Permutation Algebraic Geometry: Presentations, Algebraic Subspaces, Covering Groupoids, Cohomology, Topoi, and Fibres

Abstract

We construct permutation algebraic geometry from actions on presentations rather than automorphisms of one fixed space. In the affine polynomial sector, a slot permutation gives a linear operator \(T_w\) that is generally nonmultiplicative. For an ideal \(I\subseteq S\), the closure\[I^{[w]}=\langle T_w(I)\rangle\]defines a new closed subscheme, while explicit multiplicativity, closure-composition, covering, gluing, and base-change defects record every failed structural implication. Multiplicative monomial permutations are exactly coordinate permutations for polynomial rings and lattice automorphisms for Laurent polynomial rings. We construct parameter schemes, universal families, history and realization groupoids, incidence correspondences, quotient stacks, finite étale covering categories, permutation fundamental groups, Čech nerves, classifying and locally ringed topoi, and fibrewise transport. External coefficient histories and internal ambient placements form a perfect iso-comma groupoid over a common record groupoid. Joint ambient–subspace placements produce affine and projective endpoints, permutation prime spectra, Hilbert profiles, induced covers, and cartesian derived sheaves. The total topos defines intrinsic permutation cohomology with localization, descent, Leray, and history–endpoint spectral sequences. Affine incidence diagrams define derived orbit spectra, gluing-defect modules, and diagram and aggregate André–Quillen cohomology. Nonconstant endpoint records prove that permutation induction is not a relabelling of ordinary equivariant geometry. Classical schemes, quotient stacks, and effective descent are recovered on distinct coherent zero-defect loci. Formal histories alone imply neither morphisms, covers, descent, topos equivalence, nor cohomological invariance. **Keywords** Permutation algebraic geometry; coefficient permutation; polynomial ring; defect ideal; twisted Hochschild cohomology; weak ideal transform; algebraic subspace; history groupoid; perfect permutation groupoid; matching-defect cohomology; quotient stack; finite étale cover; equivariant cohomology; topos; fibre; descent.