AI 中文总结
研究特征2正则环上广义总秩猜想，证明了初等阿贝尔2-群卡尔森猜想，通过识别链模型同调与弗罗贝尼乌斯拉回、变形泰特微分比较长度，得出相关同调界及球面秩猜想。
AI 中文摘要
我们证明了特征2的正则环上的广义总秩猜想：若R是特征2的正则诺特整环，P是一个具有有限射影旗且同调群H(P)非零的微分R-模，则\(\rank_R(P)\geq2^{\codim_RH(P)}\)。特别地，我们证明了任意秩的初等阿贝尔2-群的卡尔森猜想。我们还得到了此类群的任意连续作用以及有限群代数上的完美复形的精确同调界；球面秩猜想随之成立。证明通过将P⊗RP上\(C_2\)-泰特构造的链模型的同调与H(P)的弗罗贝尼乌斯拉回进行识别，并通过变形泰特微分来比较长度。
英文摘要
We prove the generalized total rank conjecture over regular rings in characteristic $2$: if $R$ is a regular Noetherian domain of characteristic $2$ and $P$ is a differential $R$-module admitting a finite projective flag and having nonzero homology $H(P)$, then $\rank_R(P)\ge2^{\codim_RH(P)}$. In particular, we prove Carlsson's conjecture for elementary abelian $2$-groups in every rank. We also obtain sharp homology bounds for arbitrary continuous actions of such groups and for perfect complexes over finite group algebras; the sphere rank conjecture follows. The proof identifies the homology of a chain model for the $C_2$-Tate construction on $P\otimes_RP$ with the Frobenius pullback of $H(P)$, and compares lengths by deforming the Tate differential.
Comments11 pages; minor edits
