The Module Categories Of Trivial Extensions Of K-liner Categories And Their Tilting Subcategories

Tilting theory plays a center role in the research and development of representationtheory of algebras. The trivial extension of algebra is one of the important algebras.Miyachi studied the tilting modules over trivial extensions of algebras, and got anequivalent description of their structures. Recently, R.Mratínez-Villa and M.Ortiz-Morales have extended the classical tilting theory to functor categories.The notion of the k-linear category is a generalization of algebra. The aim of thispaper is to define the notion of the trivial extension of the k-linear category, anddescribe the structure of its module category, thus generalize the classical theory ofthe module category of the trivial extension of algebra. Then we can generalize theresults of Miyachi to functor categories by studying the tilting subcategory of themodule category of the trivial extension of the k-linear category.Let C be a k-linear category and M be a C-bimodule. Firstly, we defineC’=C(?)M to be the trivial extension of the k-linear category, which gives ageneralization of the trivial extension of algebra. Secondly, we describe its modulecategory, and prove that the left module category C-Mod is equivalent to the righttrivial extension category (C-Mod)(M(?)_C-). Since the k-linear triangularmatrix category can be seen as a special case of the trivial extension of a k-linearcategory, we can prove that its module category is equivalent to the category of triplesafter applying the above results. Finally, we study the tilting subcategory of themodule category of the trivial extension of a k-linear category, and give adescription on its structure. After applying the above results to the torsion theoryinduced by the tiling subcategory, we can get the description of its structure,generalizing the classical tilting theory over trivial extensions of algebra.