Cohomology of Schematic Algebras Original Research Article

LetRbe a positively graded Noetheriank-algebra and let Proj R=(R, κ+)−grbe the quotient category of the category of gradedR-modulesR−grmodulo those of finite length. IfRis schematic (cf. [F. Van Oystaeyen and L. Willaert,J. Pure Appl. Algebra104(1995), 109–122]) and has finite global dimension, we show that the cohomology groups ofR(as defined in [M. Artin and J. J. Zhang,Adv. in Math.109(1994), 228–287]) are finite dimensional. Furthermore, we introduce imageech cohomology groups for schematic algebras and prove that they coincide with the usual graded cohomology groups. These generalized imageech cohomology groups are still manageable; we illustrate this with an example.