A Generic Identification Theorem for Groups of Finite Morley Rank, Revisited

Uniqueness Cases in Odd Type Groups of Finite Morley Rank, Revisited

The paper contains versions of the Strong Embedding Theorem and the Uniqueness Subgroup Theorem for groups of finite Morley rank and odd type which are needed for the study of permutations actions and modules in the finite Morley rank category.

Şiş Kebap 定理：有限 Morley 秩群的通用识别定理

The Şiş Kebap Theorem: A Generic Identification Theorem for Groups of Finite Morley Rank

本文提出有限 Morley 秩“通用”群的最终识别定理的新版本，发展了已有研究成果，为 Hrushovski 计划中有限 Morley 秩无限单群的分类提供支持。

Signalizers and balance in groups of finite Morley rank

We show that a minimal counter example to the Cherlin-Zilber Algebraicity Conjecture for simple groups of finite Morley rank has Prufer 2-rank at most two. This article covers the signalizer functor theory and identifies the groups of Lie rank at least three; leaving the uniqueness case analysis to previous articles. This result signifies the end of the general methods used to handle large groups; hereafter each individual group PSL$_2$, PSL$_3$, PSp$_4$, and G$_2$ will require its own identification theorem.

The geometry of involutions in ranked groups with a TI-subgroup

We revisit the geometry of involutions in groups of finite Morley rank. Our approach unifies and generalises numerous results, both old and recent, that have exploited this geometry; though in fact, we prove much more. We also conjecture that this path leads to a new identification theorem for $\operatorname{PGL}_2(\mathbb{K})$.

Affine representability of quadrics revisited

The quadric $\operatorname{Q}_{2n}$ is the ${\mathbb Z}$-scheme defined by the equation $\sum_{i=1}^n x_i y_i = z(1-z)$. We show that $\operatorname{Q}_{2n}$ is a homogeneous space for the split reductive group scheme $\operatorname{SO}_{2n+1}$ over ${\mathbb Z}$. The quadric $\operatorname{Q}_{2n}$ is known to have the ${\mathbb A}^1$-homotopy type of a motivic sphere and the identification as a homogeneous space allows us to give a characteristic independent affine representability statement for motivic spheres. This last observation allows us to give characteristic independent comparison re…