KM Algebraic Geometry: Algebraic Screening, Representation Defects, Homological Methods, and Derived Reconstruction A Special Monograph in KM Theory

Abstract

We develop KM algebraic geometry as a typed comparison theory for algebraic structures before and after screening. The objects under consideration include rings, modules, associative and graded algebras, representations, chain complexes, sheaves, schemes, algebraic cycles, deformation problems, moduli groupoids, and derived algebraic realizations. The central datum is not an isolated algebraic object but a pair of compatible or potentially incompatible realizations connected by declared screening morphisms. For algebraic objects \(A\) and \(B\), screening maps \[ \kappa_A:A\longrightarrow\mathcal K A, \qquad \kappa_B:B\longrightarrow\mathcal K B \] are compared with a morphism \(f:A\to B\) and a screened candidate \(f_{\mathcal K}:\mathcal K A\to\mathcal K B\). In an additive category the resulting morphism defect is \[ \Delta_f=\kappa_Bf-f_{\mathcal K}\kappa_A. \] In nonadditive, derived, or higher-categorical settings, this difference is replaced by a typed comparison square, equalizer defect, mapping cone, homotopy fiber, natural transformation, or higher comparison cell. No algebraic construction is assumed to commute with screening merely because the same notation is available in the two realizations. The first layer develops algebraic screening for rings, modules, ideals, quotients, localizations, completions, tensor products, symmetric and exterior powers, filtered objects, Rees constructions, and differential graded algebras. Exact criteria separate preservation of injectivity, surjectivity, exactness, primality, reducedness, integrality, flatness, projectivity, finite presentation, and Noetherianity. Quotient screening and subobject screening are treated as different variance problems. Localization and completion are compared through explicit universal maps, while tensor comparison records distinguish algebraic tensor products from completed analytic tensor products. Fitting ideals, Gröbner bases, elimination ideals, free resolutions, and Smith-type certificates are used only when their dependence on generators, bases, presentations, and coefficient rings has been controlled. The representation-theoretic layer treats group, algebra, Lie-algebra, groupoid, quiver, and tensor-category representations. It introduces representation defects, invariant and coinvariant defects, character and trace defects, induction and restriction comparisons, projective cocycle obstructions, block and central-character comparisons, and deformation complexes of representations. Equality of dimensions, characters, Grothendieck classes, or Jordan–Hölder factors is not identified with equivalence of representations. Tannakian reconstruction is admitted only from a declared rigid tensor category together with a faithful exact fiber functor and the coherence required to transport screening through tensor products, duals, internal Homs, and natural automorphism groups. The homological layer compares screening before and after taking homology, resolutions, derived tensor products, derived Hom, Tor, Ext, hypercohomology, truncation, and totalization. The natural comparison \[ \mathcal K H_n(C)\longrightarrow H_n(\mathcal K C) \] carries separate kernel and cokernel defects. Mapping cones provide universal additive defect objects, and a seven-term exact sequence controls the defect of a composite comparison. Projective, injective, and flat resolutions are not transported without hypotheses ensuring that the screened resolution remains exact and belongs to the relevant resolving class. Chain isomorphism, chain-homotopy equivalence, quasi-isomorphism, derived equivalence, and Morita equivalence remain distinct notions. Filtered and double complexes produce comparison spectral sequences. An isomorphism on one page propagates only under the functorial kernel-and-quotient construction defining later pages. Identification of abutments additionally requires convergence, completeness, exhaustiveness, separation, and finite or otherwise controlled reconstruction of filtrations. Equality of \(E_\infty\)-terms does not canonically identify the unfiltered objects. Parameter families require uniform convergence data, while rank jumps produce coherent homology sheaves rather than vector bundles unless local freeness is proved. The geometric layer constructs screened affine and projective schemes, quasi-coherent and coherent sheaves, relative spectra, Proj constructions, divisors, line bundles, normalization, blowups, formal completions, and finite, flat, smooth, proper, and étale morphisms. Because affine geometry is contravariant, every ring comparison is checked against the direction of the induced scheme morphism. Zariski, Nisnevich, étale, and fpqc descent are governed by separate effectiveness and cocycle conditions. Objectwise local lifts do not produce a global algebraic object without compatible overlap isomorphisms and higher coherence. Algebraic cycles and intersection theory are compared through proper pushforward, flat pullback, refined Gysin morphisms, normal cones, Segre classes, excess-intersection terms, Chern classes, Grothendieck groups, and algebraic \(K\)-theory. Internal KM cycles, topological classes, differential forms, and characteristic classes do not determine classical algebraic cycles without a supplied cycle comparison or algebraization theorem. Betti, de Rham, Deligne, differential, and Chow realizations are assembled into typed comparison records. Their kernels and cokernels measure loss and failure of realization rather than being suppressed by a common notation. Deformation theory is organized by derivations, Kähler differentials, square-zero extensions, cotangent complexes, deformation groupoids, and obstruction classes. Under the standard cotangent-complex hypotheses, first-order deformation classes and primary obstructions are controlled by appropriate Ext groups. Pointwise vanishing of obstruction classes does not imply the existence of a global family. Moduli problems retain automorphisms and stabilizers, so a stack or groupoid is not replaced by a coarse moduli space without recording the information lost by that passage. Geometric invariant theory is compared through stable loci, semistable loci, orbit closures, stabilizers, and quotient defects. Derived algebraic geometry enters through explicitly declared simplicial, differential graded, model-categorical, or infinity-categorical packages. Derived fiber products record nontransverse intersection information, while perfect complexes, derived moduli problems, Hochschild and cyclic invariants, singularity categories, matrix factorizations, and derived Morita comparison are introduced only when their existence and functoriality have been established in the selected enhancement. Higher notation is not treated as an existence proof. Reverse reconstruction is governed by obstruction theory. For a linear comparison \(c:V\to W\), a target element \(w\) lifts precisely when \[ w+\operatorname{im}(c)=0 \qquad\text{in }\operatorname{coker}(c). \] When nonempty, its lift fiber is a torsor under \(\ker(c)\). Nonlinear and moduli-theoretic lifts are instead organized by groupoids, homotopy fibers, or derived fibers. Vanishing of an obstruction establishes existence, not uniqueness, naturality, or canonicality. The final chapters treat finite algebraic models, singular geometric examples, nonflat base-change counterexamples, rank-jumping sheaves, nontransverse intersections, moduli problems with stabilizers, and computational certificates. The resulting theory consolidates the algebraic interfaces of the preceding KM manuscripts while remaining a separately citable thematic monograph. Its guiding principles are that no algebraic construction commutes with screening by notation alone and that every reverse arrow requires its own lifting, descent, or reconstruction theorem. ## Keywords KM algebraic geometry; algebraic screening; representation defect; homological algebra; derived functor; spectral sequence; descent defect; algebraization; reconstruction obstruction; deformation theory; moduli stack; intersection theory; algebraic \(K\)-theory; derived algebraic geometry.