spectral cookbook


Typically spectra of categories involve utilization of some preordering and looking for β€œalmost final” objects in that preordering and declaring them points of spectra. For example, one can look at some class of topologizing subcategories and equip that class with a preordering. A deeper inspection shows that an additional functor may be involved.

Historical motivation

While the spectrum of a commutative ring RR is obtained just from studying the ideal in a ring, that is RR-submodules in RR, the structure of various sets of ideals in a noncommutative ring usually is too small and otherwise inadequate for geometric purposes. One needs to consider not only ideals, but the whole category of (say left) RR-modules, thus not necessarily submodules of RR. A similar thing is with the reconstruction of commutative schemes: the whole abelian category of quasicoherent sheaves of π’ͺ\mathcal{O}-modules, not only the quasicoherent π’ͺ\mathcal{O}-submodules of π’ͺ\mathcal{O}, is needed for the reconstruction.

The development of spectral machinery for categories

Pierre Gabriel introduced his spectrum of indecomposable injectives to reconstruct Noetherian separated schemes from their categories of qausicoherent sheaves; now it is often called the Gabriel spectrum. Later many other spectra of abelian categories were invented, including the 1980-s A. L. Rosenberg’s spectrum used for the reconstruction of quasicompact quasiseparated schemes. Around 2000, Rosenberg noticed that the zoo of many spectra has their common feature and that all the spectra can be produced in analogous way. This pattern for producing spectra is introduced as a spectral cookbook in

  • (RosenbergSpectraNSp) A. L. Rosenberg, Spectra of noncommutative spaces, MPIM2003-110 ps dvi (2003)

The cookbook

Local categories and local spectrum

(RosenbergSpectraNSp 1.1) A category CC is local if the full subcategory generated by all objects which are not initial, has itself an initial object. In particular, every local category has initial objects.

(RosenbergSpectraNSp 1.2) The local spectrum of an arbitrary small category CC is the full subcategory 𝔖𝔭𝔒𝔠 1(C)β†ͺC\mathfrak{Spec}^1(C)\hookrightarrow C whose objects are all xx in ObCOb C such that the undercategory x\Cx\backslash C is local.

Support of an object

(RosenbergSpectraNSp 1.3) If xx is an object in CC, its support in CC is the full subcategory 𝔖𝔲𝔭𝔭 C(x)βŠ‚C\mathfrak{Supp}_C(x)\subset C whose objects are all objects yy in CC such that C(x,y)=βˆ…C(x,y)=\emptyset.

𝔖𝔭𝔒𝔠 0\mathfrak{Spec}^0

(RosenbergSpectraNSp 1.4) 𝔖𝔭𝔒𝔠 0(C)βŠ‚C\mathfrak{Spec}^0(C)\subset C is the full subcategory of CC generated by those objects xx in CC whose support 𝔖𝔲𝔭𝔭 C(x)\mathfrak{Supp}_C(x) has a final object, x^\hat{x} (in particular the support is nonempty).

(RosenbergSpectraNSp 1.4.4) A choice of a final object x^\hat{x} for every object xx in 𝔖𝔭𝔒𝔠 0(C)\mathfrak{Spec}^0(C) extends to a functor ΞΈ c:𝔖𝔭𝔒𝔠 0(C)β†’C\theta_c : \mathfrak{Spec}^0(C)\to C.

When CC is a preorder

(RosenbergSpectraNSp 1.4.6) Now specialize to the case where CC is a preorder category having finite coproducts (equivalently supremum for every pair of objects). Then the functor ΞΈ c:𝔖𝔭𝔒𝔠 0(C)β†’C\theta_c : \mathfrak{Spec}^0(C)\to C factors through the embedding 𝔖𝔭𝔒𝔠 1(C)β†’C\mathfrak{Spec}^1(C)\to C. Consequently, it corestricts to a functor 𝔖𝔭𝔒𝔠 0(C)→𝔖𝔭𝔒𝔠 1(C)\mathfrak{Spec}^0(C)\to \mathfrak{Spec}^1(C) which may also be denoted as ΞΈ C\theta_C.

Relative spectra

(RosenbergSpectraNSp 1.6) Given a functor between small categories F:Cβ†’DF: C\to D the relative spectra 𝔖𝔭𝔒𝔠 0(C,F),𝔖𝔭𝔒𝔠 1(C,F)\mathfrak{Spec}^0(C,F), \mathfrak{Spec}^1(C,F) are defined as isocomma objects (β€œ2-categorical pullbacks”)

𝔖𝔭𝔒𝔠 0(C,F) ⟢θ F C Ο€ F 0 ↓ ↓ F 𝔖𝔭𝔒𝔠 0(C) ⟢θ C D𝔖𝔭𝔒𝔠 1(C,F) ⟢θ F 1 C Ο€ F 1 ↓ ↓ F 𝔖𝔭𝔒𝔠 1(C) ⟢θ C 1 D\array{ \mathfrak{Spec}^0(C,F)&\stackrel{\theta_F}\longrightarrow&C\\ {\pi_F^0}_\mathrlap{}\downarrow&&\downarrow{}_\mathrlap{F}\\ \mathfrak{Spec}^0(C)&\underset{\theta_C}\longrightarrow&D } \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \array{ \mathfrak{Spec}^1(C,F)&\stackrel{\theta^1_F}\longrightarrow&C\\ {\pi_F^1}_\mathrlap{}\downarrow&&\downarrow{}_\mathrlap{F}\\ \mathfrak{Spec}^1(C)&\underset{\theta^1_C}\longrightarrow&D }

It appears that, for a fixed target category DD, the constructions (C,F)↦𝔖𝔭𝔒𝔠 0(C,F)(C,F)\mapsto \mathfrak{Spec}^0(C,F) and (C,F)↦𝔖𝔭𝔒𝔠 1(C,F)(C,F)\mapsto \mathfrak{Spec}^1(C,F) extend to pseudofunctors Cat/Dβ†’CatCat/D\to Cat. If DD is a preorder with finite coproducts then every functor F:Cβ†’DF: C\to D induces a canonical functor

ΞΈ C,F:𝔖𝔭𝔒𝔠 0(C,F)βŸΆπ”–π”­π”’π”  1(C,F) \theta_{C,F} : \mathfrak{Spec}^0(C,F)\longrightarrow \mathfrak{Spec}^1(C,F)

which are the components of a natural transformation of pseudofunctors Cat/D→CatCat/D\to Cat

ΞΈ:𝔖𝔭𝔒𝔠 0βŸΆπ”–π”­π”’π”  1 \theta : \mathfrak{Spec}^0\longrightarrow \mathfrak{Spec}^1

Applications to spectra of Abelian categories


Starting with an Abelian category AA we proceed in three steps. In the first step we construct some preorder C AC_A from AA, or of a functor F A:C A→D AF_A: C_A\to D_A with a preorder category D AD_A and then apply the spectral cookbook. The objects of C AC_A may be some special subcategories of AA (e.g. some class of topologizing subcategories), of multiplicative systems of morphisms, as used in the localization theory and so on. In a third step, one often passes to the equivalence classes of objects in the obtained spectra.

Additionally, one may add construction of some sort of topology or additional β€œstructure stack” to obtained spectrum using possibly supplemental knowledge about the input data.

Remark on monoidal case

Some reconstruction theorems (like the theorems of Balmer and of Garkusha) consider abelian symmetric monoidal categories instead; their spectra are very analogous but the subcategories used to construct the intermediate preorder category are monoidal subcategories as well and various constructions respect the monoidal structure. The pattern is still the same.

Revised on March 6, 2013 19:07:36 by Zoran Ε koda (