Monomorphism Categories

The research of monomorphism categories dates back to1930s, monomorphism categories have gained great attention and stimulated people’s research enthusiasm in the following decades. There is plenty of evidence that monomorphism categories have deep relationship with other branches of mathematics, the quiver involved are from type of A2to type of An and to finite acyclic ones, our understanding of monomorphism cat-egories becomes gradually profound. In this thesis we mainly research monomorphism categories of arbitrary finite acyclic quivers.Our main results are the following.In Chapter3, we generalize the main results of [51] of quiver of An type to arbitrary finite acyclic quivers. We get the following:(1) we give a reciprocity of the monomorphism operator Mon(Q,-) and the left perpendicular operator⊥.(2) we pove that Mon(Q,A) is functorially finite in Rep(Q,A), and Mon(Q,A) has Auslander-Reiten sequences..(3) We give a sufficient and necessary condition for Mon(Q,A) being of finite type.In Chapter4, we prove that monomorphism categories have enough injective ob-jects, and we give the concrete form of the indecomposable injective objects.In Chapter5, we generalize the stable monomorphism category of A2in [15] to the stable monomorphism categories of arbitrary finite acyclic quivers. We obtain the following:(1) We prove that the stable category of a self-injective algebra can be embedded in the stable monomorphism categories as a triangulated subcategory in many ways.(2) We prove that a tilting object in the stable category of a self-injective algebra induces a natural tilting object in the stable monomorphism categories.(3) We realize the stable monomorphism categories to singularity categories. In Chapter6, we research the categories of short exact sequences with projective middle term. We have the following:(1) we prove that the category of short exact sequences with projective middle terms is contravariantly finite in the category of short exact sequences, it is functorially finite provided the involved algebra is self-injective, the it has Auslander-Reiten sequences.(2) we prove that the homotopic category of the category of short exact sequences with projective middle terms is equal to the stable category of the category of short exact sequences with projective middle terms.(3) We prove that the stable category is equivalent to the stable category of the category of short exact sequences with projective middle terms as triangulated categories provided the involved algebra is self-injective.References[15] X. W. Chen, The stable monomorphism category of Probenius category, Math. Research Letters,2011,18(1):141-150.[51] P. Zhang, Monomorphism categories, cotilting theory, and Gorenstein-projective modules, J. Algebra,2011,339(1):181-202.