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For a quiver Q, a k-algebra A, and a full subcategory X of A-mod, the monomorphism category Mon(Q, X ) is introduced. The main result says that if T is an A-module such that there is an exact sequenceup to multiplicities of indecomposable direct summands, such that Mon(Q, ⊥ T ) = ⊥ (kQ ⊗ k T ).As applications, the category of the Gorenstein-projective (kQ ⊗ k A)-modules is characterized as Mon(Q, GP(A)) if A is Gorenstein; the contravariantly finiteness of Mon(Q, X ) can be described; and a sufficient and necessary condition for Mon(Q, A) being of finite type is given.

In this article, we generalize the concept of torsion pairs and study its structure. As a trial of obtaining all torsion pairs, we decompose torsion pairs by projective modules and injective modules. Then we calculate torsion pairs on the algebra KA n and tub categories. At last we try to find all torsion pairs on the module categories of finite dimensional hereditary algebras.

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