Algebra in a Category

Transcription

1 Algebr in Ctegory Dniel Muret Otober 5, 2006 In the topos Sets we build lgebri strutures out o sets nd opertions (morphisms between sets) where the opertions re required to stisy vrious xioms. One we hve trnslted these xioms into digrmmti orm, we n opy the deinition into ny tegory with inite produts. The dvntge o suh onstrutions is tht mny o the omplited onepts turn out to be simple objets (suh s rings or groups) built up inside tegory other thn Sets. We begin this note by explining how to build belin groups, rings nd modules in ny tegory with inite produts. We then show tht the tegories o these objets hve vrious nie properties. Throughout we will be deling with produts A n nd sine in generl tegory there my be no nonil produt, there is some potentil mbiguity. Our stnding ssumptions bout the underlying set theory inlude strong orm o the xiom o hoie, whih llows us to selet produt or every set o objets (inluding terminl objet). Throughout ll tegories re nonempty. Contents 1 Deinitions Abelin Groups Rings Right Modules Let Modules Sheves o Groups, Rings nd Modules 6 1 Deinitions 1.1 Abelin Groups An belin group is set A with binry opertion (ddition), unry opertion (dditive inverse), nd nullry opertion (zero). These re morphisms : A A A, (x, y) x + y v : A A, x x u : 1 A, 0 (1) stisying ertin identities, whih n be restted s the ommuttivity o the ollowing digrms: Assoitivity o Addition A A 1

2 Additive Identity A 1 A 1 A 1 u A u 1 Additive Inverses A ( 1 v ) ( v 1 ) A 1 u A 1 u Commuttivity τ A where τ is deined by tking the projetions, p 2 : A A A nd interhnging them - so τ = ( p2 ). Deinition 1. Let C be tegory with inite produts. We deine n belin group objet (or just belin group) in C to be n objet A together with three rrows : A A A, v : A A, u : 1 A or whih the our digrms bove ommute. A morphism o belin groups (A,, v, u) (B,, v, u ) in C is morphism : A B whih ommutes with the struture mps. Tht is, the ollowing three digrms ommute A A A A v A 1 u A A A A A v A 1 u A This deines the tegory Ab(C) o belin groups in C. Note tht Ab(Sets) is isomorphi to the norml tegory o belin groups, nd we will denote both by Ab. Remrk 1. The deinition o n belin group in C inludes morphism A A A. Here the domin is the objet seleted s prt o the ssignment o speii produt C C to every objet C o C, reerred to in the introdution. I we were to mke dierent seletion o produts, the resulting tegory o belin groups in C would be isomorphi. The reder who is unomortble with strong hoie my preer to leve the hoie o produts rbitrry, so tht n belin group in C is n objet A together with morphism : A A A rom some produt o A with itsel to A. This leves open the possibility o the sme belin group ourring more thn one in Ab(C), in the sense o the identity 1 A : A A being n isomorphism o the two dierent objets, without the struture morphisms being the sme! This hppens beuse the morphism : A A A is only determined up to nonil isomorphism, in the sense tht we ould hoose nother produt A A nd would uniquely determine : A A A. In this pproh the tegory Ab(Sets) would only be equivlent to Ab, not isomorphi. One wy round this is to deine n equivlene reltion on group objets tht sys (A,, v, u) (A,, v, u ) i. 1 A is n isomorphism o group objets. A r esier wy is to ept strong xiom o hoie, whih is in ny se essentil to mny other spets o tegory theory. 2

3 1.2 Rings Throughout the rest o this setion, C will be tegory with inite produts. A ring objet (or just ring) in C is n belin group (A,, v, u) together with nother two morphisms m : A A A (multiplition) nd q : 1 A (multiplitive identity) whih mke the ollowing our digrms ommute: Assoitivity o Multiplition A 3 m 1 1 m m A m Multiplitive Identity A 1 A 1 A 1 q m A m To desribe digrms or let nd right distributivity, let, p 2, p 3 : A 3 A be the projetions rom the produt nd onsider the mps φ = p 2 p 3 : A3 A 4, φ = Then we require tht the ollowing two digrms ommute: Let Distributivity A 4 φ A 3 p 2 p 3 1 q 1 : A3 A 4 m m A Right Distributivity A 4 A 3 φ 1 m m A A morphism o rings (A,, v, u, m, q) (B,, v, u, m, q ) is morphism : A B o belin groups whih in ddition mkes the ollowing two digrms ommute: A A m A 1 q A A A m A 1 q A 3

