Morita Theory

Why do the ring of 2×2 matrices and the ring itself have "the same algebra"? Morita theory answers: they are equivalent, their module categories are indistinguishable. This is a foundational concept in noncommutative geometry: a "point" is not just a ring but a Morita equivalence class.

• Quantum mechanics: noncommutative C*-algebras of operators; Morita equivalence = physical duality between theories
• String theory: D-branes are described by modules over algebras; Morita equivalence = T-duality
• Connes' noncommutative geometry: a "space" is a Morita class of a C*-algebra

Предварительные знания

• Algebraic Geometry: An Introduction

The Module Category R-Mod

Kiiti Morita's 1958 paper introduced an equivalence that ignores the ring R itself yet preserves all module-theoretic information: any 2 rings R, S with R-Mod ≃ S-Mod share K-theory, Hochschild cohomology, and Brauer class. **R-Mod** is the category of left R-modules: objects are left R-modules, morphisms are R-homomorphisms. The category R-Mod has rich structure: - **Additive category:** Hom(M, N) is an abelian group; composition is bilinear. - **Abelian category:** kernels, cokernels, and images exist; exact sequences make sense. - **Functors:** ⊗_R is right exact; Hom_R(P, ·) is left exact when P is projective. **Projective modules:** P is projective iff Hom_R(P, ·) is exact iff P is a direct summand of a free module R^n iff there exists Q with P⊕Q ≅ R^n. **Injective modules:** I is injective iff Hom_R(·, I) is exact iff (Baer criterion) for every ideal J⊆R and f: J→I, f extends to R→I. **Flat modules:** M is flat iff M⊗_R· is exact. Projective implies flat; the converse fails in general.

**Global dimension theorem:** gl.dim(R) = sup{pd(M) : M ∈ R-Mod}, where pd(M) is the projective dimension (length of the shortest projective resolution). Fields: gl.dim = 0. Hereditary rings: gl.dim = 1. k[x]: gl.dim = 1. k[x₁,...,xₙ]: gl.dim = n, this is Hilbert's syzygy theorem!

Morita Equivalence

Two rings R and S are **Morita equivalent** (R ~_M S) if the categories R-Mod and S-Mod are equivalent as additive categories. **Morita's theorem:** R ~_M S iff there exists a finitely generated projective generator P ∈ R-Mod with S ≅ End_R(P)^{op}. **Generator:** P generates R-Mod if every R-module is a quotient of P^{(I)} (a direct sum of copies of P). **Main example:** R ~_M Mₙ(R) for any n ≥ 1. Functor: M ↦ R^n ⊗_R M = Mⁿ (sends R-Mod to Mₙ(R)-Mod). Mₙ(R)-modules are "the same" as R-modules, but "packaged in bundles of n". **Morita invariants:** Properties preserved under equivalence: - Center Z(R) ≅ Z(S) - Ideals of R ↔ ideals of S - Simplicity R ↔ S - Global dimension gl.dim(R) = gl.dim(S) **Not Morita invariants:** R commutative ≠ S commutative (R ~_M Mₙ(R) is noncommutative).

**Arveson-Morita theorem for C*-algebras:** Morita equivalence extends to C*-algebras (algebras of operators on Hilbert space): A ~_M B iff there exists a Hilbert A-B-bimodule. This is a central concept in Connes' noncommutative geometry: a "point" of a noncommutative space is a Morita class of a C*-algebra.

Azumaya Algebras and the Brauer Group

A **central simple algebra (CSA)** over a field k is a finite-dimensional k-algebra A with: - Z(A) = k (center consists only of scalars) - A is simple (no proper two-sided ideals) Examples: Mₙ(k), quaternions ℍ over ℝ, Hamilton's algebra. **Wedderburn's theorem:** Every simple Artinian k-algebra ≅ Mₙ(D), where D is a division algebra over k. **Azumaya algebra** over a commutative ring R: an R-algebra A such that A ⊗_R A^{op} ≅ End_R(A) via a⊗b ↦ (x ↦ axb). **Brauer group** Br(k): Morita-equivalence classes of CSAs over k. A group! Multiplication: [A]·[B] = [A⊗B]. Inverse: [A^{op}]. Identity: [k]. Br(ℝ) = Z/2Z: elements [ℝ] and [ℍ]. Br(finite field) = 0 (Chevalley-Warning theorem). Br(ℚ) is a rich group, deeply connected to number theory.

**Wedderburn-Artin theorem:** Every finite-dimensional semisimple algebra over a field is isomorphic to a direct product of matrix algebras: A ≅ Mₙ₁(D₁) × ... × Mₙₖ(Dₖ), where each Dᵢ is a division algebra. This is the precise analog of the spectral decomposition theorem for symmetric matrices, extended to abstract algebras.

Key Ideas

• R-Mod is an abelian category with projective, injective, and flat objects
• R ~_M S iff R-Mod ≅ S-Mod iff there exists a finitely generated projective generator P with End(P) ≅ S
• R ~_M Mₙ(R), the main example of Morita equivalence
• Invariants: center, ideals, simplicity, gl.dim; NOT invariant: commutativity

Further Directions

Morita theory is the gateway to noncommutative geometry. Derived categories generalize Morita equivalence to "derived Morita equivalence" (Rickard). Group cohomology and Azumaya algebras are connected through the Brauer group.

• Derived Categories — D^b(R-Mod) generalizes Morita equivalence: Rickard's derived equivalence D^b(R) ≅ D^b(S) is strictly stronger than Morita equivalence
• Group Cohomology — H²(G, k*) = Br(k^G/k), the Brauer group and Galois cohomology are connected through the theory of crossed products

Вопросы для размышления

• Prove R ~_M Mₙ(R) for any n by constructing explicit functors F: R-Mod → Mₙ(R)-Mod and G inverse, and verifying F∘G ≅ Id and G∘F ≅ Id.
• Is ℤ ~_M ℤ[1/2]? If not, what is the obstacle? Investigate whether Morita equivalence preserves torsion modules.
• Compute Br(ℝ): prove ℍ ⊗_ℝ ℍ ≅ M₄(ℝ), i.e., [ℍ]² = [ℝ] in Br(ℝ).

Связанные уроки

• la-01-vectors-intro