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Date Created: 09/19/15
Joseph Lipman ue University Department of Mathematics ipmari mathpurdueedu February 27 2009 lnuoduction I39m going to talk eventually about a problem in automated reasoning growing out of Grothendieck Duality theory a problem which I39ve been raising informally from time to time over the past eighteen years but haven39t thought about intensively enough I hope someday someone will be moved to do better To motivate the problem indeed to state it properly I need to describe the formalism of Grothendieck duality This involves the very rich formalism of relations among Grothendieck39s siX operations Actually I39ll only talk about five of them To make it more agreeable perhaps this formalism will be discussed in a simple commutative algebra context However from time to time it will be noted that the very same formalism plays a central role in the more complicated context of Grothendieck duality 0 Extension and restriction of scalars 0 Extension and restriction of scalars Tensor and Hom closed categories 0 Extension and restriction of scalars Tensor and Hom closed categories 9 Compatibilities among the four operations 0 Extension and restriction of scalars Tensor and Hom closed categories 9 Compatibilities among the four operations a Let the games begin commutativities growing from the axiomatic soil 9 Extension and restriction of scalars Tensor and Hom closed categories 9 Compatibilities among the four operations a Let the games begin commutativities growing from the axiomatic soil 9 A fifth operation 9 Extension and restriction of scalars Tensor and Hom closed categories 9 Compatibilities among the four operations a Let the games begin commutativities growing from the axiomatic soil 9 A fifth operation 3 Twisted inverse image the basic pseudofunctor more commutativities 9 Extension and restriction of scalars Tensor and Hom closed categories 9 Compatibilities among the four operations a Let the games begin commutativities growing from the axiomatic soil 9 A fifth operation 3 Twisted inverse image the basic pseudofunctor more commutativities 0 Coherence mastering commutative diagrams in closed categories 9 Extension and restriction of scalars Tensor and Hom closed categories 9 Compatibilities among the four operations a Let the games begin commutativities growing from the axiomatic soil 9 A fifth operation 6 Twisted inverse image the basic pseudofunctor more commutativities 0 Coherence mastering commutative diagrams in closed categories Summing up how to deal with diagrams built out of Grothendieck operations 7 Details in Notes on Derived Functors and Grothendieck Duality SLN 1960 Available as are the slides for this lecture at lt httpwwwmathpurdueedu lipman gt R category of commutative rings J For f R a 5 in R set R R modules7 S 5 modules7 1 R H 5 extension of scalars for M E R fM M R 5 E S covariant functor S a R restriction of scalars for N E S N N E R contravariant functor Adjointness of f and Hom5fE F HomRE F E e R F e S behavior vis vis composr For an identity map 1 R a R we have 1 identity of R For R L S i T in R 3 a natural transitivity isomorphism of functors dgf gf igt g satisfying dlf d 1 identity and associative sort of in that for each triple of maps 0 L o i o i o the following commutes dhgf hng gt hgf dhgfl id h gfr hg f hwy And analogously with arrows reversed for restriction of scalars I l As with any adjunction there is a functorial unit map 5 1 A 121 adjoint to the identity map fquot A 2 One checks that for any Rmodule M and m E M 6M MA M R 5 takes m to m 1 For 0 L o i o in R the following natural diagram of functors involving three different unit maps and two transitivity isomorphisms commutes 1 gt gt fggf gfgf L 4ng gt 1ng Equivalently categorically the dual diagram commutes 1 lt gm lt gf g l gfgf gfgf Ef e I l X and Hom over a ring R are instances of two more Grothendieck operations Their axiomatic properties are summarized in the notion of symmetric monoidal cosed category Definition Eilenberg Kelly 1965 A symmetric monodal category F R ROx7p739y consists of a category R a product functor X R x R a R an object R of R and functorial isomorphisms A7 B7 C in R a A B C Lgt A B C R ALgtA pA RgtA 39y A B gt B A associativity left and right units symmetry such that 39y2 1 and the following diagrams commute Definition continued A R BLgtA R B R ALgtA R p 1l imam Al ix A B A B A A A B C D L A B C D igtA BQ gtC D a ll imam A B C D A B C D A B C igtA BQltgtC L B C A wall i0 B A C L B A C amp B C A Definition continued A closed category is a symmetric monoidal category F as above together with a functor called internal hom 77 R0p x R a R where R039 is the dual category of R and a functorial isomorphism 7r HomRA B7 C Lgt HomRA7 BC The isomorphism 7139 just expresses tensor hom adjunction Example Monoids as monoidal categories As a category having only identity maps a monoid G together with its multiplication is a monoidal category symmetric if G is commutative If G is a group closed under inverses then the operation yz zy 1 makes G into a closed category Homgxyz 7 j ltgt xy z ltgt X zy 1 ltgt Homgx7 yz 7 j I l 39lction via Ll t an couni t ma For later purposes it is better to reformulate the preceding adjunction 7139 in computer friendly terms via the unit and counit maps of 7139 with ClgtA A X B and lJC 8 C these are the functorial maps 6 AHW A8A 8 e7ridentity map ofA 8 7 ClgtJC 