4 This deines the tegory nrng(c) o (not neessrily ommuttive) rings in C. One gin, the tegory nrng(sets) is isomorphi to the usul tegory o not neessrily ommuttive rings, nd we denote both by nrng. We sy ring R nrng(c) is ommuttive i the ollowing digrm ommutes Commuttivity τ m A Here τ is deined by tking the projetions, p 2 : A A A nd interhnging them - so τ = ( p2 ). Let Rng(C) denote the ull subtegory o nrng(c) onsisting o the ommuttive ring objets. The tegory Rng(Sets) is isomorphi to the usul tegory o ommuttive rings, nd we denote both by Rng. 1.3 Right Modules A right module objet (or just right module) over ring (R,, v, u, m, q) in C is group (A,, v, u ) together with morphism : A R A mking the ollowing digrms ommute. Digrms re lbelled with the usul lw they generlise: m ( + b) r = r + b r R 1 A R A R A R A 1 = A 1 1 q A R A (r + s) = r + s A R 2 1 A R A R A R A ( r) s = (rs) A R 2 1 m A R 1 A R A 4

5 A morphism o modules (A,, v, u, ) (B,, v, u, ) is morphism o belin groups whih in ddition mkes the ollowing digrm ommute: A R A 1 B R B This deines the tegory o right modules over the ring objet R nrng(c), whih is denoted Mod(C; R) or more oten just ModR. To void n explosion in nottion we will reer to group, ring nd modules by the objet, nd only reer to the struture morphisms when neessry. 1.4 Let Modules A let module objet (or just let module) over ring (R,, v, u, m, q) in C is group (A,, v, u ) together with morphism : R A A mking the ollowing digrms ommute. Digrms re lbelled with the usul lw they generlise: r ( + b) = r + r b R R A R A 1 R A A 1 = 1 A q 1 R A A (r + s) = r + s R 2 A 1 R A R A R A A s (r ) = (sr) R 2 A m 1 R A 1 R A A A morphism o modules (A,, v, u, ) (B,, v, u, ) is morphism o belin groups whih in ddition mkes the ollowing digrm ommute: R A A 1 R B B 5

6 This deines the tegory o let modules over the ring objet R nrng(c), whih is denoted (C; R)Mod or more oten just RMod. 2 Sheves o Groups, Rings nd Modules In this setion the reder will need to know something bout topoi, in prtiulr bout grothendiek topologies. However, the reder only interested in sheves on topologil spe n keep in mind the entrl exmple o smll site whih we will desribe in moment. For nonempty smll tegory C we denote by P (C) the topos o ontrvrint untors C Sets. We ll the objets o Ab(P (C)) presheves o belin groups, the objets o nrng(p (C)) presheves o rings, nd the objets o Rng(P (C)) presheves o ommuttive rings. Lemm 1. For ny smll tegory C there is nonil isomorphism o tegories E : Ab(P (C)) Ab Cop E(F,, v, u)(c) = (F (C), C, v C, u C ) Proo. Let (F,, v, u) be n belin group objet in P (C). Then or C C, (F (C), C, v C, u C ) is n belin group in Sets, sine digrms in untor tegories ommute i. they ommute pointwise. Similrly, i φ : C C then F (φ) is morphism o the pproprite groups, sine, v, u re nturl. Also, i : F G is morphism o group objets, then it is lso nturl trnsormtion o the group vlued untors. In the opposite diretion, i F : C op Ab then the struture morphisms on eh F (C) (whih re untions between sets) piee together to orm nturl trnsormtions, v, u (o setvlued untors), nd the tuple (F,, v, u) is in Ab(P (C)). A morphism F G o untors is nturlly morphism o the group objets produed rom F, G. One we hek tht these two ssignments re in t inverse, we see tht we hve the desired isomorphism. Lemm 2. For ny smll tegory C there re nonil isomorphisms o tegories E : nrng(p (C)) nrng Cop E : Rng(P (C)) Rng Cop E(F,, v, u, m, q)(c) = (F (C), C, v C, u C, m C, q C ) Using the previous two Lemms, we heneorth identiy the tegories Ab(P (C)) nd Ab Cop, with similr onvention holding or rings nd ommuttive rings. Corollry 3. Let C be smll tegory. Then Ab(P (C)) is omplete grothendiek belin tegory, Rng(P (C)) is omplete nd oomplete nd nrng(p (C)) is omplete. Proo. These sttements ollow immeditely rom the t tht Ab is omplete grothendiek belin, Rng is omplete nd oomplete nd nrng is omplete. Deinition 2. A smll site is pir (C, J) onsisting o smll tegory C nd grothendiek topology J on C. Let X be topologil spe nd C the tegory o open sets o X. Then the open over topology J is grothendiek topology, nd the pir (C, J) is n exmple o smll site. A preshe o sets on C is she in the usul sense i nd only i it is J-she. The reder not milir with topoi n substitute this exmple every time we mention smll sites. Deinition 3. Let (C, J) be smll site. A she o belin groups on C is n objet A Ab(P (C)) whih is J-she when onsidered s preshe o sets. Similrly we deine sheves o rings nd sheves o ommuttive rings on C. Sine limits in Sh J (C) re omputed pointwise, the objets o Ab(Sh J (C)) re preisely the sheves o belin groups, nd similrly or rings nd ommuttive rings. Thereore Ab(Sh J (C)) is ull subtegory o Ab(P (C)), nd similrly or nrng nd Rng. 6

