MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question

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

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) .

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.