Cohomology of Schematic Algebras

Author: van Oystaeyen F.   Willaert L.  

Publisher: Academic Press

ISSN: 0021-8693

Source: Journal of Algebra, Vol.185, Iss.1, 1996-10, pp. : 74-84

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Abstract

Let R be a positively graded Noetherian k -algebra and let Proj R =( R , kappa + )- gr be the quotient category of the category of graded R -modules R - gr modulo those of finite length. If R is schematic (cf. [F. Van Oystaeyen and L. Willaert, J. Pure Appl. Algebra 104 (1995), 109-122]) and has finite global dimension, we show that the cohomology groups of R (as defined in [M. Artin and J. J. Zhang, Adv. in Math. 109 (1994), 228-287]) are finite dimensional. Furthermore, we introduce Cech cohomology groups for schematic algebras and prove that they coincide with the usual graded cohomology groups. These generalized Cech cohomology groups are still manageable; we illustrate this with an example.