Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

More documents

Recommendations

Info

252Proposition A.12 ([ELGO2, Proposition <strong>1.</strong>2]). Let C be an A-coring. Then thefollowing are equivalent(a) A C is flat.(b) (Mod-A) C is an abelian category and the forgetful functor U : (Mod-A) C →Mod-A is left exact (and hence exact).(c) (Mod-A) C is a Grothendieck category and the forgetful functor U : (Mod-A) C →Mod-A is left exact (and hence exact).Proof. By Lemma A.10, the category (Mod-A) C has coproducts and cokernels.(a) ⇒ (c) By Lemma A.11 has kernels. Consider the following diagram( ) Ker (f) , ρKer(f) k() M, ρMf ( ) N, ρNχ ( ) Coker (f) , ρCoker(f)χ ′ ρ k′( ) ( )Coker (k) , ρCoker(k) Ker (χ) , ρKer(χ).¯fin (Mod-A) C . Then we get the diagramKer (f)k M χ ′ Coker (k)fρ¯f Nk ′ Ker (χ) .χ Coker (f)in Mod-A. Since Mod-A is preabelian, f is an isomorphism in Mod-A and hence alsoin (Mod-A) C . Thus also the category (Mod-A) C is preabelian and moreover abelian(there exist products of every finite family of objects in the category). Moreover,by Lemma A.10 and Lemma A.11 U is left exact. Further, the direct limits areexacts for module categories and thus also for (Mod-A) C . We now have to find agenerator for (Mod-A) C . Let ( M, ρ M) ∈ (Mod-A) C and let p : A (M) → M is theusual epimorphism of right A-modules. Let us consider the epimorphism l given bythe following compositel : C (M) −→ A (M) ⊗ A C p⊗ AC−→ M ⊗ A C −→ 0where the first arrow is the canonical isomorphism and the second one is the usualepimorphism so that l ( (c m ) m∈M)=∑m∈M m ⊗ A c m where c m are almost all zero.Then we have the following diagramρ0Pgρ M0MC (M) l M ⊗ A C 00
253where (P, ρ, g) is the pullback of ( l, ρ M) . Recall that P is the submodule ofdefined by settingC (M) × MP = { (x, m) ∈ C (M) × M | l (x) = ρ M (m) }and ρ : P → C (M) is defined by setting ρ ((x, m)) = x while g : P → M bysetting g ((x, m)) = m. Since ρ M is mono, ρ is also mono (thus ρ (P ) = H is asubcomodule of C (M) ) and since l is epi, g is epi. Denote by h M F : C(F ) → C (M) thecanonical inclusion for any F ⊆ M. Let m ∈ M, then there exist n ∈ N and F m ={y 1 , y 2 , . . . , y n } ⊆ M such that ρ M (m) = ∑ ))y∈F my ⊗ A c y = l(h M F m((c y ) y∈Fm.Then, for every m ∈ M, there exists z ∈ P such that m = g (z) = g ((ρ (z) , m))where ρ (z) = h M F m((c y ) y∈Fm)∈ h M F m(C(F m) ) ⊆ C (M) . Thus, for every m ∈ M, thereexists x m = (c y ) y∈Fm∈ such that m = g (( h M F m(x m ) , m )) . Then we have definedthe following homomorphismsuch thatν xm : x m A −→ Mx m ↦→ g (( h M F m(x m ) , m )) .m = ν xm (x m ) .Since x m A ⊆ C (Fm) , we deduce that the subcomodules of C (N) form a set of generatorsfor (Mod-A) C ⊕i.e. H is a generator for (Mod-A) C .H⊆C (N)(c) ⇒ (b) Obvious.(b) ⇒ (a) By Example 4.3 and Definition 4.10 F : Mod-A → (Mod-A) C is a rightadjoint of U and then F is left exact. Then using the hypothesis that U is left exact,we deduce that U ◦ F : Mod-A → Mod-A is also left exact, i.e. A C is flat. □Definition A.13. Let A be a Grothendieck category. An object A ∈ A is calledfinitely generated if, for every direct family of subobjects {A i } i∈Iof A such thatA = ∑ A i , there exists an index i 0 ∈ I such that A = A i0 .i∈IProposition A.14. Let A be a Grothendieck category. An object A ∈ A is finitelygenerated if and only if, for every family of subobjects {A i } i∈Iof A such that ∑ i∈IA i =∑A, there exists a finite number of subobjects A 1 , . . . , A n such that A = n A i .Proof. (⇐) Let {A i } i∈Ibe a direct family of subobjects of A closed under sumsand such that A = ∑ A i . By hypothesis there exists a finite number of subobjectsi∈I∑A 1 , . . . , A n such that A = n A i . Since the family is direct and closed under sums,i∈I∑there exists an index i 0 ∈ I such that A = n A i ⊆ A i0 . Then A is finitely generated.i∈Ii∈I
