Author

Xinan Dai

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Preprint Aug 2026

The Complex Banach Isometric Conjecture

Banach asked whether a normed space must be Hilbert if, for one fixed dimension greater than one, all subspaces of that dimension are linearly isometric. We prove the complex case. By the codimension-one reduction, the essential finite-dimensional problem is to characterize a balanced convex body in a complex $(n+1)$-s...

Xi-Nan Dai, W.-D. Deng, Yi-Dong Shi et al. · 0 citations

Preprint Jul 2026

An Exact Counterexample to Carlson's Associated-Prime Depth Conjecture from a Group of Order 128

In Question~3.1 of his 1995 paper on depth and transfer, Carlson asked whether the depth of a finite-group cohomology ring is always realized by the dimension of one of its associated primes. We give a negative answer. Let \[ G=\SG{128}{859},\qquad k=\kbar. \] An exact presentation certificate proves that $\depth H^*(G...

Xinan Dai, Wenhao Deng, Yingdong Shi et al. · 1 citation

Preprint Jul 2026

Centralizer Excess as an Obstruction to Carlson's Depth Conjecture

Let $G=\operatorname{SmallGroup}(128,859)$ and $k=\overline{k}$. The cohomology ring $H^*(G;k)$ has depth two, and we prove that the minimum quotient dimension of an associated prime is exactly three. Okuyama's theorem shows that an integer $r$ occurs as such a dimension exactly when there is an elementary abelian subg...

Xinan Dai, Kuok Fai Chao · 0 citations