim⁡A≥2. If dim⁡A=2 and A is regular, then Spec A has an A1-fibration over P1 or A1 with restrictions on the number of multiple fibers (see Theorem 3). If dim⁡A=3, we assume that A is cofinite, i.e., k[x,y,z] is a finite A-module, and A contains a coordinate x. Then either A is a polynomial ring (see Theorem 6) or the invariant subring k[x,u,v]G with respect to a small finite subgroup G of GL(2,k) (see section 4). The results in section 4 are generalized in the next section.

Generalized Centrosymmetric Matrix Algebras Induced by Automorphisms

Let [Formula: see text] be a ring with an automorphism [Formula: see text] of order two. We introduce the definition of [Formula: see text]-centrosymmetric matrices. Denote by [Formula: see text] the ring of all [Formula: see text] matrices over [Formula: see text], and by [Formula: see text] the set of all [Formula: see text]-centrosymmetric [Formula: see text] matrices over [Formula: see text] for any positive integer [Formula: see text]. We show that [Formula: see text] is a separable Frobenius extension. If [Formula: see text] is commutative, then [Formula: see text] is a cellular algebra over the invariant subring [Formula: see text] of [Formula: see text].

Degree bounds for Hopf actions on Artin–Schelter regular algebras

We study semisimple Hopf algebra actions on Artin–Schelter regular algebras and prove several upper bounds on the degrees of the minimal generators of the invariant subring, and on the degrees of syzygies of modules over the invariant subring. These results are analogues of results for group actions on commutative polynomial rings proved by Noether, Fogarty, Fleischmann, Derksen, Sidman, Chardin, and Symonds.

Morita theorem for hereditary Calabi-Yau categories

We give a structure theorem for Calabi-Yau triangulated category with a hereditary cluster tilting object. We prove that an algebraic d-Calabi-Yau triangulated category with a d-cluster tilting object T such that its shifted sum T⊕⋯⊕T[−(d−2)] has hereditary endomorphism algebra H is triangle equivalent to the orbit category Db(modH)/τ−1/(d−1)[1] of the derived category of H for a naturally defined (d−1)-st root τ1/(d−1) of the AR translation, provided H is of non-Dynkin type. We also show that heredity of H follows from that of T when d=3, that of T⊕T[−1] when d=4, and similarly from a smaller endomorphism algebra for higher dimensions under vanishing of some negative self-extensions of T. Our result therefore generalizes the established theorems by Keller–Reiten and Keller–Murfet–Van den Bergh. Furthermore, we show that enhancements of such triangulated categories are unique. Finally we apply our results to Calabi-Yau reductions of a higher cluster category of a finite dimensional algebra and of the singularity category of an invariant subring.

New approaches to finite generation of cohomology rings

In support variety theory, representations of a finite dimensional (Hopf) algebra A can be studied geometrically by associating any representation of A to an algebraic variety using the cohomology ring of A. An essential assumption in this theory is the finite generation condition for the cohomology ring of A and that for the corresponding modules.In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra A, we show that the finite generation condition on A-modules can be replaced by a condition on any affine commutative A-module algebra R under the assumption that R is integral over its invariant subring RA. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if A is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra A′ over a finite field, where A can be viewed as a deformation of A′, and prove that if the finite generation condition holds for A′, then the same condition holds for A.

Semisimple Reflection Hopf Algebras of Dimension Sixteen

For each nontrivial semisimple Hopf algebra $H$ of dimension sixteen over $\mathbb{C}$, the smallest dimension inner-faithful representation of $H$ acting on a quadratic AS regular algebra $A$ of dimension 2 or 3, homogeneously and preserving the grading, is determined. Each invariant subring $A^H$ is determined. When $A^H$ is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that $H$ is a reflection Hopf algebra for $A$.

ZARISKI’S FINITENESS THEOREM AND PROPERTIES OF SOME RINGS OF INVARIANTS

In this paper we will give a short proof of a special case of Zariski’s result about finite generation in connection with Hilbert’s 14th problem using a new idea. Our result is useful for invariant subrings of unipotent or connected semisimple groups. We will also prove an analogue of Miyanishi’s result for the ring of invariants of a $$ {\mathbbm{G}}_a $$ -action on R[X, Y, Z] for an affine Dedekind domain R using topological methods.