Page 1 and 2:
Contents1.
Page 3 and 4:
linearity and compatibility conditi
Page 5 and 6:
5and since g ◦ f is an epimorphis
Page 7 and 8:
Proof. Clearly (qP )◦(αP ) = (qP
Page 9 and 10:
Lemma 2.13 ([BM, L
Page 11 and 12:
i.e. Hom B (Y, iX) equalizes Hom B
Page 13 and 14:
13such thatd 0 ◦ v = Id Yd 1 ◦
Page 15 and 16:
f ↦→ Rfis bijective for every X
Page 17 and 18:
Remark 3.10. Let A = (A, m A , u A
Page 19 and 20:
19and fromµ A P ◦ ( µ A P A ) =
Page 21 and 22:
and thusk ◦ (u A QZ) ◦ (Qz) = h
Page 23 and 24:
and since A preserves equalizers, A
Page 25 and 26:
Conversely, let Φ be a functorial
Page 27 and 28:
Proof. Apply Proposition 3.24 to th
Page 29 and 30:
Since Q is a left A-module functor,
Page 31 and 32:
(Q BB F, p QB F ) = Coequ Fun(µBQ
Page 33 and 34:
Theorem 3.37. Let B = (B, m B , u B
Page 35 and 36:
where A UG B F : B → A is such th
Page 37 and 38:
Proposition 3.44. Let A = (A, m A ,
Page 39 and 40:
Note that, since f and g are A-bili
Page 41 and 42:
Proposition 3.54. Let (L, R) be an
Page 43 and 44:
Corollary 3.58. Let (L, R) be an ad
Page 45 and 46:
Definition 4.2. A
Page 47 and 48:
Proposition 4.13. Let C = ( C, ∆
Page 49 and 50:
Then we have(P Cx) ◦ ( ρ C P X )
Page 51 and 52:
and since C preserves coequalizers,
Page 53 and 54:
Proof. Apply Corollary 4.24 to the
Page 55 and 56:
Let( (CQ ) ()D, ι Q) C = Equ Fun
Page 58 and 59:
58F D right D-comodule functors Q :
Page 60 and 61:
60prove that C ν D : C F D → (C
Page 62 and 63:
624.2. The compari
Page 64 and 65:
64and[(Ω ◦ Γ) (ϕ)] (Y ) = (LY,
Page 66 and 67:
66for every ( X, C ρ X)∈ C A, th
Page 68 and 69:
68i.e.(44) (d ϕ K ϕ Y ) ◦ (̂η
Page 70 and 71:
70In particular(49) d ϕ(CX, ∆ C
Page 72 and 73:
72We have to prove that (LD ϕ , Ld
Page 74 and 75:
74we have that Ld ϕ K ϕ Y is mono
Page 76 and 77:
and since d is mono we get that(ε
Page 78 and 79:
78Corollary 4.63 (Beck’s Precise
Page 80 and 81:
80We compute(LRɛLY ′ ) ◦ ( LR
Page 82 and 83:
82Proof. First of all we prove that
Page 84 and 85:
84i.e. Aα is a functorial morphism
Page 86 and 87:
86Then we haveA µ CCX ◦ ( A∆ C
Page 88 and 89:
884.23) is a functor Ã : C A → C
Page 90 and 91:
90Let θ l = ( σ B P Q ) ◦ (P τ
Page 92 and 93:
925)σ A = ( ε C A ) ◦ ( Cσ A)
Page 94 and 95:
94(ii) the functorial morphism can
Page 96 and 97:
96defΦ= ( QP A µ Q)◦(QP σ A Q
Page 98 and 99:
98AU A can AA F = can AA F = ( CσA
Page 100 and 101:
100Similarly, one can prove the sta
Page 102 and 103:
102(b) A comonad C = ( C, ∆ C ,
Page 104 and 105:
104We calculateso that we getx ◦
Page 106 and 107:
106There exist functorial morphisms
Page 108 and 109:
108andsatisfying(B, y) = Coequ Fun(
Page 110 and 111:
1104) With notations of Theorem 6.2
Page 112 and 113:
112Then ν : Y → D is the unique
Page 114 and 115:
114= A µ Q ◦ ( Aε C Q ) ◦ (AC
Page 116 and 117:
116= ( Aε C Q ) ◦ ( cocan1 −1
Page 118 and 119:
118so that we getχ= (Cx) ◦ (C ρ
Page 120 and 121:
120We want to prove that Γ is an o
Page 122 and 123:
122and since Dε D is an epimorphis
Page 124 and 125:
124χ= (Cχ) ◦ (C ρ Q P Q ) ◦
Page 126 and 127:
126Now, since cocan 1 : AC → QP i
Page 128 and 129:
1287. Herds and Coherds7.1.
