Definition 111.26.3. A graded $A$-algebra is a graded $A$-module $B = \bigoplus _{n \geq 0} B_ n$ together with an $A$-bilinear map
\[ B \times B \longrightarrow B, \ (b, b') \longmapsto bb' \]
that turns $B$ into an $A$-algebra so that $B_ n \cdot B_ m \subset B_{n + m}$. Finally, a graded module $M$ over a graded $A$-algebra $B$ is given by a graded $A$-module $M$ together with a (compatible) $B$-module structure such that $B_ n \cdot M_ d \subset M_{n + d}$. Now you can define homomorphisms of graded modules/rings, graded submodules, graded ideals, exact sequences of graded modules, etc, etc.
Comments (0)