🔢Algebraic K-Theory Unit 6 – Localization and Mayer-Vietoris Sequences

Key Concepts and Definitions

• Localization of a category $C$ with respect to a class of morphisms $W$ denoted as $C[W^{-1}]$ obtained by formally inverting the morphisms in $W$
• Mayer-Vietoris sequence a long exact sequence relating the homology groups of a space $X$ to the homology groups of two subspaces $A$ and $B$ whose union is $X$
• Quillen's $Q$-construction a functor from the category of exact categories to the category of spaces, used to define higher $K$-theory
• Waldhausen categories categories with a notion of cofibrations and weak equivalences, used to define $K$-theory for more general categories
  • Satisfy certain axioms (e.g., pushouts along cofibrations exist and are cofibrations)
• Verdier localization a construction that allows one to "invert" certain morphisms in a triangulated category, analogous to localization in algebra
• $K$-theory spectrum a spectrum whose homotopy groups are the $K$-theory groups of a category
• Homotopy cofiber the dual notion to the homotopy fiber in a model category or $\infty$-category

Localization: Theory and Applications

• Localization of a category formalizes the idea of "inverting" certain morphisms, making them isomorphisms in the localized category
• Useful for studying the behavior of a category "up to" certain classes of morphisms (e.g., weak equivalences in a model category)
• Can be applied to various categories in algebra and topology, such as modules over a ring, chain complexes, or spectra
• Localization of a ring $R$ at a multiplicative subset $S$ denoted as $S^{-1}R$, obtained by formally inverting the elements of $S$
  • Elements of $S^{-1}R$ are equivalence classes of fractions $\frac{r}{s}$ with $r \in R$ and $s \in S$
• Localization of a module $M$ over a ring $R$ at a multiplicative subset $S$ denoted as $S^{-1}M$, obtained by tensoring with $S^{-1}R$
• Localization of a category can be viewed as a special case of the calculus of fractions, which allows one to formally invert morphisms in a category satisfying certain conditions
• Bousfield localization a technique for localizing a model category with respect to a homology theory or a class of maps

Mayer-Vietoris Sequences: Basics

• Mayer-Vietoris sequence relates the homology (or cohomology) of a space $X$ to the homology (or cohomology) of two subspaces $A$ and $B$ whose union is $X$
• Provides a way to compute the homology of a space by breaking it down into simpler pieces
• Arises from the long exact sequence associated to a short exact sequence of chain complexes
  • Short exact sequence: $0 \to C_*(A \cap B) \to C_*(A) \oplus C_*(B) \to C_*(A + B) \to 0$
• Can be generalized to more than two subspaces using the spectral sequence of a cover
• Mayer-Vietoris sequence for $K$-theory relates the $K$-theory of a scheme $X$ to the $K$-theory of an open cover $\{U, V\}$ of $X$
  • Sequence: $\cdots \to K_i(U \cap V) \to K_i(U) \oplus K_i(V) \to K_i(X) \to K_{i-1}(U \cap V) \to \cdots$
• Can be used to compute the $K$-theory of schemes by reducing to the case of affine schemes, where $K$-theory is easier to understand

Connecting Localization and Mayer-Vietoris

• Localization and Mayer-Vietoris sequences can be combined to study the behavior of $K$-theory under certain operations, such as open covers or blow-ups
• Thomason-Trobaugh theorem states that the $K$-theory of a scheme $X$ can be computed using the Mayer-Vietoris sequence associated to a suitable open cover of $X$, and the localization theorem for $K$-theory
  • Localization theorem relates the $K$-theory of a scheme $X$ to the $K$-theory of the complement of a closed subscheme $Y$ and the $K$-theory of $Y$
• Quillen-Grayson theorem uses localization and Mayer-Vietoris sequences to show that the $K$-theory of a regular ring $R$ is homotopy equivalent to the $K$-theory of its field of fractions $Q(R)$
• Localization and Mayer-Vietoris sequences can be used to study the behavior of $K$-theory under geometric operations such as blow-ups, where the Mayer-Vietoris sequence relates the $K$-theory of the blow-up to the $K$-theory of the base and the exceptional divisor
• Can be used to prove excision theorems for $K$-theory, which relate the $K$-theory of a scheme to the $K$-theory of a closed subscheme and its complement

