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2016 | OriginalPaper | Chapter

1. Introduction and Survey

Authors : David Eisenbud, Irena Peeva

Published in: Minimal Free Resolutions over Complete Intersections

Publisher: Springer International Publishing

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Abstract

We begin the chapter with some history of the results that form the background of this book. We then define higher matrix factorizations, our main focus. While classical matrix factorizations are factorizations of a single element, higher matrix factorizations deal directly with sequences of elements. In Sect. 1.3, we outline our main results. Throughout the book, we use the notation introduced in Sect. 1.4.

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Literature
1.
go back to reference P. Aspinwall, Some applications of commutative algebra to string theory, in Commutative Algebra, ed. by I. Peeva (Springer, Berlin, 2013), pp. 25–56CrossRef P. Aspinwall, Some applications of commutative algebra to string theory, in Commutative Algebra, ed. by I. Peeva (Springer, Berlin, 2013), pp. 25–56CrossRef
5.
go back to reference L. Avramov, Infinite free resolutions, in Six Lectures on Commutative Algebra. Modern Birkhauser Classics (Birkhäuser, Basel, 2010), pp. 1–118 L. Avramov, Infinite free resolutions, in Six Lectures on Commutative Algebra. Modern Birkhauser Classics (Birkhäuser, Basel, 2010), pp. 1–118
6.
go back to reference L. Avramov, R.-O. Buchweitz, Homological algebra modulo a regular sequence with special attention to codimension two. J. Algebra 230, 24–67 (2000)MathSciNetCrossRefMATH L. Avramov, R.-O. Buchweitz, Homological algebra modulo a regular sequence with special attention to codimension two. J. Algebra 230, 24–67 (2000)MathSciNetCrossRefMATH
9.
go back to reference J. Backelin, J. Herzog, B. Ulrich, Linear maximal Cohen-Macaulay modules over strict complete intersections. J. Pure Appl. Algebra 71, 187–202 (1991)MathSciNetCrossRefMATH J. Backelin, J. Herzog, B. Ulrich, Linear maximal Cohen-Macaulay modules over strict complete intersections. J. Pure Appl. Algebra 71, 187–202 (1991)MathSciNetCrossRefMATH
11.
go back to reference M. Ballard, D. Favero, L. Katzarkov, A category of kernels for graded matrix factorizations and its implications for Hodge theory. Publ. Math. l’IHES 120, 1–111 (2014)MathSciNetCrossRefMATH M. Ballard, D. Favero, L. Katzarkov, A category of kernels for graded matrix factorizations and its implications for Hodge theory. Publ. Math. l’IHES 120, 1–111 (2014)MathSciNetCrossRefMATH
12.
go back to reference R.-O. Buchweitz, G.-M. Greuel, F.-O. Schreyer, Cohen-Macaulay modules on hypersurface singularities II. Invent. Math. 88, 165–182 (1987)MathSciNetCrossRefMATH R.-O. Buchweitz, G.-M. Greuel, F.-O. Schreyer, Cohen-Macaulay modules on hypersurface singularities II. Invent. Math. 88, 165–182 (1987)MathSciNetCrossRefMATH
13.
go back to reference J. Burke, Complete intersection rings and Koszul duality (in preparation) J. Burke, Complete intersection rings and Koszul duality (in preparation)
16.
go back to reference A. Cayley, On the theory of elimination. Camb. Dublin Math. J. 3, 116–120 (1848). Collected papers: vol. I (Cambridge University Press, Cambridge, 1889), pp. 370–374 A. Cayley, On the theory of elimination. Camb. Dublin Math. J. 3, 116–120 (1848). Collected papers: vol. I (Cambridge University Press, Cambridge, 1889), pp. 370–374
18.
go back to reference H. Dao, C. Huneke, Vanishing of ext, cluster tilting and finite global dimension of endomorphisms of rings. Am. J. Math. 135, 561–578 (2013)MathSciNetCrossRefMATH H. Dao, C. Huneke, Vanishing of ext, cluster tilting and finite global dimension of endomorphisms of rings. Am. J. Math. 135, 561–578 (2013)MathSciNetCrossRefMATH
23.
go back to reference A. Efimov, L. Positselski, Coherent analogues of matrix factorizations and relative singularity categories. Alg. and Number Theory 9, 1159–1292 (2015)MathSciNetCrossRefMATH A. Efimov, L. Positselski, Coherent analogues of matrix factorizations and relative singularity categories. Alg. and Number Theory 9, 1159–1292 (2015)MathSciNetCrossRefMATH
25.
go back to reference D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations. Trans. Am. Math. Soc. 260, 35–64 (1980)MathSciNetCrossRefMATH D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations. Trans. Am. Math. Soc. 260, 35–64 (1980)MathSciNetCrossRefMATH
28.
