Abstract
Given a noether algebra with a noncommutative resolution, a general construction of new noncommutative resolutions is given. As an application, it is proved that any finite length module over a regular local or polynomial ring gives rise, via suitable syzygies, to a noncommutative resolution.
Highlights
The focus of this article is on constructing endomorphism rings with finite global dimension
The statement of Theorem 1 is inspired by a recollement type inequality (1.1) that yields that the finiteness of global dimensions of eAe and A/(e) implies that of A, provided pdA(A/(e)) is finite
The hypotheses in Theorem 1 enable us to apply this fact to A = EndR(M ⊕ ΩcX) and the idempotent e ∈ A corresponding to the direct summand M
Summary
The focus of this article is on constructing endomorphism rings with finite global dimension. For a noetherian ring R which is not necessarily commutative, and a finitely generated faithful R-module M , the ring EndR(M ) is a noncommutative resolution (abbreviated to NCR) if its global dimension is finite; see [6]. If M is a d-torsionfree generator giving an NCR, and gldim EndR(X) is finite, for any integer 0 c < min{d, gradeR X}, the following statements hold.
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