The Noether numbers for cyclic groups of prime order

Invariants of normal local rings by p-cyclic group actions

Let $B$ be a Noetherian normal local ring, and $G\subset\Aut(B)$ a cyclic group of local automorphisms of prime order. Let $A$ be the ring of $G$-invariants of $B$, assume that $A$ is Noetherian. We study the invariant morphism; in particular, we prove that $B$ is a monogenous $A$-algebra if and only if the augmentation ideal of $B$ is principal. If in particular $B$ is regular, we prove that $A$ is regular if the augmentation ideal of $B$ is principal.

Gorenstein isolated quotient singularities of odd prime dimension are cyclic

Gorenstein isolated quotient singularities of odd prime dimension are cyclic. In the case where the dimension is bigger than 1 and is not an odd prime number, then there exist Gorenstein isolated non-cyclic quotient singularities.

Groups with large Noether bound

The finite groups having an indecomposable polynomial invariant whose degree is at least half of the order of the group are classified. Apart from four sporadic exceptions these are exactly the groups having a cyclic subgroup of index at most two.

Noether's problem for the groups with a cyclic subgroup of index 4

Let $G$ be a finite group and $k$ be a field. Let $G$ act on the rational function field $k(x_g:g\in G)$ by $k$-automorphisms defined by $g\cdot x_h=x_{gh}$ for any $g,h\in G$. Noether's problem asks whether the fixed field $k(G)=k(x_g:g\in G)^G$ is rational (i.e. purely transcendental) over $k$. Theorem 1. If $G$ is a group of order $2^n$ ($n\ge 4$) and of exponent $2^e$ such that (i) $e\ge n-2$ and (ii) $ζ_{2^{e-1}} \in k$, then $k(G)$ is $k$-rational. Theorem 2. Let $G$ be a group of order $4n$ where $n$ is any positive integer (it is unnecessary to assume that $n$ is a power of 2). Assume …

Noether's problem for some p-groups

Let K be any field and G be a finite group. Noether's problem asks whether the fixed field is rational (=purely transcendental) over K. We will prove that if G is a non-abelian p-group of order p^n containing a cyclic subgroup of index p and K is any field containing a primitive p^{n-2}-th root of unity, then K(G) is rational over K.

On Commutative Rings Whose Prime Ideals Are Direct Sums of Cyclics

In this paper we study commutative rings $R$ whose prime ideals are direct sums of cyclic modules. In the case $R$ is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring $(R, \cal{M})$, the following statements are equivalent: (1) Every prime ideal of $R$ is a direct sum of cyclic $R$-modules; (2) ${\cal{M}}=\bigoplus_{λ\in Λ}Rw_λ$ and $R/{\rm Ann}(w_λ)$ is a principal ideal ring for each $λ\in Λ$;(3) Every prime ideal of $R$ is a direct sum of at most $|Λ|$ cyclic $R$-modules; and (4) Every p…

Cyclic p-roots of prime lengths p and related complex Hadamard matrices

In this paper it is proved, that for every prime number p, the set of cyclic p-roots in C^p is finite. Moreover the number of cyclic p-roots counted with multiplicity is equal to (2p-2)!/(p-1)!^2. In particular, the number of complex circulant Hadamard matrices of size p, with diagonal entries equal to 1, is less or equal to (2p-2)!/(p-1)!^2.

分离与普通诺特数

Separating versus ordinary Noether numbers

本文研究了有限群的分离诺特数与普通诺特数的关系，发现当群具有特定子群时，分离诺特数与普通诺特数相等，并确定了某些非交换群的分离诺特数特性。

计数半素数阶循环群上的Schur环

Counting Schur Rings over Cyclic Groups of Semi-prime Order

本文继续研究循环群上的Schur环计数问题，重点分析半素数阶循环群pq及4p阶循环群的结构与性质。

A study of cofiniteness through minimal associated primes

In this paper we shall investigate the concepts of cofiniteness of local cohomology modules and Abelian categories of cofinite modules over arbitrary Noetherian rings. Then we shall improve some of the results given in the literature.

Ideal Theory in Rings (Translation of "Idealtheorie in Ringbereichen" by Emmy Noether)

This paper is a translation of the paper "Idealtheorie in Ringbereichen", written by Emmy Noether in 1920, from the original German into English. It in particular brings the language used into the modern world so that it is easily understandable by the mathematicians of today. The paper itself deals with ideal theory, and was revolutionary in its field, that is modern algebra. Topics covered include: the representation of an ideal as the least common multiple of irreducible ideals; the representation of an ideal as the least common multiple of maximal primary ideals; the association of prime i…

Coinvariants for modular representations of cyclic groups of prime order

We consider the ring of coinvariants for modular representations of cyclic groups of prime order. For all cases for which explicit generators for the ring of invariants are known, we give a reduced Gröbner basis for the Hilbert ideal and the corresponding monomial basis for the coinvariants. We also describe the decomposition of the coinvariants as a module over the group ring. For one family of representations, we are able to describe the coinvariants despite the fact that an explicit generating set for the invariants is not known. In all cases our results confirm the conjecture of Harm Derks…

Retract rationality and Noether's problem

Let K be any field and G be a finite group. We will prove that, if K is any field, p an odd prime number, and G is a non-abelian group of exponent p with |G|=p^3 or p^4 satisfying [K(ζ_p):K] <= 2, then K(G) is rational over K. We will also show that K(G) is retract rational if G belongs to a much larger class of p-groups. In particular, generic G-polynomials of G-Galois extensions exist for these groups.

循环群的 Burnside Tambara 函子的谱

The Spectrum of the Burnside Tambara Functor of a Cyclic Group

本文研究有限群 Burnside Tambara 函子的素理想谱，构造了一族素理想，并证明对循环群该族构成完整素谱，同时给出向更广泛群类推广的部分结果。