D ECONSTRUCTING A USLANDER ’ S F ORMULAS Alex Martsinkovsky Functor Categories, Model Theory, and Constructive Category Theory Pärnu July 17, 2019 A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 1 / 46
P LAN OF THE TALK Part 1. Auslander’s formulas Part 2. Zeroth derived functors Part 3. Stabilization of additive functors Part 4. An extension of the Auslander-Reiten formula Part 5. R 0 F and L 0 F revisited A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 2 / 46
A USLANDER ’ S FORMULAS Part 1. Auslander’s formulas A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 3 / 46
A USLANDER ’ S FORMULAS T HE A USLANDER TRANSPOSE Let Λ be a ring and A a finitely presented Λ -module. Choose a projective presentation P 1 Ý Ñ P 0 Ý Ñ A Ý Ñ 0 and dualize it into Λ . The resulting exact sequence Ñ A ˚ Ý Ñ P ˚ Ñ P ˚ 0 Ý 0 Ý 1 Ý Ñ Tr A Ý Ñ 0 defines the Auslander transpose Tr A of A . Dualizing again, we recover A , up to projective equivalence. N.B. The last statement is not true if A is not finitely presented. A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 4 / 46
A USLANDER ’ S FORMULAS T HE A USLANDER -R EITEN FORMULA Λ := artin algebra over a commutative artinian ring k , I := the injective envelope of k viewed as a module over itself, D : “ Hom k p , I q , A 1 and B := finitely generated Λ -modules. T HEOREM There is a bifunctorial isomorphism D Ext 1 p A 1 , B q » p B , D Tr A 1 q . R EMARK The only needed assumption here is that A 1 be finitely presented. If A 1 is not finitely presented, the proof breaks down. A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 5 / 46
A USLANDER ’ S FORMULAS T HE BIDUALIZATION MAP Let Λ be an arbitrary ring, A a finitely presented Λ -module, and Ñ A ˚˚ the canonical map. Then there is an exact sequence e A : A Ý e A � A ˚˚ � Ext 1 p Tr A , Λ q � A � Ext 2 p Tr A , Λ q � 0 0 This is the Λ -component of the exact sequence of functors ρ A b � p A ˚ , � Ext 1 p Tr A , � Ext 2 p Tr A , � 0 � A b 0 q q q R EMARK This sequence is not exact if A is an infinite projective, even over a field (cardinality!). A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 6 / 46
A USLANDER ’ S FORMULAS T HE COMPANION TRANSFORMATION µ p A , q Assume again that A is finitely presented. Then there is an exact sequence µ p A , � A ˚ b q � p A , � Tor 2 p Tr A , � Tor 1 p Tr A , � 0 0 q q q R EMARK This sequence is not exact if A is an infinite projective, even over a field (the identity map cannot factor through a finitely generated projective!). A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 7 / 46
A USLANDER ’ S FORMULAS T HE GOAL OF THIS TALK The goal of today’s talk is to extend the above formulas to arbitrary modules over arbitrary rings. A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 8 / 46
Z EROTH DERIVED FUNCTORS Part 2. Zeroth derived functors A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 9 / 46
Z EROTH DERIVED FUNCTORS P ROJECTIVE RESOLUTIONS Blanket assumption : all functors are from modules to abelian groups F : Mod p Λ q Ý Ñ Ab and are additive. Given a module M , an exact sequence . . . Ý Ñ P 1 Ý Ñ P 0 Ý Ñ M Ý Ñ 0 (excluding M itself), where the P i are projective, is called a projective resolution of M . T HEOREM Any two projective resolutions of M are homotopy equivalent. A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 10 / 46
Z EROTH DERIVED FUNCTORS I NJECTIVE RESOLUTIONS Given a module M , an exact sequence Ñ I 0 Ý Ñ I 1 Ý 0 Ý Ñ M Ý Ñ . . . (excluding M itself), where the I i are injective, is called an injective resolution of M . T HEOREM Any two injective resolutions of M are homotopy equivalent. A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 11 / 46
Z EROTH DERIVED FUNCTORS N OMENCLATURE : DERIVED FUNCTORS Given an additive functor F , apply it to a resolution. Since resolutions are homotopically unique, the homology groups of the resulting complex are unique up to isomorphism. Projective resolutions Injective resolutions R i F Covariant F L i F L i F Contravariant F R i F N.B. For a contravariant F , subscripts and superscripts are flipped. A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 12 / 46
