This document discusses organizing structures for understanding the category of modules Mod-R over a ring R. It describes how the Zariski spectrum constructed from R gives insight into Mod-R, and defines the rep-Zariski spectrum ZarR of R using indecomposable pure-injective R-modules. It then constructs a sheaf of categories LDefR over ZarR to associate a "categoried space" to R, providing more structure than just considering injective R-modules. Finally, it notes an equivalence between small abelian categories and deinable categories.

1. Spectra of categories of modules
Mike Prest
Department of Mathematics
University of Manchester
Manchester M13 9PL
UK
mprest@maths.man.ac.uk
November 14, 2006
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2. 1 Mod-R
2 The Zariski spectrum through the category of modules
3 Functor categories
4 The structure sheaf
5 Varieties and “abelian spaces”
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3. Mod-R
Mod-R
Can we understand the category Mod-R of (right) modules over a
ring R?
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4. Mod-R
Mod-R
Can we understand the category Mod-R of (right) modules over a
ring R?
Or find some organising structures which at least give some insight
into the category and how it hangs together?
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5. The Zariski spectrum through the category of modules
The Zariski spectrum through the category of modules
Consider R (commutative noetherian) → Spec(R), OSpec(R)
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6. The Zariski spectrum through the category of modules
The Zariski spectrum through the category of modules
Consider R (commutative noetherian) → Spec(R), OSpec(R)
This is an invariant of Mod-R.
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7. The Zariski spectrum through the category of modules
The Zariski spectrum through the category of modules
Consider R (commutative noetherian) → Spec(R), OSpec(R)
This is an invariant of Mod-R.
P (prime) → E (R/P), the injective hull of R/P.
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8. The Zariski spectrum through the category of modules
The Zariski spectrum through the category of modules
Consider R (commutative noetherian) → Spec(R), OSpec(R)
This is an invariant of Mod-R.
P (prime) → E (R/P), the injective hull of R/P.
D(r ) = {P : r ∈ P}
/
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9. The Zariski spectrum through the category of modules
The Zariski spectrum through the category of modules
Consider R (commutative noetherian) → Spec(R), OSpec(R)
This is an invariant of Mod-R.
P (prime) → E (R/P), the injective hull of R/P.
D(r ) = {P : r ∈ P} → {E : (R/rR, E ) = 0}
/
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10. The Zariski spectrum through the category of modules
The Zariski spectrum through the category of modules
Consider R (commutative noetherian) → Spec(R), OSpec(R)
This is an invariant of Mod-R.
P (prime) → E (R/P), the injective hull of R/P.
D(r ) = {P : r ∈ P} → {E : (R/rR, E ) = 0}
/
Taking the [A] = {E : (A, E ) = 0} where A is a finitely presented
R-module as basic open sets gives the same topology.
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11. The Zariski spectrum through the category of modules
Let C be a locally coherent category.
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12. The Zariski spectrum through the category of modules
Let C be a locally coherent category. We define the Gabriel-Zariski
spectrum, Zar(C), of C.
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13. The Zariski spectrum through the category of modules
Let C be a locally coherent category. We define the Gabriel-Zariski
spectrum, Zar(C), of C.
The points of Zar(C) are the (isomorphism classes of)
indecomposable injective objects of C. Let inj(C) denote the set of
these.
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14. The Zariski spectrum through the category of modules
Let C be a locally coherent category. We define the Gabriel-Zariski
spectrum, Zar(C), of C.
The points of Zar(C) are the (isomorphism classes of)
indecomposable injective objects of C. Let inj(C) denote the set of
these.
And the basic open sets are the [A] = {E ∈ inj(C) : (A, E ) = 0} as A
ranges over finitely presented objects of C.
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15. Functor categories
Functor categories
Mod-R = (R op , Ab)
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16. Functor categories
Functor categories
Mod-R = (R op , Ab)
Replace R by mod-R (the category of finitely presented modules) to
obtain Fun-R = mod-R, Ab
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17. Functor categories
Functor categories
Mod-R = (R op , Ab)
Replace R by mod-R (the category of finitely presented modules) to
obtain Fun-R = mod-R, Ab
This functor category is abelian,
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18. Functor categories
Functor categories
Mod-R = (R op , Ab)
Replace R by mod-R (the category of finitely presented modules) to
obtain Fun-R = mod-R, Ab
This functor category is abelian, locally coherent,
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19. Functor categories
Functor categories
Mod-R = (R op , Ab)
Replace R by mod-R (the category of finitely presented modules) to
obtain Fun-R = mod-R, Ab
This functor category is abelian, locally coherent, Grothendieck.
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20. Functor categories
Functor categories
Mod-R = (R op , Ab)
Replace R by mod-R (the category of finitely presented modules) to
obtain Fun-R = mod-R, Ab
This functor category is abelian, locally coherent, Grothendieck. So
we may consider the space Zar(mod-R, Ab) = Zar(Fun-R).
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21. Functor categories
Theorem
(Gruson and Jensen) The functor given on objects by M → M ⊗R −
from Mod-R to Fun-(R op ) is a full embedding.
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22. Functor categories
Theorem
(Gruson and Jensen) The functor given on objects by M → M ⊗R −
from Mod-R to Fun-(R op ) is a full embedding. An object of
Fun-(R op ) is injective iff it is isomorphic to N ⊗ − for some
pure-injective R-module N.
