Exactness of the second sequence of Proposition 2.6(1) #
graded-category.tex: "Applying these arguments to the opposite module
category proves exactness of the second sequence in (1)." In (mod A)ᵒᵖ the
right radical approximations are the left meshes of mod A (biproducts of the
left almost split maps of the summands, with I → I/soc I at injective
summands), and the Krull–Schmidt normal form is the target normal form of
mod A; so the abstract Lemma 2.5 holds in (mod A)ᵒᵖ.
At a nonprojective vertex x the almost split sequence τx → E_x → x is also
the left mesh of τx. Its dual radical approximation gives, for a family
v_i : E_i → W out of the summands:
- (lowest degree)
∑ α_i v_i ∈ J²implies everyv_i ∈ J; - (exactness at the middle) if every
v_i ∈ J^{n+1}and∑ α_i v_i ∈ J^{n+3}, thenv_i ≡ β_i umoduloJ^{n+2}for oneu ∈ J^n(x, W); - (injectivity) if every
β_i u ∈ J^{n+2}, thenu ∈ J^{n+1}.
The Krull–Schmidt target normal form in mod A.
The second map of the left mesh of a module is an epimorphism.
The right radical approximation of a module in (mod A)ᵒᵖ, from its left
mesh.
Instances For
At a nonprojective vertex, the second map of the almost split sequence is a
weak cokernel of the first: the sequence is also the left mesh of τx.
The dual radical approximation of τx in (mod A)ᵒᵖ, from the almost
split sequence ending at the nonprojective vertex x.
Instances For
The map out of the middle term with components v_i.
Instances For
Proposition 2.6(1), second sequence, lowest degree: ∑ α_i v_i ∈ J²
implies v_i ∈ J for every summand.
Proposition 2.6(1), second sequence, exactness at the middle term: if
every v_i ∈ J^{n+1} and ∑ α_i v_i ∈ J^{n+3}, then there is
u ∈ J^n(x, W) with v_i - β_i u ∈ J^{n+2} for every summand.
Proposition 2.6(1), second sequence, injectivity: if every
β_i u ∈ J^{n+2}, then u ∈ J^{n+1}.