8 C X B a C 77 7r 1identity map of 8 Indeed it is standard that adjunctions between any two given functors o S 0 4 correspond oneone to the existence of maps 6 1 a lJCD and 7 lJClD a 1 such that both of the following compositions are identity maps d viae bum vian 7 w viae wcbw vian w Deductions From the axioms of monoidal categories even ignoring the symmetry isomorphism 39y one can deduce that the following diagrams commute R A BLR A B A B RLgtA B R ml i Mi ip A B A B A B A B see MacLane39s Categories for the working mathematician first exercise in Chap 7 This is done by a clever direct argument or by use of a coherence theorem ibid Chap7 2 which says roughly that all diagrams built up from the axioms must commute This is a primitive example of a basic question to be explained later underlying the present talk 39erLll39 operations For f R a 5 in R there are natural R maps Rags aA RaBi A SB ABES J read this with deleted and natural S maps se R E 5fFampfE RF EF6R 1 isjust the standard iso E R S 85 F R 5 A E 8 F lt8 5 These maps are related via f f adjointness For example 1 is adjoint to the natural composite map E R F a fE R fF i ffE 5 FF Note As this categorical characterization isn39t the usual definition of 1 there is something to check Ditto without explicit mention for subsequent examples I l Assume aXiomatically that 1 fE R F a fE 5 FF is an isomorphism This holds over rings and also in most other cases of interest One checks commutativity of the following natural diagrams commutativities which serve as axioms for the interaction of and X Interactions involving f and 7 result via adjunction asmALtisw tAMBLtiAM i if ii if7 R A 7 11A B A a fB A M fiA fB fC LE fA B fC L fA B C a law A fB fC A B C T fA B C M 0 h 0 i Abstractly speaking the preceding commutativities signify that S a R is compatible with the monoidal structures on its source and target or as we say that is a monodal functor If G and H are abeian groups viewed as closed categories then a functor j G a H is monoidal ltgt j is a group homomorphism For R L S i T in R the following natural diagrams commute R gfLT l l 15 gJ gm4 gm B gm4 B l l agtAMgtB T gA gB 7 fed4M 5 For any contravariant functor CD from R to the category of abelian groups the preceding diagrams with g Clgtf commute Let quotthe games begin The category theoretic structure we have described up to now is the formalism of adjoint monoidal closed category valued pseudofunctors 0 There is a functorial map 1A7 B a MA B corresponding under 7139 to the composed map aiAi B M i am B A a8 where mug is the counit map of the 7 adjunction o For fixed A the functorial isomorphism 1 fC X A Lgt fC X FA induces a conjugate internal adjunction isomorphism on right adjoints namely A7 B lt fAB c There is a functorial map i A7 B PEA7 fB which is adjoint to the composition 71 A B a A 3 L gm 3 Il Of course the above all turn out to be standard maps in the ring context but again this formalism applies to other contexts The original paper of Eilenberg and Kelly contains a crowd of maps and commutative diagrams all coming out of the axioms Many such diagrams force themselves on you when for example you delve into Grothendieck duality theory Here39s an example involving all four operations Exam ple Exercise Establish from axioms a natural commutative diagram Marv G F fFfG F mic fffFG fF fFG fF G lnterpret this in the context of rings Let f R a 5 be in R For A E S and B E S we have the usual isomorphism A R B Lgt A 85 5 R B This can be described in the above formalism as being the natural composed functorial projection map p igA B igA fB iA fB Assume aXiomatically that p is an isomorphism This assumption holds in the most interesting contexts D n F Jase x For any commutative square of ring homomorphisms 5 L 5 i 0 lg R T R and C E S we have the usual map R R C a 5 85 C This can be described in the above formalism as being the functorial map 9 00 UV Hgtv k adjoint to the natural composition a vv Lgt ugv Those 0 for which 00 is a functorial isomorphism play a special role For rings these arejust the fiber sum squares ie 5 E 5 R R More interestingly in the derived category context used in Grothendieck duality they turn out to be the tor independent fiber squares ie TorFR 5 0 for all i gt o Il interaction of Pi ojectiou and Base Change Exam ple Exercise For any commutative square of ring homomorphisms s L s fT 039 Tg R T R the following diagram with h vf gu and C E R D E S commutes uC u D uC D A u fC D i i9 uC gvD gvfC D pi lav gguC VD gt ghC vD gvfC vD I l For f R a 5 in R and E E S F E R there is a canonical isomorphism HomRE7 F gt Hom5E7 HomR7 Thus The functor 1 R a S taking F to HomR7 F is right adjoint to In Grothendieck duality theory the existence of a right adjoint for f is a fundamental nontrivial theorem In any case we can add the right adjoint 1 to the preceding formalism For any commutative fiber sum pushout square 5 L s e s R R flap RTR with u flat and f finite and finitely presented one has canonical isos 5 85 HomR7 F gt R R HomR7 F gt HomR7 R R F gt HomRS7 R R F This composition can be described formally as the functorial map 30 vfXF a gXuF adjoint to the composition gvfXF if fffo uF 9 The theorem that for f a proper map of noetherian schemes and u flat this base change map is an isomorphism is a pillar of Grothendieck duality a inverse im Grothendieck duality theory is concerned basically with the twisted inverse image pseudofunctor built