Exactness of Inverse Limits
نویسنده
چکیده
THEOREM I. Let X be a small category. Then the following assertions are equivalent: (1) The inverse limit proj limx: AB-^AB is exact (2) For every abelian category SÏ with exact direct products y the inverse limit proj lim* : %—»3I is exact. (3) Every connected component Y of X contains an object y together with an endomorphism eÇz Y (y, y) such that the following properties are satisfied: (i) y is a smallest object of F, i.e., for any object zÇzY there is a morphism y—>z. (ii) e equalizes any two morphisms with the same codomain and domain y, i.e., any diagram y-£±ylX is commutative.
منابع مشابه
Exact triangles in Seiberg - Witten - Floer theory . Part III : proof of exactness
6 Exactness in the middle term 35 6.1 The moduli space on V × R . . . . . . . . . . . . . . . . . . . 36 6.2 Admissible elements in M̂V×R(a∞) . . . . . . . . . . . . . . . 41 6.3 Assembling the geometric limits . . . . . . . . . . . . . . . . . 43 6.4 The relation w0 ∗ ◦ w 1 ∗ = 0 . . . . . . . . . . . . . . . . . . . . . 51 6.5 The inclusion Ker(w0 ∗) ⊂ Im(w 1 ∗) . . . . . . . . . . . . . . . . 56
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