Generalized gorensteinness and a homological determinant for preprojective algebras

The study of invariants of group actions on commutative polynomial rings has motivated many developments in commutative algebra and algebraic geometry. It has been of particular interest to understand what conditions on the group result in an invariant ring satisfying useful properties. In particular, Watanabe’s Theorem states that the invariant subring of under the natural action of a finite subgroup of is always Gorenstein. In this paper, we study this question in the more general setting of group actions on noncommutative non-connected algebras A. We develop the notion of a homological determinant of an automorphism of A, then use the homological determinant to study actions of finite groups G on A. We give a sufficient condition so that the invariant ring AG has finite injective dimension and satisfies the generalized Gorenstein condition. More precisely, let A be a noetherian -graded generalized Gorenstein algebra with finite global dimension. Suppose all elements of G fix the idempotents of A and act with trivial homological determinant. Then the invariant ring AG is generalized Gorenstein.

The symmetric signature of cyclic quotient singularities

The symmetric signature is an invariant of local domains which was recently introduced by Brenner and the first author in an attempt to find a replacement for the $F$-signature in characteristic zero. In the present note we compute the symmetric signature for two-dimensional cyclic quotient singularities, i.e. invariant subrings $k[[u,v]]^G$ of rings of formal power series under the action of a cyclic group $G$. Equivalently, these rings arise as the completions (at the irrelevant ideal) of two-dimensional normal toric rings. We show that for this class of rings the symmetric signature coincides with the $F$-signature.

Generators of truncated symmetric polynomials

Adem and Reichstein introduced the ideal of truncated symmetric polynomials to present the permutation invariant subring in the cohomology of a finite product of projective spaces. Building upon their work, I describe a generating set of the ideal of truncated symmetric polynomials in arbitrary positive characteristic, and offer a conjecture for minimal generators.

Equivariant class group. I. Finite generation of the Picard and the class groups of an invariant subring

The purpose of this paper is to define equivariant class group of a locally Krull scheme (that is, a scheme which is locally a prime spectrum of a Krull domain) with an action of a flat group scheme, study its basic properties, and apply it to prove the finite generation of the class group of an invariant subring.In particular, we prove the following.Let k be a field, G a smooth k-group scheme of finite type, and X a quasi-compact quasi-separated locally Krull G-scheme. Assume that there is a k-scheme Z of finite type and a dominant k-morphism Z→X. Let φ:X→Y be a G-invariant morphism such that OY→(φ⁎OX)G is an isomorphism. Then Y is locally Krull. If, moreover, Cl(X) is finitely generated, then Cl(G,X) and Cl(Y) are also finitely generated, where Cl(G,X) is the equivariant class group.In fact, Cl(Y) is a subquotient of Cl(G,X). For actions of connected group schemes on affine schemes, there are similar results of Magid and Waterhouse, but our result also holds for disconnected G. The proof depends on a similar result on (equivariant) Picard groups.

Generalized F-signature of invariant subrings

It is known that a certain invariant subring R has finite F-representation type. Thus, we can write the R-module Re as a finite direct sum of finitely many R-modules. In such a decomposition of Re, we pay attention to the multiplicity of each direct summand. For the multiplicity of free direct summand, there is the notion of F-signature defined by C. Huneke and G. Leuschke and it characterizes some singularities. In this paper, we extend this notion to non-free direct summands and determine their explicit values.

Equivariant Total Ring of Fractions and Factoriality of Rings Generated by Semi-Invariants

Let F be an affine flat group scheme over a commutative ring R, and S an F-algebra (an R-algebra on which F acts). We define an equivariant analogue Q F (S) of the total ring of fractions Q(S) of S. It is the largest F-algebra T such that S ⊂ T ⊂ Q(S), and S is an F-subalgebra of T. We study some basic properties. Utilizing this machinery, we give some new criteria for factoriality (unique factorization domain property) of (semi-)invariant subrings under the action of affine algebraic groups, generalizing a result of Popov. We also prove some variations of classical results on factoriality of (semi-)invariant subrings. Some results over an algebraically closed base field are generalized to those over an arbitrary base field.