by pasting together the pseudofunctors 1 over proper maps and f over tale maps via the preceding basechange isomorphism The pasting is possible because any finitetype separated map of noetherian schemes factors as properoopen immersion Nagata39s compactification theorem Suresh Nayak showed recently that the process extends to essentially finite type separable scheme maps For simplicity we state the defining theorem only for quasi finite maps of noetherian rings essentially finitetype maps with finite fibers A formally similar statement holds in Grothendieck duality with proper in place of finite A quasi finite map whose ring theoretic fibers are finite products of separable field extensions is called tale Theorem On the category of quasi finite maps of noetherian rings there is a pseudofunctor that is uniquely determined up to isomorphism by the following three properties i The pseudofunctor restricts on the subcategory of finite maps to a right adjoint of 7 ii The pseudofunctor restricts on etae maps to 7 iii For any commutative fiber sum pushout square 5 s g RR 4 a p RTR With u hence v etae and f hence g finite the base change map 30 is vf39 v39f39 igt vf39 gu39 igt g39u39 glu Construction of f In the quasi finite context the necessary compactification is given by Zariski39s main theorem Any quasi finite map f R a 5 factors as R i T i 5 with p finite and e tale Choose such a factorization and set f39F epXF5 T HomRT F feR The problem is to show 1quot independent of choice of factorization This means showing that various diagrams commute Now explore various compatibilities of f39 with f X and 7 More commutative diagrams two natural transformations 04 3 between these functors also built up from the basic data we have 04 3 provided that in the construction of these transformations there is no functor of the form T7 R With T a nonconstant functor Example where a 75 3 Fix a closed category R as above For A E R there is a natural map rA A a A7 R7 R corresponding under 7139 to the evaluation map A X A7 R E A7 R A a R where the latter map is the unit of tensor hom adjuction The composition A R E A R R R M A R J is NOT always the identity eg for oo dim39l vector spaces over a field I I li lww was 3 31 Putting D A X B C one gets from the isomorphisms HomD A B C lgtHomD X A B C i HomD A X B C L HomD A BC1gt HomD A B internal tensor hom adjunction pointless over rings but not schemes 7r 7rA B C A B C Lgt A B CH Using the description via unit and counit of the tensor hom adjunction in closed categories one gets from the coherence theorem that the following functorial diagram commutes A B C7Dl L A B7lchlll L A Byl Dill 0 lim A B CD A7B CyD It is instructive to prove this commutativity directly from the axioms I l Aeratcm and clallenge One would like ideally to have a similar coherence theorem for the full formalism of adjoint monoidal closed category pseudofunctors This would obviously be very useful But though there have been some results beyond Kelley Mac Lane none to my knowledge can handle say the two earlier exercises involving adjoint functors between closed categories How hard can it be to do one diagram To do a whole class of diagrams This is the challenge u m m ary o The formalism of Grothendieck39s six operations of which we considered only five readily illustrated in the context of commutative rings appears in many other contexts For example it forms the natural framework around which to build Grothendieck duality for schemes with Zariski topology or with tale topology or classical topology 0 The formalism is very rich leading to many diagrams whose commutativity is basic to the various domains of application 0 Proving these diagrams commute usually means decomposing them via definitions of the maps involved into smaller diagrams which are known to commute Ultimately this process has to lead back to the axioms of the formalism Finding such a decomposition can be tedious even for one diagram let alone many which soon tax the limits of human patience Summary continued 0 Experience suggests that the level of complexity of this process is somewhere between that of solving Rubik39s cube and proving theorems from axioms Subjectively it seems that the number of techniques used is rather small suggesting that a computer could be taught how to do it or in case no definitive algorithm can exist at least to be of significant use as an assistant However despite some small efforts I am unable to teach a computer how to find a solution even to the relatively simple exercise mentioned before namely to prove from the axioms of monoidal categories that the following diagrams commute R A BLgtR A B A B RLgtA B R A 1l lA 1 Pl if A B A B A B A B Il Summary continued It is not hard to teach the computer the axioms and the formal deduction rules But then in my experience the computer just starts to generate endless trivialities without ever approaching a solution On the other hand there has been significant work done in automated theorem proving so there may be some techniques for preventing such vacuous deduction If there are any experts in the audience I would be glad to hear from them 0 Best of all would be a coherence theorem giving practical criteria for testing commutativity o I suspect that working seriously on this problem more so than I have would lead to substantial new results in artificial intelligence andor logic
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