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23. Functor categories
Theorem
(Gruson and Jensen) The functor given on objects by M → M ⊗R −
from Mod-R to Fun-(R op ) is a full embedding. An object of
Fun-(R op ) is injective iff it is isomorphic to N ⊗ − for some
pure-injective R-module N. In particular this functor induces a
bijection between the set, inj(Fun-(R op )), of (iso classes of)
indecomposable injective functors and the set pinjR of (iso classes
of) indecomposable pure-injective R-modules.
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24. Functor categories
Theorem
(Gruson and Jensen) The functor given on objects by M → M ⊗R −
from Mod-R to Fun-(R op ) is a full embedding. An object of
Fun-(R op ) is injective iff it is isomorphic to N ⊗ − for some
pure-injective R-module N. In particular this functor induces a
bijection between the set, inj(Fun-(R op )), of (iso classes of)
indecomposable injective functors and the set pinjR of (iso classes
of) indecomposable pure-injective R-modules.
Thus the Gabriel-Zariski topology on inj(Fun-(R op )) induces a
topology on pinjR , which we denote by ZarR and refer to as the
rep-Zariski spectrum of R.
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25. Functor categories
By a duality between the functor categories for left and right
modules, the sets [F ] = {N ∈ pinjR : FN = 0} for F ∈ fun-R form a
basis of open sets for ZarR .
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26. Functor categories
By a duality between the functor categories for left and right
modules, the sets [F ] = {N ∈ pinjR : FN = 0} for F ∈ fun-R form a
basis of open sets for ZarR . Here fun-R denotes the subcategory of
finitely presented objects in Fun-R
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27. Functor categories
By a duality between the functor categories for left and right
modules, the sets [F ] = {N ∈ pinjR : FN = 0} for F ∈ fun-R form a
basis of open sets for ZarR . Here fun-R denotes the subcategory of
finitely presented objects in Fun-R and “FN = 0” really means
→
− →
−
F N = 0 where F is the (unique) extension of F to a functor on
Mod-R which commutes with direct limits.
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28. Functor categories
By a duality between the functor categories for left and right
modules, the sets [F ] = {N ∈ pinjR : FN = 0} for F ∈ fun-R form a
basis of open sets for ZarR . Here fun-R denotes the subcategory of
finitely presented objects in Fun-R and “FN = 0” really means
→
− →
−
F N = 0 where F is the (unique) extension of F to a functor on
Mod-R which commutes with direct limits.
This gives a much richer structure than what we would get by looking
just at injective R-modules. A good pair of examples is R = k[T ]
(the affine line) and R = k A1 (more or less the projective line).
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29. The structure sheaf
The structure sheaf
Given a ring R we define a presheaf (defined on a basis) over ZarR by
[F ] = {N : FN = 0} → End((R, −)/ F ), where F denotes the
result of “localisation at F ”.
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30. The structure sheaf
The structure sheaf
Given a ring R we define a presheaf (defined on a basis) over ZarR by
[F ] = {N : FN = 0} → End((R, −)/ F ), where F denotes the
result of “localisation at F ”. Note this is localisation “one
representation level up” i.e. in the functor category.
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31. The structure sheaf
The structure sheaf
Given a ring R we define a presheaf (defined on a basis) over ZarR by
[F ] = {N : FN = 0} → End((R, −)/ F ), where F denotes the
result of “localisation at F ”. Note this is localisation “one
representation level up” i.e. in the functor category.
This presheaf-on-a-basis, Def R , is separated (this follows from the
interpretation of End((R, −)/ F ) as a certain ring of functions on a
certain module) so embeds in its sheafification, which we denote by
LDef R and refer to as the sheaf of locally definable scalars (for
R-modules).
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32. The structure sheaf
More generally:
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33. The structure sheaf
More generally:
[F ] → fun-R/ F , the whole category of finitely presented functors
localised at F .
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34. The structure sheaf
More generally:
[F ] → fun-R/ F , the whole category of finitely presented functors
localised at F .
- a presheaf of small abelian categories (all localisations of fun-R). It
is separated and embeds in its sheafification, LDef R , the “sheaf of
categories of locally definable scalars”. Thus we associate to R a
“categoried space” ZarR , LDef R .
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35. The structure sheaf
More generally:
[F ] → fun-R/ F , the whole category of finitely presented functors
localised at F .
- a presheaf of small abelian categories (all localisations of fun-R). It
is separated and embeds in its sheafification, LDef R , the “sheaf of
categories of locally definable scalars”. Thus we associate to R a
“categoried space” ZarR , LDef R .
And there’s no reason to stop there: we may replace Mod-R by any
functor category Mod-R = (Rop , Ab) or even by any definable
subcategory of such a category (that is, a subcategory closed under
products, direct limits and pure subobjects).
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36. Varieties and “abelian spaces”
Varieties and “abelian spaces”
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37. Varieties and “abelian spaces”
Varieties and “abelian spaces”
Theorem
There is an equivalence between the 2-category of small abelian
categories with exact functors and the 2-category of definable
categories with functors which commute with direct limits and direct
products.
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38. Varieties and “abelian spaces”
Varieties and “abelian spaces”
Theorem
There is an equivalence between the 2-category of small abelian
categories with exact functors and the 2-category of definable
categories with functors which commute with direct limits and direct
products.
R (commutative) → Ab(R (op) ) = A (small abelian category)
→ Ex(A, Ab) = D (definable category, in fact Mod-R)
→ Zar(D), LDef (D) (= ZarR , LDef R ): a rather large sheaf of
categories, but it contains, as a subsheaf, over the subspace of
injective points of ZarR , the thread consisting of localisations of
(R, −) and that is exactly Spec(R), OSpec(R) , the usual structure
sheaf of R.
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