Group actions on rings and the Čech complex

We have previously shown that, when a finite group G acts on a polynomial ring S in n variables over a finite field k, only finitely many isomorphism classes of indecomposable kG-modules occur as summands of S. We have also shown that the regularity of the invariant subring SG is at most zero, which has various consequences, for example SG is generated in degrees at most n(|G|−1) (provided n,|G|≥2). Both of these theorems depend on the Structure Theorem of Karagueuzian and the author, which is proved by means of a long and complicated calculation. The aim of this paper is to prove these results using a more conceptual method.

Conjugacy Classes and Invariant Subrings of R-Automorphisms of R[x

We consider the group G of R-automorphisms of the polynomial ring R[x] in the case where the ring R has nonzero nilpotent elements. Little is known about G in this case, and because of the importance of G in understanding questions involving the polynomial ring R[x], we initiate here several lines of investigation. We do this by examining in detail examples involving the ring of integers modulo n. If R is a local ring with maximal ideal m such that R/m = ℤ2 and m 2 = (0), we describe more explicitly the structure of G and determine all rings of invariants of R[x] with respect to subgroups of G.

Basic polynomial invariants, fundamental representations and the Chern class map

Consider a crystallographic root system together with its Weyl group W acting on the weight lattice \Lambda . Let \mathbb Z[\Lambda]^W and S(\Lambda)^W be the W -invariant subrings of the integral group ring \mathbb Z[\Lambda] and the symmetric algebra S(\Lambda) respectively. A celebrated result by Chevalley says that \mathbb Z[\Lambda]^W is a polynomial ring in classes of fundamental representations \rho_1,...,\rho_n and S(\Lambda)^W\otimes{Q} is a polynomial ring in basic polynomial invariants q_1,...,q_n . In the present paper we establish and investigate the relationship between \rho_i 's and q_i 's over the integers. As an application we provide estimates for the torsion of the Grothendieck \gamma -filtration and the Chow groups of some twisted flag varieties up to codimension 4.

Good filtrations and F-purity of invariant subrings

Let $k$ be an algebraically closed field of positive characteristic, $G$ a reductive group over $k$, and $V$ a finite dimensional $G$-module. Let $B$ be a Borel subgroup of $G$, and $U$ its unipotent radical. We prove that if $S=\Sym V$ has a good filtration, then $S^U$ is $F$-pure.

A Note on the Classical Dickson Invariants

Let 𝔽q be the finite field with q elements, where q is a power of some prime p. In the classical paper [2], Dickson defined the polynomial [Formula: see text], where e1,…, en are non-negative integers. He observed that any [e1,…, en] is divisible by [0,1,…,n-1], and the quotient is GL n(𝔽q)-invariant. He then went on to show that one can pick n of these invariants to generate the entire invariant subring of GL n(𝔽q). In this paper, we answer the following two natural questions: How to express any given [e1,…, en]/[0,1,…,n-1] in terms of the fundamental generators? In general, when does [e1,…, en] divide [f1,…, fn]? The answers are Theorems 1.2 and 1.3, respectively.

Strong Lefschetz Elements of the Coinvariant Rings of Finite Coxeter Groups

For the coinvariant rings of finite Coxeter groups of types other than H$_4$, we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups.

Lifting of the additive group scheme actions

Let $B$ be a normal affine $\boldsymbol{C}$-domain and let $A$ be a $\boldsymbol{C}$-subalgebra of $B$ such that $B$ is a finite $A$-module. Let $\delta$ be a locally nilpotent derivation on $A$. Then $\delta$ lifts uniquely to the quotient field $L$ of $B$, which we denote by $\Delta$. We consider when $\Delta$ is a locally nilpotent derivation of $B$. This is a classical subject treated in [17, 19, 16]. We are interested in the case where $A$ is the $G$-invariant subring of $B$ when a finite group $G$ acts on $B$. As a related topic, we treat in the last section the finite coverings of log affine pseudo-planes in terms of the liftings of the $\boldsymbol{A}^1$-fibrations associated with locally nilpotent derivations.