7 Let (C, J) be smll site nd P preshe o sets on C. Let P + be the preshe o sets obtined by the plus-onstrution nd η : P P + the nonil morphism o presheves o sets. I P is preshe o belin groups, then P + is preshe o belin groups with ddition {x S} + {y g g T } = {x h + y h h S T } nd η is morphism o presheves o belin groups. Similrly i P is preshe o rings, so is P +, with the multiplition {x S} {y g g T } = {x h y h h S T } nd η is morphism o presheves o rings. I P is preshe o ommuttive rings, so is P +. I φ : P Q is morphism o presheves o belin groups, rings or ommuttive rings, so is φ + : P + Q +. Thereore pplying this onstrution twie gives untors : Ab(P (C)) Ab(Sh J (C)) : nrng(p (C)) nrng(sh J (C)) : Rng(P (C)) Rng(Sh J (C)) Fix one o the three types o struture nd let i denote the inlusion. Then P P + (P + ) + gives nturl trnsormtion η : 1 i whih is esily seen to be the unit o n djuntion i. Sine produts nd pullbks in Sh J (C) re omputed pointwise, nd pointwise quire group nd ring strutures, the untors ll preserve inite limits. Thereore Ab(Sh J (C)) is Girud subtegory o Ab(P (C)) nd similrly or rings nd ommuttive rings. Remrk 2. The sheiition untor just deined is the one typilly introdued in the theory o topoi. In the speil se where our smll site is the open over topology on topologil spe, the sheiition untor is usully desribed dierently. Both onstrutions must yield nturlly equivlent untors, so this dierene is nothing to worry bout. Corollry 4. Let (C, J) be smll site. Then Ab(Sh J (C)) is omplete grothendiek belin tegory, Rng(Sh J (C)) is omplete nd oomplete nd nrng(sh J (C)) is omplete. In prtiulr this is true or sheves o belin groups, rings nd ommuttive rings on topologil spe X. Deinition 4. I (C, J) is smll site nd P she o belin groups on C, subshe o belin groups is monomorphism η : Q P o sheves o belin groups with the property tht η C : Q(C) P (C) is the inlusion o subset or every C C. Every subobjet o P is equivlent to subshe. I Q, Q re subsheves o belin groups o P then Q Q i nd only i Q(p) Q (p) or every p C. Corollry 5. Let (C, J) be smll site. The strutures on the belin tegory Ab(Sh J (C)) re desribed s ollows Zero The zero objet is the preshe Z(p) = 0. Kernel I φ : M N is morphism o sheves o belin groups, then K(p) = Ker(φ p ) deines she o belin groups, nd the inlusion K M is the kernel o φ. Cokernel I φ : M N is morphism o sheves o belin groups, then C(p) = N(p)/Im(φ p ) deines preshe o belin groups, nd the nonol morphism N C C is the okernel o φ. Limits I D is digrm o presheves o belin groups, then deine L(p) to be the limit o the digrm D(p) o belin groups. This beomes she o belin groups nd the projetions L D i re limit or the digrm. Colimits I D is digrm o presheves o belin groups, then deine C(p) to be the olimit o the digrm D(p) o belin groups. This beomes preshe o belin groups nd the morphisms D i C C re olimit or the digrm. 7

