Mutations of noncommutative crepant resolutions: Quasi-symmetric representation and GIT quotient

Table of Links

• Abstract and Intro
• Exchanges and Mutations of modifying modules
• Quasi-symmetric representation and GIT quotient
• Main results
• Applications to Calabi-Yau complete intersections
• Appendix A. Matrix factorizations
• Appendix B. List of Notation
• References

3. Quasi-symmetric representation and GIT quotient

3.1. Quasi-symmetric representations and magic windows. This section recalls fundamental properties of derived categories of GIT quotients arising from quasi-symmetric representations, which are established in [HSa] and [SV1]. We freely use notation from Section 1.6.

and then it associates the GIT quotient stack [Xss(ℓ)/G].

Proposition 3.10 ([HSa, Proposition 6.2]). There is an equivalence of groupoids

Proposition 3.13 ([HSa, Proposition 6.5]). There is an equivalence

extending the equivalence in Proposition 3.10.

(3) This follows from (2).

The following is elementary, but we give a proof for the convenience of the reader

Proof. If W is trivial, the results are obvious. Thus, assume that W ̸= 1

The following result proves that this map is bijective.