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# Ziegler's Spectrum

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### Ziegler's Spectrum

1. 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 () November 14, 2006 1 / 11
2. 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” () November 14, 2006 2 / 11
3. 3. Mod-R Mod-R Can we understand the category Mod-R of (right) modules over a ring R? () November 14, 2006 3 / 11
4. 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? () November 14, 2006 3 / 11
5. 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) () November 14, 2006 4 / 11
6. 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. () November 14, 2006 4 / 11
7. 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. () November 14, 2006 4 / 11
8. 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} / () November 14, 2006 4 / 11
9. 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} / () November 14, 2006 4 / 11
10. 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. () November 14, 2006 4 / 11
11. 11. The Zariski spectrum through the category of modules Let C be a locally coherent category. () November 14, 2006 5 / 11
12. 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. () November 14, 2006 5 / 11
13. 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. () November 14, 2006 5 / 11
14. 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. () November 14, 2006 5 / 11
15. 15. Functor categories Functor categories Mod-R = (R op , Ab) () November 14, 2006 6 / 11
16. 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 () November 14, 2006 6 / 11
17. 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, () November 14, 2006 6 / 11
18. 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, () November 14, 2006 6 / 11
19. 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. () November 14, 2006 6 / 11
20. 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). () November 14, 2006 6 / 11
21. 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. () November 14, 2006 7 / 11
22. 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. () November 14, 2006 7 / 11
23. 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. () November 14, 2006 7 / 11
24. 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. () November 14, 2006 7 / 11
25. 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 . () November 14, 2006 8 / 11
26. 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 () November 14, 2006 8 / 11
27. 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. () November 14, 2006 8 / 11
28. 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). () November 14, 2006 8 / 11
29. 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 ”. () November 14, 2006 9 / 11
30. 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. () November 14, 2006 9 / 11
31. 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). () November 14, 2006 9 / 11
32. 32. The structure sheaf More generally: () November 14, 2006 10 / 11
33. 33. The structure sheaf More generally: [F ] → fun-R/ F , the whole category of ﬁnitely presented functors localised at F . () November 14, 2006 10 / 11
34. 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 . () November 14, 2006 10 / 11
35. 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). () November 14, 2006 10 / 11
36. 36. Varieties and “abelian spaces” Varieties and “abelian spaces” () November 14, 2006 11 / 11
37. 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. () November 14, 2006 11 / 11
38. 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. () November 14, 2006 11 / 11