论文标题
在单层代数和Segre扩展
On Serre dimension of monoid algebras and Segre extensions
论文作者
论文摘要
让$ r $成为尺寸$ d $和$ m $的noetherian戒指，是$ r $ r $的$ d $ $ d $，$ nocutative $，$ nocelative $。然后$ s $ - $ dim（r [m]）\ leq max \ {1，dim（r [m]）-1 \} = max \ {1，d+r-1 \} $。此外，$，$，我们定义一类单型$ \ {\ Mathfrak {m} _n \} _ {n \ geq 1} $，以便如果$ m \ in \ mathfrak {m mathfrak {m} _n $ is eminoral $，eminoral $，emmoral $，$ s $ s $ s $ s $（r [m] $ dim（r [m]） n \ leq r $。作为一个应用程序，我们证明，对于Segre Extension $ S_ {Mn}（r）$ of $ r，$ $ $ s $ - $ dim（s_ {mn}（r））\ leq dim（s_ {mn}（mn}（r）） - \ big [\ frac {m+n-1} {min \ {m，n \}}} \ big] $。
Let $R$ be a commutative noetherian ring of dimension $d$ and $M$ be a commutative$,$ cancellative$,$ torsion-free monoid of rank $r$. Then $S$-$dim(R[M]) \leq max\{1, dim(R[M])-1 \} = max\{1, d+r-1 \}$. Further$,$ we define a class of monoids $\{\mathfrak{M}_n\}_{n \geq 1}$ such that if $M \in \mathfrak{M}_n$ is seminormal$,$ then $S$-$dim(R[M]) \leq dim(R[M]) - n= d+r-n,$ where $1 \leq n \leq r$. As an application, we prove that for the Segre extension $S_{mn}(R)$ over $R,$ $S$-$dim(S_{mn}(R)) \leq dim(S_{mn}(R)) - \Big[\frac{m+n-1}{min\{m,n\}}\Big] = d+m+n-1 - \Big[\frac{m+n-1}{min\{m,n\}}\Big]$.