Advanced Techniques and Proofs

• Quillen's devissage theorem a powerful tool for computing $K$-theory in the presence of a filtration by subobjects, using localization and Mayer-Vietoris sequences
  • Allows one to reduce the computation of $K$-theory to the case of "simple" objects (e.g., fields in the case of rings)
• Waldhausen's $S_{\bullet}$-construction a generalization of Quillen's $Q$-construction that allows one to define $K$-theory for categories with cofibrations and weak equivalences
  • Produces a simplicial space whose geometric realization is the $K$-theory space
• Waldhausen's approximation theorem states that the $K$-theory of a Waldhausen category can be computed using a "good" approximation by another Waldhausen category
  • Allows one to reduce the computation of $K$-theory to simpler categories
• Thomason-Trobaugh theorem on the $K$-theory of schemes uses a combination of localization, Mayer-Vietoris sequences, and devissage to compute the $K$-theory of a scheme in terms of the $K$-theory of its open subschemes and closed subschemes
• Motivic homotopy theory a framework that combines techniques from algebraic geometry and homotopy theory to study invariants such as motivic cohomology and algebraic $K$-theory
  • Provides a powerful set of tools for computing $K$-theory using localization, Mayer-Vietoris sequences, and other techniques from homotopy theory

Real-world Applications

• $K$-theory has applications in various areas of mathematics, including algebraic geometry, number theory, and topology
• In algebraic geometry, $K$-theory is used to study vector bundles and coherent sheaves on schemes
  • Grothendieck group $K_0(X)$ of a scheme $X$ classifies vector bundles on $X$ up to stable equivalence
• In number theory, $K$-theory is used to study the structure of rings of integers in number fields and their generalizations
  • Quillen-Lichtenbaum conjecture relates the $K$-theory of a ring of integers to its étale cohomology, which has important applications in the study of special values of $L$-functions
• In topology, $K$-theory is used to study the classification of manifolds and the structure of their diffeomorphism groups
  • Waldhausen's $A$-theory, a variant of $K$-theory for topological spaces, is closely related to the study of pseudoisotopies and the space of concordances
• $K$-theory has applications in mathematical physics, particularly in the study of quantum field theories and string theory
  • $K$-theory of $C^*$-algebras is used to classify $D$-brane charges in string theory
• Algebraic $K$-theory has applications in computer science, particularly in the study of algorithms and complexity theory
  • $K$-theory of exact categories is used to study the complexity of algorithms for solving linear systems over rings

Common Pitfalls and How to Avoid Them

• Forgetting to localize can lead to incorrect computations of $K$-theory
  • Always consider whether localization is necessary for the problem at hand
• Misapplying the Mayer-Vietoris sequence can lead to errors in computations
  • Make sure that the hypotheses of the Mayer-Vietoris sequence are satisfied (e.g., the space is the union of two open subspaces)
• Confusing the $K$-theory of a scheme with the $K$-theory of its category of vector bundles
  • The two are related but not identical; use the correct definition for the problem at hand
• Forgetting to consider the higher $K$-groups can lead to incomplete results
  • $K$-theory is not just about the Grothendieck group $K_0$; consider the higher $K$-groups as well
• Misunderstanding the functoriality of $K$-theory can lead to incorrect statements
  • $K$-theory is a functor, but it is not always compatible with certain operations (e.g., taking quotients)
• Confusing Quillen's $Q$-construction with Waldhausen's $S_{\bullet}$-construction
  • The two are related but serve different purposes; use the correct construction for the problem at hand
• Forgetting to consider the $K$-theory of non-regular schemes can lead to incomplete results
  • The $K$-theory of singular schemes is more complicated but still important; use the correct tools (e.g., the Thomason-Trobaugh theorem) to study it

Further Reading and Resources

• Quillen's original paper "Higher algebraic $K$-theory: I" lays the foundations for the study of algebraic $K$-theory and introduces the $Q$-construction
• Waldhausen's paper "Algebraic $K$-theory of spaces" introduces the $S_{\bullet}$-construction and the study of $K$-theory for categories with cofibrations and weak equivalences
• Thomason and Trobaugh's paper "Higher algebraic $K$-theory of schemes and of derived categories" is a comprehensive study of the $K$-theory of schemes using localization and Mayer-Vietoris sequences
• Weibel's book "The $K$-book: An introduction to algebraic $K$-theory" is a standard reference for the study of algebraic $K$-theory, covering both the classical and modern approaches
• Rosenberg's book "Algebraic $K$-theory and its applications" provides a more advanced treatment of algebraic $K$-theory and its applications in various areas of mathematics
• The Stacks Project (https://stacks.math.columbia.edu/) provides a comprehensive online reference for algebraic geometry, including a chapter on algebraic $K$-theory
• The $n$Lab (https://ncatlab.org/) is a collaborative online encyclopedia for mathematics, physics, and philosophy, with extensive coverage of algebraic $K$-theory and related topics