8 Imge Let φ : M N be morphism o sheves o belin groups nd I the subshe o belin groups o N deined by n I(p) i nd only i there exists T J(p) suh tht n h Im(φ q ) or every h : q p in T. Then I N is the imge o φ. Inverse Imge Let φ : M N be morphism o sheves o belin groups nd T N subshe o belin groups o N. Then the inverse imge φ 1 T is the subshe o belin groups o M deined by (φ 1 T )(p) = φ 1 p (T (p)). Proo. Sine Ab(Sh J (C)) is Girud subtegory o Ab(P (C)) the lims re ll esily heked (AC,Setion 3). Deinition 5. Let C be smll tegory nd R preshe o rings on C. A preshe o right modules over R is n objet o Mod(P (C); R). This onsists o the ollowing dt: preshe o belin groups M : C op Ab together with right R(C)-module struture on M(C) or every C C, with the property tht (m r) = m r or : D C, r R(C) nd m M(C). We use the nottion or M()( ) nd R()( ) to mke the nottion neter. Similrly preshe o let modules over R is n objet o (P (C); R)Mod, whih onsists o preshe o belin groups M together with let R(C)-module struture on M(C) or every C C, with the property tht (r m) = r m. Deinition 6. Let (C, J) be smll site nd R she o rings on C. A she o right modules over R is n objet o Mod(Sh J (C); R), whih is just preshe o right modules over R whih hppens to be J-she. Thereore Mod(Sh J (C); R) is ull subtegory o Mod(P (C); R). Similrly she o let modules over R is n objet o (Sh J (C); R)Mod, whih is preshe o let R-modules tht is J-she, so (Sh J (C); R)Mod is ull subtegory o (P (C); R)Mod. Let (C, J) be smll site nd R she o rings on C. I P is preshe o right R-modules, then the preshe o belin groups P + obtined rom the plus-onstrution is preshe o right R-modules with tion {x S} r = {x r S} Similrly i P is preshe o let R-modules, then P + beomes preshe o let R-modules. The morphism P P + is then morphism o presheves o modules. I φ : P Q is morphism o presheves o modules, then so is φ +. Thereore pplying this onstrution twie gives untors : Mod(P (C); R) Mod(Sh J (C); R) : (P (C); R)Mod (Sh J (C); R)Mod In both ses P P + P ++ gives nturl trnsormtion η : 1 i whih is esily seen to be the unint o n djuntion i. As or sheves o groups, one heks tht preserves ll inite limits, nd thereore Mod(Sh J (C); R) is Girud subtegory o Mod(P (C); R) nd similrly or let modules. Now let (C, J) be smll site nd R preshe o rings on C. I P is preshe o right R- modules, then the preshe o belin groups P + beomes preshe o right R + -modules, where R + is the preshe o rings obtined rom R, vi the tion {x S} {r g g T } = {x h r h h S T } Similrly i P is preshe o let R-modules, then P + beomes preshe o let R-modules. The morphism P P + is then morphism o presheves o belin groups omptible with the morphism o presheves o rings R R +. I φ : P Q is morphism o presheves o modules, then φ + is morphism o presheves o R + -modules. Thereore pplying this onstrution twie, nd denoting the she o rings R ++ by R, we hve untors : Mod(P (C); R) Mod(Sh J (C); R) : (P (C); R)Mod (Sh J (C); R)Mod 8

