duality between algebra and geometry






In mathematics, one often has an equivalence of categories between algebra-like objects and space-like objects. Such an idea has many incarnations: Stone duality, Gelfand duality, etc., and in this article we make some observations that are common to these dualities.

Given an algebra-like object AA, we assign to it its poset of ideals (typically defined as kernels of homomorphisms ABA\to B), which is interpreted as the poset of opens of some space SS.

The technical term for such posets is locale, which is a notion very closely related to topological spaces. In particular, from any locale one can canonically extract a topological space, and this is the topological space SS produced in many classical Stone-type dualities. The points of SS are ideals corresponding to morphisms AkA\to k, where kk is often a particularly simple algebra. These often turn out to be maximal ideals in AA.

Conversely, given a space-like object SS, we assign to it the algebra of morphisms SkS\to k, where kk is often the “same” algebra kk as above, only this time its underlying object is a space, not just a set.

Some examples from general topology, measure theory, differential geometry, algebraic geometry, and complex geometry (the list is very much incomplete):

Boolean algebrahomomorphismZ/2\mathbf{Z}/2idealcompact totally disconnected Hausdorff space (Stone space)continuous mapStone duality
complete Boolean algebracomplete homomorphismZ/2\mathbf{Z}/2closed idealcompact extremally disconnected Hausdorff space (Stonean space)open continuous mapStonean duality
localizable Boolean algebracomplete homomorphismZ/2\mathbf{Z}/2closed idealhyperstonean spaceopen continuous map
localizable Boolean algebracomplete homomorphismZ/2\mathbf{Z}/2closed idealcompact strictly localizable enhanced measurable spacemeasurable map
commutative von Neumann algebranormal *-homomorphismC\mathbf{C}closed *-idealcompact strictly localizable enhanced measurable spacemeasurable map
commutative unital C*-algebra*-homomorphismC\mathbf{C}closed *-idealcompact Hausdorff spacecontinuous mapGelfand duality
commutative algebra over kkhomomorphismkkidealcoherent space / affine schemecontinuous map / morphism of schemesZariski duality
finitely generated germ-determined C^∞-ringC^\infty-homomorphismR\mathbf{R}germ-determined idealsmooth locus (e.g., smooth manifold)smooth mapMilnor duality
finitely presented complex EFC-algebraEFC-homomorphismC\mathbf{C}idealglobally finitely presented Stein spaceholomorphic mapStein duality


duality between \;algebra and geometry

A\phantom{A}geometryA\phantom{A}A\phantom{A}categoryA\phantom{A}A\phantom{A}dual categoryA\phantom{A}A\phantom{A}algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand-KolmogorovAlg op\overset{\text{<a href="">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}A\phantom{A}comm. C-star-algebraA\phantom{A}
A\phantom{A}noncomm. topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}A\phantom{A}general C-star-algebraA\phantom{A}
A\phantom{A}algebraic geometryA\phantom{A}A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}A\phantom{A}almost by def.TopAlg fin op\overset{\text{<a href="">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} A\phantom{A}A\phantom{A}fin. gen.A\phantom{A}
A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}noncomm. algebraicA\phantom{A}
A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}A\phantom{A}fin. gen.
A\phantom{A}associative algebraA\phantom{A}A\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}A\phantom{A}Milnor's exerciseTopAlg comm op\overset{\text{<a href="">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}A\phantom{A}SuperSpaces Cart n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}A\phantom{A}Milnor's exercise Alg 2AAAA op C ( n) q\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }A\phantom{A}A\phantom{A}supercommutativeA\phantom{A}
A\phantom{A}formal higherA\phantom{A}
A\phantom{A}(super Lie theory)A\phantom{A}
ASuperL Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}AALada-MarklA sdgcAlg op CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}A\phantom{A}differential graded-commutativeA\phantom{A}
A\phantom{A} (“FDAs”)

in physics:

