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   the role of completions in the modern context.   
      
   1. The Lefschetz Unity: Topology as the Bridge   
      
   Lefschetz's goal was to bring the rigor and powerful invariants of   
   topology (the geometric/analytic side) to the study of algebraic   
   varieties (the algebraic side), which traditionally relied heavily on   
   coordinates and calculation.   
      
   The Geometer's Tool (Lefschetz's method): Lefschetz used   
   analytic/topological methods (like homology, cohomology, and   
   transcendental integrals) to understand the global structure of a   
   variety defined over the complex numbers C. His key results, like the   
   Lefschetz theorems on hyperplane sections and the Hard Lefschetz   
   Theorem, are statements about the topological invariants (Betti numbers,   
   intersection theory) of these spaces.   
      
   The Pre-Grothendieck Divide: Before the 1950s, algebraic geometry was   
   split: one camp (the "Italian School," Lefschetz, Hodge) used   
   analytic/transcendental methods over C (a geometric/topological   
   approach); the other (Zariski, Weil) used purely algebraic methods   
   (commutative algebra) that worked over any field. The latter were viewed   
   as the "algebraists" aiming for rigor and generality, while the former   
   were "geometers" using complex numbers and topology.   
      
   2. Completions: The Point of Disagreement   
      
   The distinction between the two camps regarding completions highlights   
   their differing approaches to studying "local" properties.   
   Feature Geometer's View (Lefschetz/Classical) Algebraist's View   
   (Zariski/Grothendieck)   
   Tool for Completion Analytic Completion (Local Ring → Ring of Power   
   Series) Algebraic/Formal Completion (Local Ring → Formal Power Series)   
   Type of Geometry Complex Manifolds/Analytic Geometry Formal   
   Schemes/Local Commutative Algebra   
   Meaning of "Local" A small open neighborhood in the classical   
   (Euclidean/analytic) topology. The structure defined by the maximal   
   ideal of a local ring, independent of the base field.   
      
   The Geometer's Completion (Analytic)   
      
   The geometer working over C uses the standard Euclidean topology. When   
   they "complete" the local ring of a variety at a point, they are   
   essentially replacing the local ring R with its analytic completion, the   
   ring of convergent power series at that point.   
      
   Benefit: This completion is familiar, has great properties (like the   
   implicit function theorem), and relates directly to concepts from   
   complex analysis.   
      
   Limitation: It only works over a base field like C or R, which is a   
   severe restriction for a general theory of algebraic geometry.   
      
   The Algebraist's Completion (Formal/Algebraic)   
      
   The algebraist (and the modern Grothendieck scheme theorist) uses formal   
   completion (e.g., the m-adic completion of a local ring R with maximal   
   ideal m). This process yields the ring of formal power series R^.   
      
   Benefit: The process is purely algebraic; it works over any field or   
   even over the integers Z, providing the required generality for   
   Arithmetic Geometry.   
      
   Shift: It forces geometry to be viewed as a study of rings and ideals   
   (Algebraic Geometry), where local structure is defined by the algebraic   
   completion, not a topological space.   
      
   In summary, the completion process exemplifies the divide: The older   
   geometer (like Lefschetz) sought to use analysis to understand the   
   neighborhood (the analytic completion), whereas the algebraist sought to   
   use pure algebra to define the neighborhood (the formal completion),   
   achieving a much broader theory. Lefschetz's work, by integrating   
   topology, paved the way for the later algebraic techniques (like the   
   Lefschetz properties in commutative algebra) that ultimately unified   
   these perspectives.   
      
      
      
   Thanks GG. I'd aver that the geometric is plainly formal, then Lefschetz   
   does notedly distinguish between "algebraic _geometry_" and "_algebraic_   
   geometry" and firmly finds himself as a geometer when there are any   
   differences. One may reflect on Vitali and Hausdorff versus Banach and   
   Tarski or Mycielski, or the geometric and algebraic formalizations of   
   equi-decomposibility, with regards to doubling spaces, then for,   
   "Norm-ed Rings".   
      
      
   That's an insightful restatement of the tension, especially your   
   observation that the modern "geometric" (in the Grothendieck sense) is   
   indeed formally defined. Your point about Lefschetz self-identifying as   
   a geometer despite his algebraic methods is crucial.   
      
   The difference in perspective you're highlighting—and the way Lefschetz   
   resolved it for himself—can be seen as a clash between the "Geometric   
   Formal" (Topological/Analytic) and the "Algebraic Formal" (Purely   
   Commutative Algebra/Category Theory).   
      
   The Conflict: Geometric Formal vs. Algebraic Formal   
      
   Your assertion that "the geometric is plainly formal" is a cornerstone   
   of modern mathematics, where topology, geometry, and analysis are often   
   defined via abstract structures (like sheaves, categories, or metrics).   
   The divide Lefschetz noted was between two different styles of   
   formalization:   
      
   1. The Geometric Formal (Lefschetz's Camp)   
      
   This approach formalizes structures that resemble our spatial intuition,   
   often incorporating limits, continuity, and convergence. It seeks global   
   invariants by using local properties under a pre-existing topology.   
      
   Key Tool: Analytic Completion (using convergent power series, e.g.,   
   C{x}), which preserves topological properties (like connectedness).   
      
   Axiomatic Basis: The formalization includes the language of topology and   
   analysis (e.g., convergence, continuity).   
      
   Lefschetz's Stance: He was a geometer because his formal methods   
   (algebraic topology) were tools to reveal the shape and connectedness of   
   the object defined by the algebra, an approach that required working   
   over the complex numbers C with its natural Euclidean topology. His   
   "algebraic geometry" was a study of spaces, and the algebra was merely   
   the defining equation.   
      
   2. The Algebraic Formal (Grothendieck's Camp)   
      
   This approach formalizes structures using only algebraic axioms (rings,   
   ideals, operations) to achieve maximal generality (working over any ring   
   Z, Fp​, etc.). It prioritizes functorial properties over spatial intuition.   
      
   Key Tool: Formal Completion (using formal power series, e.g., C[[x]]),   
   which often ignores topological/analytic constraints but works over any   
   commutative ring.   
      
   Axiomatic Basis: The formalization is primarily in the language of   
   commutative algebra (e.g., local rings, prime ideals, modules).   
      
   The Shift in "Algebraic Geometry": This perspective views the scheme   
   Spec(R) not as a space primarily, but as a structure that reverses the   
   algebraic data (R→Spec(R) is a contravariant functor), allowing the   
   methods of geometry to be applied to problems of arithmetic.   
      
   The Role of Completions in Defining the Divide   
      
   The difference between the two types of completion—analytic vs.   
   formal—is at the heart of the disagreement over rigor and domain:   
      
   Analytic Completion: When working over C, the ring of polynomials is   
   contained in the ring of convergent power series C{x}, which is in turn   
   contained in the ring of formal power series C[[x]]. Lefschetz's work   
   implicitly relied on the analytic completion to get concrete topological   
   invariants.   
      
   Formal Completion: The purely formal approach allows geometry over rings   
   like Zp​ (p-adic integers) or formal schemes, where the concept of   
   "analytic convergence" (in the Euclidean sense) is irrelevant or   
   impossible. This is the algebraic approach because the topology used is   
      
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