process

Torsion classes of finite type and spectra 403Note that A 0 is a commutative ring with 1 2 A 0 , that all summands M j are A 0 -modules, and that M D L j 2Z M j is a direct sum decomposition of M as an A 0 -module.Let A be a graded ring. The category of graded A-modules, denoted by Gr A, hasas objects the graded A-modules. A morphism of graded A-modules f W M ! N isan A-module homomorphism satisfying f.M j / N j for all j 2 Z. An A-modulehomomorphism which is a morphism in Gr A will be called homogeneous.Let M be a graded A-module and let N be a submodule of M . Say that N is agraded submodule if it is a graded module such that the inclusion map is a morphismin Gr A. The graded submodules of A are called graded ideals. Ifd is an integer thetail M d is the graded submodule of M having the same homogeneous components.M d / j as M in degrees j d and zero for j < d. We also denote the ideal A 1by A C .For n 2 Z, GrA comes equipped with a shift functor M 7! M.n/ where M.n/ isdefined by M.n/ j D M nCj . Then Gr A is a Grothendieck category with generatingfamily fA.n/g n2Z . The tensor product for the category of all A-modules induces atensor product on Gr A: given two graded A-modules M; N and homogeneous elementsx 2 M i ;y 2 N j , set deg.x ˝ y/ WD i C j . We define the homomorphism A-moduleHom A .M; N / to be the graded A-module which is, in dimension n 2 Z, the groupHom A .M; N / n of graded A-module homomorphisms of degree n, i.e.,Hom A .M; N / n D Gr A.M; N.n//:We say that a graded A-module M is finitely generated if it is a quotient of a freegraded module of finite rank L nsD1 A.d s/ where d 1 ;:::;d s 2 Z. Say that M is finitelypresented if there is an exact sequencemMA.c t / !tD1nMA.d s / ! M ! 0:sD1The full subcategory of finitely presented graded modules will be denoted by gr A. Notethat any graded A-module is a direct limit of finitely presented graded A-modules, andtherefore Gr A is a locally finitely presented Grothendieck category.Let E be any indecomposable injective graded A-module (we remind the reader thatthe corresponding ungraded module, L n E n, need not be injective in the category ofungraded A-modules). Set P D P.E/to be the sum of the annihilator ideals ann A .x/of non-zero homogeneous elements x 2 E. Observe that each ideal ann A .x/ is homogeneous.Since E is uniform the set of annihilator ideals of non-zero homogeneouselements of E is upwards closed so the only issue is whether the sum, P.E/, of themall is itself one of these annihilator ideals.Given a prime homogeneous ideal P , we use the notation E P to denote the injectivehull, E.A=P/, ofA=P . Notice that E P is indecomposable. We also denote the set ofisomorphism classes of indecomposable injective graded A-modules by Inj A.