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In noncommutative projective geometry, there is a counterpart of dualizing complex in commutative world. It seems to me that they are called either a balanced dualizing complex or rigid dualizing complex. I am aware that they are different objects but cannot really understand how they are different. I would appreciate it if someone kindly explain to me the difference.

P.S. I am reading this paper by M. van den Bergh

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1 Answer 1

The difference is mainly that "balanced" pertains only to connected graded noncommutative rings, whereas "rigid" makes sense for any noncommutative ring. (Always over a base field.)

This is explained in the lecture notes:

Dualizing Complexes over Noncommutative Rings
http://www.math.bgu.ac.il/~amyekut/lectures/dc-rings-1-notes.pdf.

See Section 2 there, and especially Definitions 2.4 and 2.5.

There is also a version of "balanced" for complete local noncommutative rings, in the paper:

Dualizing Complexes over Noncommutative Local Rings, by Wu and Zhang
Journal of Algebra 239, 513–548 (2001) .

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