论文标题
关于具有应用于Leavitt路径代数的代数函子的同构条件
On isomorphism conditions for algebra functors with applications to Leavitt path algebras
论文作者
论文摘要
我们将某些函数从交换戒指（及相关类别）类别介绍到$ \ Mathbb {z} $ - 代数（不一定是关联或交换性）的类别。激励人物之一是leavitt路径代数函数$ r \ mapsto l_r（e）$ for给定图形$ e $。我们的目标是找到类型的“下降”同构结果：如果$ \ mathfrak {f}，\ Mathcal {g} $是algebra functors和$ k \ subset k'$ a场扩展，在什么条件下，在什么条件下，同组$ \ \ \ \ \ \ m athfrak {f}（k''）暗示存在同构的存在$ \ mathfrak {f}（k）\ cong \ mathcal {g}（k）$ $ k $ -algebras？我们为所谓的“扩展不变函数”找到了一些积极的答案，其中包括与Leavitt Path代数相关的函子，Steinberg代数，路径代数，组代数，代数，进化代数等。出于我们的目的，我们采用了希尔伯特的无效定理的延伸，以无限多个变量作为我们的主要工具之一。我们还指出，对于扩展不变型函数$ \ mathfrak {f}，\ mathcal {g} $，同构$ \ mathfrak {f} {f}（h）\ cong \ mathcal \ mathcal {g}（g}（h）$ $ \ mathfrak {f}（s）\ cong \ mathcal {g}（s）$用于任何交换和Unital $ k $ -Algebra $ s $。
We introduce certain functors from the category of commutative rings (and related categories) to that of $\mathbb{Z}$-algebras (not necessarily associative or commutative). One of the motivating examples is the Leavitt path algebra functor $R\mapsto L_R(E)$ for a given graph $E$. Our goal is to find "descending" isomorphism results of the type: if $\mathfrak{F},\mathcal{G}$ are algebra functors and $K\subset K'$ a field extension, under what conditions an isomorphism $\mathfrak{F}(K')\cong \mathcal{G}(K')$ of $K'$-algebras implies the existence of an isomorphism $\mathfrak{F}(K)\cong\mathcal{G}(K)$ of $K$-algebras? We find some positive answers to that problem for the so-called "extension invariant functors" which include the functors associated to Leavitt path algebras, Steinberg algebras, path algebras, group algebras, evolution algebras and others. For our purposes, we employ an extension of the Hilbert's Nullstellensatz Theorem for polynomials in possibly infinitely many variables, as one of our main tools. We also remark that for extension invariant functors $\mathfrak{F},\mathcal{G}$, an isomorphism $\mathfrak{F}(H)\cong\mathcal{G}(H)$, for some Hopf $K$-algebra $H$, implies the existence of an isomorphism $\mathfrak{F}(S)\cong\mathcal{G}(S)$ for any commutative and unital $K$-algebra $S$.
