Using the internal language of toposes in algebraic geometry (2111.03685v1)

Abstract: Any scheme has its associated little and big Zariski toposes. These toposes support an internal mathematical language which closely resembles the usual formal language of mathematics, but is "local on the base scheme": For example, from the internal perspective, the structure sheaf looks like an ordinary local ring (instead of a sheaf of rings with local stalks) and vector bundles look like ordinary free modules (instead of sheaves of modules satisfying a local triviality condition). The translation of internal statements and proofs is facilitated by an easy mechanical procedure. We investigate how the internal language of the little Zariski topos can be exploited to give simpler definitions and more conceptual proofs of the basic notions and observations in algebraic geometry. To this end, we build a dictionary relating internal and external notions and demonstrate its utility by giving a simple proof of Grothendieck's generic freeness lemma in full generality. We also employ this framework to state a general transfer principle which relates modules with their induced quasicoherent sheaves, to study the phenomenon that some properties spread from points to open neighborhoods, and to compare general notions of spectra. We employ the big Zariski topos to set up the foundations of a synthetic account of scheme theory. This account is similar to the synthetic account of differential geometry, but has a distinct algebraic flavor. Central to the theory is the notion of synthetic quasicoherence, which has no analogue in synthetic differential geometry. We also discuss how various common subtoposes of the big Zariski topos can be described from the internal point of view and derive explicit descriptions of the geometric theories which are classified by the fppf and by the surjective topology.

• The paper demonstrates that using the internal language of toposes streamlines definitions and proofs in algebraic geometry.
• It translates classical algebraic constructs into topos-theoretic terms, simplifying the study of schemes and sheaf properties.
• Modal operators and internal characterizations are employed to clarify property spreading and enhance understanding of scheme theory.

An Expert Overview of "Using the Internal Language of Toposes in Algebraic Geometry"

The paper "Using the Internal Language of Toposes in Algebraic Geometry" by Ingo Blechschmidt offers a thorough examination of algebraic geometry through the lens of categorical logic, particularly utilizing the internal language of toposes. This approach provides a coherent and simplified framework for various concepts and constructions in algebraic geometry.

Key Contributions

1. Internal Language of Toposes: The paper begins by outlining the utility of topos theory in algebraic geometry. Blechschmidt emphasizes the translation of external mathematical statements into an internal language, which mirrors set-theoretic operations in topos-theoretic terms. This language provides a robust framework where intricate algebraic properties of schemes and sheaves can be addressed formally and systematically.
2. Simplification of Algebraic Concepts: By leveraging the internal language, the author proposes simpler definitions and proofs for basic algebraic geometry notions. Notably, this method allows classical theorems in algebra to be imported into the sheaf-theoretic setting, highlighting the naturality and conceptual clarity brought about by the internal perspective.
3. Application to Scheme Theory: Blechschmidt demonstrates how the internal language elucidates scheme-theoretic fundamentals, such as quasicoherence, flatness, and finiteness conditions. Through internal characterization, the paper provides insights into the behavior of these properties across both affine and general schemes.
4. Modal Operators and Property Spreading: The research utilizes modal operators to explore property spreading from points to neighborhoods. This innovative approach allows for a modal logic perspective on questions like whether properties holding at a point propagate to an open neighborhood. This perspective is particularly beneficial for understanding the properties' behavior in non-trivial topological spaces.
5. Characterization of Sheaves: Another significant contribution is the characterization of different classes of sheaves, such as $\neg\neg$-sheaves and $\Box$-sheaves. The work extends to explore extensions of sheaf properties, leveraging modalities to give internal accounts for sheaves on open and closed subspaces, thereby enriching the understanding of sheaf encapsulations and the spectrum of schemes.

Numerical and Theoretical Implications

The paper presents various theoretical implications, such as the internal characterization of normality and integrality, contributing to an understanding that extends beyond classical equivalences into intuitionistic frameworks. By showing that the internal language can faithfully reflect these properties, Blechschmidt's work facilitates a deeper connection between topos theory and algebraic geometry.

Future Prospects and Developments

Blechschmidt speculates on further developments, particularly the potential of this framework to simplify foundational questions in scheme theory and its compatibility with synthetic accounts of algebraic geometry. The extension into other logical modalities and their applications to different geometrical configurations could open new avenues in both categorical logic and algebraic geometry.

Conclusion

Ingo Blechschmidt's work offers a uniquely categorical dimension to understanding algebraic geometry. By leveraging the internal language of toposes, the paper not only provides an elegant framework for redefining and understanding algebraic constructs but also sets the stage for future explorations into synthetic and modal interpretations of geometry. This exposition stands as a testament to the power of categorical logic as a tool for refining and advancing modern mathematical thought.