9 Deinition 7. Let C be smll tegory nd R preshe o rings on C. Deine nother preshe o rings R op : C op nrng by R op (C) = R(C) op, the opposite ring o R(C). We let R op gree with R on morphisms. Clerly R = (R op ) op nd R is ommuttive i nd only i R = R op. I (C, J) is smll site then R is she o rings i nd only i R op is. Lemm 6. Let C be smll tegory nd R preshe o rings on C. Then there is nonil isomorphism o tegories T : (P (C); R)Mod Mod(P (C); R op ) T (A,, v, u, ) = (A,, v, u, τ) where τ : A R R A is the nonil twist. I (C, J) is smll site nd R she o rings on C, then there is nonil isomorphism o tegories T : (Sh J (C); R)Mod Mod(Sh J (C); R op ) T (A,, v, u, ) = (A,, v, u, τ) Proo. It suies to hek the irst sttement. Using the pointwise riterion or the dt to determine preshe o right modules, it is esy to see tht T is well-deined untor, whih is n isomorphism sine the twist is invertible. Deinition 8. Let (C, J) be smll site. A she o Z-grded rings on C is she o rings R together with set o subsheves o belin groups R d, d Z suh tht the morphisms R d R indue n isomorphism o sheves o belin groups d Z R d = R nd or d, e Z, C C nd s R d (C), t R e (C) we must hve st R d+e (C). We lso require tht 1 R 0 (C). We sy R is positive or is she o grded rings i R d = 0 or ll d < 0. Deinition 9. Let (C, J) be smll site nd R she o Z-grded rings. A she o grded let R-modules is she o let R-modules M together with set o subsheves o belin groups {M n } n Z suh tht the morphisms M n M indue n isomorphism o sheves o belin groups n Z M n = M nd suh tht or C C, d, n Z nd s R d (C), m M n (C) we hve s m M n+d (C). A morphism o sheves o grded let R-modules is morphism o sheves o let R-modules M N whih rries M n into N n or ll n Z. This mkes the sheves o grded let R-modules into predditive tegory, denoted (Sh J (C); R)GrMod. Similrly she o grded right R-modules is she o right R-modules M together with set o subsheves o belin groups {M n } n Z suh tht the morphisms M n M indue n isomorphism o sheves o belin groups n Z M n = M nd suh tht or C C, d, n Z nd s R d (C), m M n (C) we hve m s M n+d (C). A morphism o sheves o grded right R-modules is morphism o sheves o right R-modules preserving grde. This mkes the sheves o grded right R-modules into predditive tegory, denoted GrMod(Sh J (C); R). I φ : M N is morphism o sheves o grded R-modules (right or let) then or n Z the mps φ p : M(p) N(p) restrit to give morphisms o belin groups M n (p) N n (p) whih deine morphism o sheves o belin groups φ n : M n N n. Then φ = n φ n in the sense tht φ is the unique morphism o sheves o belin groups mking the ollowing digrm ommute or every n Z M φ N M n φ n N n Let (C, J) be smll site, S she o Z-grded rings nd onsider the oprodut o presheves o belin groups A = n Z S n. For p C we deine ring struture on the belin group A(p) = n S n(p) by {(s n )(t n )} i = s x t y x+y=i 9

10 using the produt in the ring S(p). This mkes A into preshe o rings on C (lerly A(p) is ommuttive i S(p) is). Let Q be the she o rings on C obtined by sheiying A, whih is Z-grded ring with the subsheves given by the imges o S d A Q. The nonil morphism o presheves o rings A S given pointwise by the morphism n S n(p) S(p) indued by the inlusions S n (p) S(p) indues n isomorphism o sheves o Z-grded rings ϕ : Q S. This shows tht she o Z-grded rings is determined by the groups S n (p) or n Z, p C nd the multiplition rules or homogenous elements. For more long this line see (LC,Setion 3). Deinition 10. Let (C, J) be smll site nd R she o Z-grded rings. Then the she o rings R op is she o Z-grded rings with the sme grding, whih we ll the opposite Z-grded ring. Lemm 7. Let (C, J) be smll site nd R she o Z-grded rings. Then there is nonil isomorphism o tegories T : (Sh J (C); R)GrMod GrMod(Sh J (C); R op ) Proo. Given she o grded let R-modules M, let T (M) be the nonil she o right R op - modules s deined bove. The subsheves M n mke T (M) into she o grded right R-modules. The untor T ts s the identity on morphisms, nd it is obvious tht T is n isomorphism. Reerenes [1] B. Mithell, Theory o Ctegories, Ademi Press (1965). 10