Page 130 and 131:
130◦ ( σ A QQQ ) ◦ (A µ Q P Q
Page 132 and 133:
132= µ B Q ◦ (A µ Q B ) ◦ ( A
Page 134 and 135:
134Assume now that there is another
Page 136 and 137:
136and hence we get(160) x ◦ (χP
Page 138 and 139:
138Proposition 7.7. In the setting
Page 140 and 141:
140We calculateA µ Q ◦ ( σ A Q
Page 142 and 143:
142x=and=δ C=(l= QlQ ̂QQ)◦ (QP
Page 144 and 145:
144(◦ ρ D ̂QQ)Q◦ (QlQ) ◦ (Q
Page 146 and 147:
146given byWe computeσ B = m B ◦
Page 148 and 149:
148andy= ′m B ◦ (ν B B) ◦ (y
Page 150 and 151:
150Now we compute(hQ) ◦ ( Qχ )
Page 152 and 153:
152Thus we obtainσ B ◦ ( ) (P µ
Page 154 and 155:
154Thus hQ is an isomorphism with i
Page 156 and 157:
156) ( )l=(pb Q AQ B ◦ ̂QA µ QB
Page 158 and 159:
158In fact we haveTherefore we dedu
Page 160 and 161:
160χ= h 1 ◦ (P xQ B ) ◦ (P QP
Page 162 and 163:
162so that we obtain:(190)We comput
Page 164 and 165:
164(194)=) )(p QB ̂QA ◦(Qpb Q◦
Page 166 and 167:
166= Ξ ◦ (A A U A λ) ◦ (xx A
Page 168 and 169:
168)(155)= k 2 ◦(Qpb Q◦ (Ql A U
Page 170 and 171:
170) ) (χ= ρ ◦(p QB ̂QA ◦(Qp
Page 172 and 173:
172Theorem 8.13. Let A and B be cat
Page 174 and 175:
174l = eC ρ L : L = − ⊗ B A
Page 176 and 177:
and[µBQ ◦ ( Qσ B)] (− ⊗ T x
Page 178 and 179:
178so that− ⊗ R 1 A ⊗ R c = (
Page 180 and 181:
180− ⊗ T x ⊗ R 1 A ⊗ A f
Page 182 and 183:
182(208)(209)(210)(211)(h1 ) 0 ⊗
Page 184 and 185:
184= abd 0 ⊗ d 1 1 ⊗ d 2 1b⊗d
Page 186 and 187:
186so that h 1 ⊗ h 2 ⊗ a ∈ A
Page 188 and 189:
188= 〈( h (1) y (1))εH ( h (2) y
Page 190 and 191:
190H C is faithfully coflat. Assume
Page 192 and 193:
192=(Qε C H C) ( ∑ )−□ C k i
Page 194 and 195:
194Following Theorem 6.29, we now c
Page 196 and 197:
196)û E(ε C H C (h) = û E (π (h
Page 198 and 199:
198Letandα l = (ϕ ⊗ H) ( (x ⊗
Page 200 and 201:
200This map is well-defined, in fac
Page 202 and 203: 202We now have to prove that this m
Page 204 and 205: 2041) − ⊗ B Σ A preserves the
Page 206 and 207: 206functorial isomorphism. In parti
Page 208 and 209: 208coaction ρ C Σ : Σ → Σ ⊗
Page 210 and 211: 210Now, we consider a particular ca
Page 212 and 213: 212Definition 9.27. Let k be a comm
Page 214 and 215: 214∆coass= a ⊗ c (1) ⊗ A 1 A
Page 216 and 217: 216Definition 9.32.</strong
Page 218 and 219: 218Let us compute, for every d ∈
Page 220 and 221: 220• 2-cells: monad functor trans
Page 222 and 223: 222We now want to prove that ρ Q·
Page 224 and 225: 224Proof. Let us consider the follo
Page 226 and 227: 226and since p Q•B Q ′ ,Q ′
Page 228 and 229: 228(241)= (1 Q • B l Q ′) ζ Q,
Page 230 and 231: 230On the other hand, we can first
Page 232 and 233: 232so that we define the map φ F (
Page 234 and 235: 234Since we have(B • B (Q · A) ,
Page 236 and 237: 2362-cells. This means that a comon
Page 238 and 239: 238defined by settingu Q·A = ( u (
Page 240 and 241: 240the unique A-bimodule morphism s
Page 242 and 243: 242Let F be a finite subset of Hom
Page 244 and 245: 244Lemma A.4. Let A be an abelian c
Page 246 and 247: 246We haveT (ζ) ◦ ξ ◦ T H (p)
Page 248 and 249: 248where k : Ker (Coker (f ◦ p))
Page 250 and 251: 250be the codiagonal map of the ρ
Page 254 and 255: 254(⇒) Let {A i } i∈Ibe a famil
Page 256 and 257: 256We will prove that h : ∐ B i
Page 258 and 259: 258Proposition A.19. Let (T, H) be
Page 260 and 261: 260Since P is finite Hom A (P, P )
Page 262 and 263: 262andP (J ′ )e f ′−→ P (I
Page 264 and 265: 264hence there exists a unique morp
Page 266: 266[RW] R. Rosebrugh, R.J. Wood, Di
show all

Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?