A\phantom{A}Poisson algebraA\phantom{A}A\phantom{A}Poisson manifoldA\phantom{A}
A\phantom{A}deformation quantizationA\phantom{A}A\phantom{A}geometric quantizationA\phantom{A}
A\phantom{A}algebra of observablesA\phantom{A}space of statesA\phantom{A}
A\phantom{A}Heisenberg pictureA\phantom{A}Schrödinger pictureA\phantom{A}
A\phantom{A}higher algebraA\phantom{A}A\phantom{A}higher geometryA\phantom{A}
A\phantom{A}Poisson n-algebraA\phantom{A}A\phantom{A}n-plectic manifoldA\phantom{A}
A\phantom{A}En-algebrasA\phantom{A}A\phantom{A}higher symplectic geometryA\phantom{A}
A\phantom{A}BD-BV quantizationA\phantom{A}A\phantom{A}higher geometric quantizationA\phantom{A}
A\phantom{A}factorization algebra of observablesA\phantom{A}A\phantom{A}extended quantum field theoryA\phantom{A}
A\phantom{A}factorization homologyA\phantom{A}A\phantom{A}cobordism representationA\phantom{A}

The duality relevant to the spectral theory is the duality between commutative von Neumann algebras and compact strictly localizable enhanced measurable spaces.

Given a normal operator TT on a Hilbert space HH, TT generates a commutative von Neumann algebra AA inside B(H)B(H), i.e., bounded operators on HH. (This is precisely the point where normality is crucial; without the relation T *T=TT *T^*T=TT^* the algebra generated by TT will be noncommutative.)

By the cited duality, the commutative von Neumann algebra AA is dual to a compact strictly localizable enhanced measurable space SpecASpec A. This is indeed the spectrum of TT in the usual sense. Under this equivalence, the element TAT\in A corresponds to the measurable map SpecACSpec A\to \mathbf{C} given by inclusion of SpecASpec A into C\mathbf{C}.

One may ask whether we can recover the full spectral theorem for a normal operator in this manner. This is possible once Stone duality is upgraded to Serre–Swan-type duality between modules and vector bundle-like objects (including, e.g., sheaves of modules etc.).

Given a vector bundle-like object VSV\to S, we assign to it its module of sections, which is a module over the algebra of maps SkS\to k. Conversely, given a module MM over AA, the corresponding vector bundle-like object VSV\to S over S=SpecAS=Spec A has as its fiber over some point sSs\in S the vector space M/IMM/IM, where II is the ideal corresponding to ss. (Many details are necessarily omitted in this brief sketch.)

Typically, genuine vector bundles correspond to dualizable modules (dualizable with respect to the tensor product over AA). Non-dualizable module tend to correspond to sheaves of modules that are not vector bundles, e.g., skyscraper sheaves etc.

modulevector bundle-like object
module over a Boolean algebrasheaf of Z/2\mathbf{Z}/2-vector spaces
Hilbert W*-module over a commutative von Neumann algebrameasurable field of Hilbert spaces
W*-representations of a commutative von Neumann algebra on a Hilbert spacemeasurable field of Hilbert spaces
Hilbert C*-module over a commutative unital C*-algebracontinuous field of Hilbert spaces
module over a commutative ringquasicoherent sheaf of 𝒪\mathcal{O}-modules over an affine scheme
dualizable module over a commutative algebra over kkalgebraic vector bundle
dualizable module over a finitely generated germ-determined C^\infty-ringsmooth vector bundle
dualizable module over finitely presented complex EFC-algebraholomorphic vector bundle

The duality relevant to the spectral theory is the duality between representations of a commutative von Neumann algebras on a Hilbert space and measurable fields of Hilbert spaces.

Given a normal operator TT on a Hilbert space HH, TT generates a commutative von Neumann algebra AA inside B(H)B(H), whose spectrum SpecASpec A is a compact strictly localizable enhanced measurable space.

Furthermore, the inclusion of AA into B(H)B(H) is a representation of AA on HH. As such, it corresponds under the Serre–Swan-type duality to a measurable field of Hilbert spaces over AA. This is precisely the measurable field produced by the classical spectral theorem.

Under the duality, the operator TT corresponds to the operator that multiplies a given section of this measurable field of Hilbert spaces by the complex-valued function SpecACSpec A\to \mathbf{C} produced above. Thus, we recovered the entire content of the classical spectral theorem.

In fact, the above considerations work equally well to establish the spectral theorem for an arbitrary family (not necessarily finite) of commuting normal operators.