� � � � Z EROTH DERIVED FUNCTORS T HE CASE i “ 0 : L 0 F Zeroth derived functors are of special interest to us: � F p P 1 q � F p P 0 q L 0 F p M q 0 . . . D ! λ F p M q F p M q P ROPOSITION L 0 F is right-exact (i.e., preserves cokernels). C OROLLARY λ F : L 0 F Ý Ñ F is an isomorphism if and only if F is right-exact. A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 13 / 46
� � Z EROTH DERIVED FUNCTORS T HE CASE i “ 0 : R 0 F F p M q D ! ρ F p M q � R 0 F p M q � F p I 0 q � F p I 1 q � . . . 0 P ROPOSITION R 0 F is left-exact (i.e., preserves kernels). C OROLLARY Ñ R 0 F is an isomorphism if and only if F is left-exact. ρ F : F Ý A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 14 / 46
Z EROTH DERIVED FUNCTORS T HE CASE i “ 0 : A SUMMARY Projective resolutions Injective resolutions λ F ρ F Ñ R 0 F Covariant F L 0 F Ý Ñ F F Ý ρ F λ F L 0 F Ý Ñ R 0 F Ý Ñ F Contravariant F F The λ F are isomorphisms if and only if F is right-exact; The ρ F are isomorphisms if and only if F is left-exact. A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 15 / 46
Z EROTH DERIVED FUNCTORS Q UESTION : WHY IS THE CASE i “ 0 INTERESTING ? Projective resolutions Injective resolutions λ F ρ F Ñ R 0 F Covariant F Ý Ñ F Ý L 0 F F ρ F λ F L 0 F Contravariant F F Ý Ñ R 0 F Ý Ñ F Answer : because all arrows are universal: the λ F – with respect to natural transformations from right-exact functors to F , the ρ F – with respect natural transformations from F to left-exact functors. A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 16 / 46
� � Z EROTH DERIVED FUNCTORS T HE UNIVERSAL PROPERTY OF λ Rex D ! @ λ F � F L 0 F R EMARK This diagram shows that the subcategory of right-exact functors is coreflective in the category of all additive functors and λ is a coreflector, i.e., a counit of the adjunction ι % L 0 A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 17 / 46
� Z EROTH DERIVED FUNCTORS T HE UNIVERSAL PROPERTY OF ρ ρ F � R 0 F F @ � D ! Lex R EMARK This diagram shows that the subcategory of left-exact functors is reflective in the category of all additive functors and ρ is a reflector, i.e., a unit of the adjunction R 0 % ι A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 18 / 46
Z EROTH DERIVED FUNCTORS A N INTRINSIC CHARACTERIZATION OF λ L EMMA If F is covariant (resp., contravariant), then λ F : L 0 F Ý Ñ F (resp., λ F : L 0 F Ý Ñ F) evaluates to an isomorphism on projectives (resp., injectives). P ROPOSITION If F is covariant (resp., contravariant), then λ F : L 0 F Ý Ñ F (resp., λ F : L 0 F Ý Ñ F) is the unique natural transformation from a right-exact functor to F which evaluates to an isomorphism on projectives (resp., injectives). A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 19 / 46
Z EROTH DERIVED FUNCTORS A N INTRINSIC CHARACTERIZATION OF ρ L EMMA Ñ R 0 F (resp., If F is covariant (resp., contravariant), then ρ F : F Ý ρ F : F Ý Ñ R 0 F) evaluates to an isomorphism on injectives (resp., projectives). P ROPOSITION Ñ R 0 F (resp., If F is covariant (resp., contravariant), then ρ F : F Ý ρ F : F Ý Ñ R 0 F) is the unique natural transformation from F to a left-exact functor which evaluates to an isomorphism on injectives (resp., projectives). A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 20 / 46
Z EROTH DERIVED FUNCTORS E XAMPLE : L 0 OF THE COVARIANT H OM E XAMPLE F : “ p A , q , where A is finitely generated . λ In this case, L 0 F Ý Ñ F is the canonical transformation A ˚ b Ý Ñ p A , q because it is an isomorphism on projectives and A ˚ b is right-exact. Hence q » Tor i p A ˚ , L i p A , q for all i . A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 21 / 46
Z EROTH DERIVED FUNCTORS E XAMPLE : R 0 OF THE TENSOR PRODUCT E XAMPLE F : “ A b , where A is finitely presented . ρ Ñ R 0 F is the canonical transformation In this case, F Ý Ñ p A ˚ , A b Ý q because it is an isomorphism on injectives and p A ˚ , q is left-exact. Hence R i p A b q » Ext i p A ˚ , q for all i . A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 22 / 46
Z EROTH DERIVED FUNCTORS R ECOGNIZING L 0 F Let F : Mod p Λ q Ý Ñ Ab be an additive covariant functor. The natural transformation τ � F F p Λ q b τ M � F p M q F p Λ q b M � F p r m qp x q x b m ✤ where x P F p Λ q , m P M , and r m : Λ Ý Ñ M : λ ÞÑ λ m , evaluates to the canonical isomorphism on Λ . Whence A LEX M ARTSINKOVSKY D ECONSTRUCTING A USLANDER ’ S F ORMULAS J ULY 17, 2019 23 / 46
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