CW复形的Nagata理想化
CW-complex Nagata Idealizations
AI总结：
本研究提出一种新颖构造，建立CW复形$P(m)$的有限子复形与双分次标准Artinian Gorenstein代数的双射关系，并将其应用于推广水平代数的Nagata理想化，研究双次数$(d_1,d_2)$多项式对应的代数。
摘要：
我们提出一种新颖构造，可将CW复形$P(m)$的骨架元素与m个变量中的单项式对应起来。由此我们推导出，$P(m)$的有限CW子复形（即有限单纯复形的商）与某些双分次标准Artinian Gorenstein代数之间存在双射，推广了文献[F:S]、[CGIM]和[G:Z]中的已有构造。我们将其应用于水平代数的Nagata理想化推广。这类代数是标准分次Artinian代数，其Macaulay对偶生成元可明确表示为双次数$(1,d)$的双分次多项式。我们研究了与双次数$(d_1,d_2)$的同类型多项式相关的代数。
英文摘要：
We introduce a novel construction which allows us to identify the elements of the skeletons of a CW-complex $P(m)$ and the monomials in $m$ variables. From this, we infer that there is a bijection between finite CW-subcomplexes of $P(m)$, which are quotients of finite simplicial complexes, and some bigraded standard Artinian Gorenstein algebras, generalizing previous constructions in \cite{F:S}, \cite{CGIM} and \cite{G:Z}. We apply this to a generalization of Nagata idealization for level algebras. These algebras are standard graded Artinian algebras whose Macaulay dual generator is given explicitly as a bigraded polynomial of bidegree $(1,d)$. We consider the algebra associated to polynomials of the same type of bidegree $(d_1,d_2)$.
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