论文标题
3级理想的分辨率和TOR代数结构定义压缩环
Resolution and Tor Algebra Structures of Grade 3 Ideals Defining Compressed Rings
论文作者
论文摘要
令$ r = k [x，y，z] $为标准分级$ 3 $ - 变量的多项式环，其中$ k $表示任何字段。我们研究$ 3 $同质理想$ i \ subseteq r $用Socle $ k（-s）^{\ ell} \ oplus k（-2s+1）$定义压缩环，其中$ s \ geq3 $和$ \ ell \ ell \ ell \ geq 1 $是整数。作者在上一篇论文中研究了$ \ ell = 1 $的案例；为所有这些理想构建了一般最小的分辨率。最近，该分辨率以（迭代）修剪复合物的幌子进行了概括。在本文中，我们表明上述形式的所有理想都通过迭代的修剪复合物来解决。此外，我们将此机械应用于构建理想$ i $，以便$ r/i $是某些固定$ r \ geq2 $的tor代数$ g（r）$的环，并且可以选择任意大型类型。特别是，这为克里斯滕森，维利希和韦曼尚未建立的阿夫拉莫夫的猜想提供了新的反例。
Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s)^{\ell} \oplus k(-2s+1)$, where $s \geq3$ and $\ell \geq 1$ are integers. The case for $\ell =1$ was studied in a previous paper by the author; a generically minimal resolution was constructed for all such ideals. More recently, this resolution is generalized in the guise of (iterated) trimming complexes. In this paper, we show that all ideals of the above form are resolved by an iterated trimming complex. Moreover, we apply this machinery to construct ideals $I$ such that $R/I$ is a ring of Tor algebra class $G (r)$ for some fixed $r \geq2$, and $R/I$ may be chosen to have arbitrarily large type. In particular, this provides a new class of counterexamples to a conjecture of Avramov not already constructed by Christensen, Veliche, and Weyman.
