Authors:
(1) Wahei Hara;
(2) Yuki Hirano. 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 5. Applications to Calabi-Yau complete intersections Therefore, (5.A) and (5.B) gives an equivalence Proposition 5.1. The restriction of (5.F to a magic window and the functor (5.G) are equivalences. Since the bottom functor is an equivalence by Theorem A.5, so is (5.G). for derived factorization categories is an equivalence by Theorem A.5. The following shows that the equivalences of magic windows generating the group action (5.D) correspond to mutation functors between noncommutative matrix factorizations. Proof. We only show that the left square commutes, since the commutativity of the right one follows from a similar argument. Consider the following diagram commutes, where the vertical equivalences are the compositions of (5.C) and (5.H). Lemma 5.5. There is an isomorphism where the first isomorphism follows from Lemma A.6. This finishes the proof. The following is a generalization of [KO, Theorem 8.5], which we prove by a similar argument as in loc. cit. Lemma 5.6. The following diagram commutes. Thus it is enough to show that there is a natural isomorphism By Lemma 5.6, there is an isomorphism Proof of Corollary 5.3. For simplicity, write Therefore, the assertion follows from Theorem 5.2. This paper is available on arxiv under CC0 1.0 DEED license. [1] Although [HSh] only discusses complexes, there are similar functors and semi-orthogonal decompositions for matrix factorizations by [BFK2], and so a similar argument as in [HSh] works in our setting. Authors: (1) Wahei Hara; (2) Yuki Hirano. Authors: Authors: (1) Wahei Hara; (2) Yuki Hirano. 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 Abstract and Intro Abstract and Intro Exchanges and Mutations of modifying modules Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Quasi-symmetric representation and GIT quotient Main results Main results Applications to Calabi-Yau complete intersections Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix A. Matrix factorizations Appendix B. List of Notation Appendix B. List of Notation References References 5. Applications to Calabi-Yau complete intersections Therefore, (5.A) and (5.B) gives an equivalence Proposition 5.1. The restriction of (5.F Proposition 5.1. The restriction of (5.F to a magic window and the functor (5.G) are equivalences. are equivalences. Since the bottom functor is an equivalence by Theorem A.5, so is (5.G). for derived factorization categories is an equivalence by Theorem A.5. The following shows that the equivalences of magic windows generating the group action (5.D) correspond to mutation functors between noncommutative matrix factorizations. Proof . We only show that the left square commutes, since the commutativity of the right one follows from a similar argument. Consider the following diagram Proof commutes, where the vertical equivalences are the compositions of (5.C) and (5.H). commutes, where the vertical equivalences are the compositions of and Lemma 5.5. There is an isomorphism Lemma 5.5. There is an isomorphism where the first isomorphism follows from Lemma A.6. This finishes the proof. The following is a generalization of [KO, Theorem 8.5], which we prove by a similar argument as in loc. cit. Lemma 5.6. The following diagram commutes. Lemma 5.6. The following diagram commutes. Thus it is enough to show that there is a natural isomorphism By Lemma 5.6, there is an isomorphism Proof of Corollary 5.3. For simplicity, write Proof of Corollary 5.3. Therefore, the assertion follows from Theorem 5.2. This paper is available on arxiv under CC0 1.0 DEED license. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv [1] Although [HSh] only discusses complexes, there are similar functors and semi-orthogonal decompositions for matrix factorizations by [BFK2], and so a similar argument as in [HSh] works in our setting.

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Mutations of noncommutative crepant resolutions: Applications to Calabi-Yau complete intersections

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Acknowledgements, Declarations, Data Availability Statement, and References

Mutations of noncommutative crepant resolutions: Exchanges & Mutations of modifying modules

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

Mutations of noncommutative crepant resolutions: Appendix A. Matrix factorizations

Mutations of noncommutative crepant resolutions: Abstract and Intro

Acknowledgements, Declarations, Data Availability Statement, and References

Mutations of noncommutative crepant resolutions: Exchanges & Mutations of modifying modules

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

Mutations of noncommutative crepant resolutions: Appendix A. Matrix factorizations

Mutations of noncommutative crepant resolutions: Abstract and Intro

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