立方根零代数的Big Finitistic维数
Big Finitistic Dimensions of Radical-Cube-Zero Algebras
- Capital Normal University（首都师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文构造十维立方根零代数，给出反例否证big finitistic维数猜想，并分类相关模及建立有限性准则，证明投射维数二类的野性。
AI中文摘要：
我们在任意域上构造了一个十维代数$A$，满足$(\operatorname{rad}A)^3=0$，使得$\operatorname{findim}A=2$，$\operatorname{Findim}A=\infty$，且$\operatorname{Findim}(A^{\mathrm{op}})=0$。一个显式的逆合冲（inverse-syzygy）构造给出了可数生成的右模，其具有任意正有限投射维数，从而否证了big finitistic维数猜想。在每一步中，我们识别出完全的投射覆盖核，并且非零Ext精确检测出解析长度。对于该代数，我们分类了投射维数至多为一且嵌入到投射模的根中的模，无生成限制，以可逆线性算子的形式给出。我们确定了投射维数为二的有限维模的所有维数向量，并分类了达到尖锐维数界的那些模，包括所有十四维的投射维数为二的模。对于任意Artin代数，我们给出了递归的充分准则，通过根零化理想和合适的角（corners）来控制有限投射解析。对于有限维初等立方根零代数，我们在每个顶点带一个环路的二顶点情形下证明了big finitistic维数的有限性，并获得了界为一或二的乘法映射准则。最后，一个精确的表示嵌入导出了投射维数二类的野性（wildness），并且在任意域上，该类中存在一个超可分解纯内射模。
英文摘要：
We construct, over any field, a ten-dimensional algebra $A$ with $(\operatorname{rad}A)^3=0$ such that $\operatorname{findim}A=2$, $\operatorname{Findim}A=\infty$, and $\operatorname{Findim}(A^{\mathrm{op}})=0$. An explicit inverse-syzygy construction gives countably generated right modules of every positive finite projective dimension, disproving the big finitistic dimension conjecture. At each step we identify the full projective-cover kernel, and nonzero Ext detects the exact resolution length. For this algebra, we classify the modules of projective dimension at most one that embed in radicals of projective modules, with no generation restriction, in terms of invertible linear operators. We determine all dimension vectors of finite-dimensional modules of projective dimension two and classify those attaining the sharp dimension bounds, including all fourteen-dimensional modules of projective dimension two. For arbitrary Artin algebras, we give recursive sufficient criteria that control finite projective resolutions through radical-annihilated ideals and suitable corners. For finite-dimensional elementary radical-cube-zero algebras, we prove finiteness of the big finitistic dimension in the two-vertex case with one loop at each vertex and obtain multiplication-map criteria for bounds of one or two. Finally, an exact representation embedding yields wildness of the projective-dimension-two class and, over every field, a superdecomposable pure-injective module in that class.
