Coherence relative to a weak torsion class

Abstract

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n-Coherence Relative to a Hereditary Torsion Theory

“…Since K n−1 is finitely generated, by 24.9 in [10], the maps ϕ 1 , ϕ 2 , and ϕ 3 are monic. Since τ is a hereditary torsion theory and X i is τ-torsion-free, by Proposition 3.2, Chap VI in [9], E(X i ) is τ-torsion-free. And so, by the above proof, ϕ 2 is an isomorphism.…”

“…We recall that a nonempty subclass T of right R-modules is called a weak torsion class [9] if T is closed under homomorphic images and extensions. Following [9], if a class T of right R-modules is a weak torsion class, then a right R-module M is called T-finitely generated (or T-FG for short) if there exists a finitely generated submodule N such that (M/N) ∈ T; a right R-module A is called T-finitely presented (or T-FP for short) if there exists an exact sequence of right R-modules 0 ⟶ K ⟶ F ⟶ A ⟶ 0 with F finitely generated free and K T-finitely generated; a right R-module A is called (T, n)-presented if there exists an exact sequence of right R-modules:…”

“…(3) If Hom(T, F) � 0 for all T ∈ T, then F ∈ F In this case, T is called a torsion class and its objects are called τ-torsion, F is called a torsion-free class, and its objects are called τ-torsion free. From, Proposition 2.1, Chap VI, in [9], a class T of right R-modules is a torsion class for some torsion theory if and only if T is closed under quotient modules, direct sums, and extensions. From Proposition 2.2, Chap VI in [9], a class F of right R-modules is a torsion-free class for some torsion theory if and only if F is closed under submodules, direct products, and extensions.…”

“…From, Proposition 2.1, Chap VI, in [9], a class T of right R-modules is a torsion class for some torsion theory if and only if T is closed under quotient modules, direct sums, and extensions. From Proposition 2.2, Chap VI in [9], a class F of right R-modules is a torsion-free class for some torsion theory if and only if F is closed under submodules, direct products, and extensions. A torsion theory τ � (T, F) is called hereditary if T is closed under submodules.…”

Review on Effect of Climate Change on Forest Ecosystem

“…Warmer temperatures change the timing of life cycle events (phenology), with earlier bud burst, leafing and flowering in trees, although there is much year-to-year and regional variation. It will affect forests also indirectly by increasing the risk of infestation [3]. Effects of climate change on forest ecosystems mainly address the role of the latter with regard to carbon emission and sequestration [4].…”