On the Hochschild homology of quantum SL(N)

We show that the quantized coordinate ring $A:=k_q[\mathit{SL}(N)]$ satisfies van den Bergh's analogue of Poincaré duality for Hochschild (co)homology with dualizing bimodule being $A_{\sigma }$, the A-bimodule which is A as k-vector space with right multiplication twisted by the modular automorphism σ of the Haar functional. This implies that ${\mathrm{H}}_{N^2?1}(A,A_{\sigma })?k$, generalizing our previous result for $k_q[\mathit{SL}(2)]$. To cite this article: T. Hadfield, U. Krähmer, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

On low-dimensional cancellation problems

A well-known cancellation problem of Zariski asks when, for two given domains (fields) $K_1$ and $K_2$ over a field k, a k-isomorphism of $K_1[t]$ ($K_1(t)$) and $K_2[t]$ ($K_2(t)$) implies a k-isomorphism of $K_1$ and $K_2$. The main results of this article give affirmative answer to the two low-dimensional cases of this problem:1. Let K be an affine field over an algebraically closed field k of any characteristic. Suppose $K(t)?k(t_1,t_2,t_3)$, then $K?k(t_1,t_2)$.2. Let M be a 3-dimensional affine algebraic variety over an algebraically closed field k of any characteristic. Let $A=K[x,y,z,w]/M$ be the coordinate ring of M. Suppose $A[t]?k[x_1,x_2,x_3,x_4]$, then $\mathrm{frac}(A)?k(x_1,x_2,x_3)$, where $\mathrm{frac}(A)$ is the field of fractions of A.In the case of zero characteristic these results were obtained by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141–154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165–171]. However, the case of finite characteristic is first settled in this article, that answered the questions proposed by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141–154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165–171].     

On Noether's problem for central extensions of symmetric and alternating groups

For a field k and a finite group G acting regularly on a set of indeterminates $\underset{?}{X}={\{X_g\}}_{g\in G}$, let $k(G)$ denote the invariant field $k{(\underset{?}{X})}^G$. We first prove for the alternating group $A_n$ that, if n is odd, then $\mathbb{Q}(A_n)$ is rational over $\mathbb{Q}(A_{n?1})$. We then obtain an analogous result where $A_n$ is replaced by an arbitrary finite central extension of either $A_n$ or $S_n$, valid over $\mathbb{Q}({\zeta }_N)$ for suitable N. Concrete applications of our results yield: (1) a new proof of Maeda's result on the rationality of $\mathbb{Q}{(X_1,\dots ,X_5)}^{A_5}/\mathbb{Q}$; (2) an affirmative answer to Noether's problem over $\mathbb{Q}$ for both $\stackrel{?}{A_5}$ and $\stackrel{?}{S_5}$; (3) an affirmative answer to Noether's problem over $\mathbb{C}$ for every finite central extension group of either $A_n$ or $S_n$ with $n?5$.     

Some extensions of the Cauchy-Davenport theorem

The Cauchy-Davenport theorem states that, if p is prime and A, B are nonempty subsets of cardinality r, s in $\mathbb{Z}/p\mathbb{Z}$, the cardinality of the sumset $A+B=\{a+b|a\in A,b\in B\}$ is bounded below by $\min (r+s-1,p)$; moreover, this lower bound is sharp. Natural extensions of this result consist in determining, for each group G and positive integers $r,s⩽|G|$, the analogous sharp lower bound, namely the function${\mu }_G(r,s)=\min \{|A+B||A,B\subset G,|A|=r,|B|=s\}.$ Important progress on this topic has been achieved in recent years, leading to the determination of ${\mu }_G$ for all abelian groups G. In this note we survey the history of earlier results and the current knowledge on this function.     

Histories in Path Graphs

For a given graph G and a positive integer r the r-path graph, $P_r(G)$, has for vertices the set of all paths of length r in G. Two vertices are adjacent when the intersection of the corresponding paths forms a path of length $r-1$, and their union forms either a cycle or a path of length $r+1$ in G. Let $P_r^k(G)$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of $P_r^k(G)$. The k-history $P_r^{-k}(H)$ is a subgraph of G that is induced by all edges that take part in the recursive definition of H. We present some general properties of k-histories and give a complete characterization of graphs that are k-histories of vertices of 2-path graph operator.     

On the distance between linear codes

Let V be an n-dimensional vector space over the finite field consisting of q elements and let ${\mathrm{\Gamma }}_k(V)$ be the Grassmann graph formed by k-dimensional subspaces of V, $1<k<n-1$. Denote by $\mathrm{\Gamma }{(n,k)}_q$ the restriction of ${\mathrm{\Gamma }}_k(V)$ to the set of all non-degenerate linear ${[n,k]}_q$ codes. We show that for any two codes the distance in $\mathrm{\Gamma }{(n,k)}_q$ coincides with the distance in ${\mathrm{\Gamma }}_k(V)$ only in the case when $n<{(q+1)}^2+k-2$, i.e. if n is sufficiently large then for some pairs of codes the distances in the graphs ${\mathrm{\Gamma }}_k(V)$ and $\mathrm{\Gamma }{(n,k)}_q$ are distinct. We describe one class of such pairs.