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Matrix factorization

Dyckeroff [ Dyc11 ] によると matrix factorizations Eisenbud [ Eis80 ] 研究 したらしい としては , Yoshino [ Yos90 ] がある Khovanov Rozansky [ KR08a ] , Eisenbud isolated hypersurface singularity maximal Cohen-Macaulay については , Eisenbud [ Eis95 ] めて いる

  • k regular local k -algebra ( R, 𝔪 ) とその superpotential w する matrix factorization dg category MF( R,w )
  • MF( R,w ) homotopy category triangulated category

String theory との Kontsevich により された これについては , Math Overflow のこの する Kevin Lin をみるのがよい Link 豊富 である Dyckerhoff Murfet [ DM12 ] 最後 には , Lurie extended topological quantum field theory との についても いてある

Knot homology との については , Khovanov Rozansky [ KR08a KR08b ] よるのだろうか この にはたくさんの がある

Carquville Runkel [ CR10 ] , matrix factorization tensor product, つまり matrix factorization category monoidal structure えている アイデア bimodule 使 うことであり , bimodule matrix factorization category monoidal structure している もあるようで ある

Matrix factorization Knörrer periodicity [ Knö87 ] topological K -theory Bott periodicity には があるらしい Michael Brown [ Bro ] して いる

Brown, Miller, Thompson, Walker [ BMTW ] , matrix factorization Grothendieck Adams operation している

としては , Positselski [ Pos11 EP15 ] , そしてその 仕事 づいた Ballard, Deliu, Favero, Isik, Katzarkov [ BDF + 16 ] されている factorization category がある

  • factorization category

References

[BDF + 16]     Matthew Ballard, Dragos Deliu, David Favero, M. Umut Isik, and Ludmil Katzarkov. Resolutions in factorization categories. Adv. Math. , 295:195–249, 2016, arXiv:1212.3264 .

[BMTW]     Michael K. Brown, Claudia Miller, Peder Thompson, and Mark E. Walker. Adams Operations on Matrix Factorizations, arXiv:1610.09907 .

[Bro]     Michael K. Brown. Knörrer Periodicity and Bott Periodicity, arXiv:1507.03329 .

[CR10]     Nils Carqueville and Ingo Runkel. On the monoidal structure of matrix bi-factorizations. J. Phys. A , 43(27):275401, 33, 2010, arXiv:0909.4381 .

[DM12]     Tobias Dyckerhoff and Daniel Murfet. The Kapustin-Li formula revisited. Adv. Math. , 231(3-4):1858–1885, 2012, arXiv:1004.0687 .

[Dyc11]     Tobias Dyckerhoff. Compact generators in categories of matrix factorizations. Duke Math. J. , 159(2):223–274, 2011, arXiv:0904.4713 .

[Eis80]     David Eisenbud. Homological algebra on a complete intersection, with an application to group representations. Trans. Amer. Math. Soc. , 260(1):35–64, 1980, http://dx.doi.org/10.2307/1999875 .

[Eis95]     David Eisenbud. Commutative algebra , volume 150 of Graduate Texts in Mathematics . Springer-Verlag, New York, 1995. With a view toward algebraic geometry.

[EP15]     Alexander I. Efimov and Leonid Positselski. Coherent analogues of matrix factorizations and relative singularity categories. Algebra Number Theory , 9(5):1159–1292, 2015, arXiv:1102.0261 .

[Knö87]     Horst Knörrer. Cohen-Macaulay modules on hypersurface singularities. I. Invent. Math. , 88(1):153–164, 1987, http://dx.doi.org/10.1007/BF01405095 .

[KR08a]     Mikhail Khovanov and Lev Rozansky. Matrix factorizations and link homology. Fund. Math. , 199(1):1–91, 2008, arXiv:math/0401268 .

[KR08b]     Mikhail Khovanov and Lev Rozansky. Matrix factorizations and link homology. II. Geom. Topol. , 12(3):1387–1425, 2008, arXiv:math/0505056 .

[Pos11]     Leonid Positselski. Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence. Mem. Amer. Math. Soc. , 212(996):vi+133, 2011, arXiv:0905.2621 .

[Yos90]     Yuji Yoshino. Cohen-Macaulay modules over Cohen-Macaulay rings , volume 146 of London Mathematical Society Lecture Note Series . Cambridge University Press, Cambridge, 1990, http://dx.doi.org/10.1017/CBO9780511600685 .