ABSTRACT
The duality theorem of Greenlees and May relating local cohomology with support on an ideal I and the left derived functors of I-adic completion [GM92] holds for rather general ideals in commutative rings. Here, simple formulas are provided for both local cohomology and derived functors of https://www.w3.org/1998/Math/MathML"> Z n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429175633/5bc16269-a792-4117-a208-ca03c55a3a8d/content/eq7161.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> -graded completion, when I is a monomial ideal in the https://www.w3.org/1998/Math/MathML"> Z n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429175633/5bc16269-a792-4117-a208-ca03c55a3a8d/content/eq7162.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> -graded polynomial ring https://www.w3.org/1998/Math/MathML"> k x 1 , … , x n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429175633/5bc16269-a792-4117-a208-ca03c55a3a8d/content/eq7163.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Greenlees-May duality for this case is a consequence. A key construction is the combinatorially defined Čech hull operation on https://www.w3.org/1998/Math/MathML"> Z n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429175633/5bc16269-a792-4117-a208-ca03c55a3a8d/content/eq7164.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> -graded modules [Mil98, Mil00, Yan00]. A simple self-contained proof of GM duality in the derived category is presented for arbitrarily graded noetherian rings, using methods motivated by the Čech hull.