go back to reference D. Eisenbud, I. Peeva, F.-O. Schreyer, Tor as a module over an exterior algebra (preprint), 2016 D. Eisenbud, I. Peeva, F.-O. Schreyer, Tor as a module over an exterior algebra (preprint), 2016
31.
go back to reference T. Gulliksen, A change of ring theorem with applications to Poincaré series and intersection multiplicity. Math. Scand. 34, 167–183 (1974)MathSciNetMATH T. Gulliksen, A change of ring theorem with applications to Poincaré series and intersection multiplicity. Math. Scand. 34, 167–183 (1974)MathSciNetMATH
32.
go back to reference D. Hilbert, Über die Theorie der algebraischen Formen. Maht. Ann. 36, 473–534 (1890); Ges. Abh., vol. II (Springer, Berlin, 1933 and 1970), pp. 199–257 D. Hilbert, Über die Theorie der algebraischen Formen. Maht. Ann. 36, 473–534 (1890); Ges. Abh., vol. II (Springer, Berlin, 1933 and 1970), pp. 199–257
33.
go back to reference M. Hochster, The dimension of an intersection in an ambient hypersurface, in Algebraic Geometry. Lecture Notes in Mathematics, vol. 862 (Springer, Berlin, 1981), pp. 93–106 M. Hochster, The dimension of an intersection in an ambient hypersurface, in Algebraic Geometry. Lecture Notes in Mathematics, vol. 862 (Springer, Berlin, 1981), pp. 93–106
36.
go back to reference H. Kajiura, K. Saito, A. Takahashi, Matrix factorization and representations of quivers. II. Type ADE case. Adv. Math. 211, 327–362 (2007)MathSciNetMATH H. Kajiura, K. Saito, A. Takahashi, Matrix factorization and representations of quivers. II. Type ADE case. Adv. Math. 211, 327–362 (2007)MathSciNetMATH
37.
go back to reference A. Kapustin, Y. Li, D-branes in Landau-Ginzburg models and algebraic geometry. J. High Energy Phys. 12, 1–43 (2003)MathSciNet A. Kapustin, Y. Li, D-branes in Landau-Ginzburg models and algebraic geometry. J. High Energy Phys. 12, 1–43 (2003)MathSciNet
41.
go back to reference J. McCullough, I. Peeva, Infinite free resolutions, in Commutative Algebra and Noncommutative Algebraic Geometry, ed. by Eisenbud, Iyengar, Singh, Stafford, Van den Bergh (Cambridge University Press, Cambridge), pp. 101–143 J. McCullough, I. Peeva, Infinite free resolutions, in Commutative Algebra and Noncommutative Algebraic Geometry, ed. by Eisenbud, Iyengar, Singh, Stafford, Van den Bergh (Cambridge University Press, Cambridge), pp. 101–143
43.
go back to reference D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Tr. Mat. Inst. Steklova 246 (2004). Algebr. Geom. Metody, Svyazi i Prilozh, 240–262; translation in Proc. Steklov Inst. Math. 246, 227–248 (2004) D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Tr. Mat. Inst. Steklova 246 (2004). Algebr. Geom. Metody, Svyazi i Prilozh, 240–262; translation in Proc. Steklov Inst. Math. 246, 227–248 (2004)
44.
go back to reference D. Orlov, Triangulated categories of singularities, and equivalences between Landau-Ginzburg models (Russian. Russian summary). Mat. Sb. 197, 117–132 (2006); translation in Sb. Math. 197, 1827–1840 (2006) D. Orlov, Triangulated categories of singularities, and equivalences between Landau-Ginzburg models (Russian. Russian summary). Mat. Sb. 197, 117–132 (2006); translation in Sb. Math. 197, 1827–1840 (2006)
45.
46.
go back to reference D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, in Algebra, Arithmetic, and Geometry: In Honor of Yu.I. Manin, vol. II. Progress in Mathematics, vol. 270 (Birkhäuser, Boston, MA, 2009), pp. 503–531 D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, in Algebra, Arithmetic, and Geometry: In Honor of Yu.I. Manin, vol. II. Progress in Mathematics, vol. 270 (Birkhäuser, Boston, MA, 2009), pp. 503–531
50.
go back to reference A. Polishchuk, A. Vaintrob, Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations. Duke Math. J. 161, 1863–1926 (2012)MathSciNetCrossRefMATH A. Polishchuk, A. Vaintrob, Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations. Duke Math. J. 161, 1863–1926 (2012)MathSciNetCrossRefMATH
51.
go back to reference A. Polishchuk, A. Vaintrob, Matrix factorizations and cohomological field theories. arXiv:1105.2903 (2014) A. Polishchuk, A. Vaintrob, Matrix factorizations and cohomological field theories. arXiv:1105.2903 (2014)
52.
go back to reference F. Reid, Modular representations of elementary abelian p-groups (in preparation). F. Reid, Modular representations of elementary abelian p-groups (in preparation).
53.
go back to reference E. Segal, Equivalences between GIT quotients of Landau-Ginzburg B-models. Commun. Math. Phys. 304, 411–432 (2011)CrossRefMATH E. Segal, Equivalences between GIT quotients of Landau-Ginzburg B-models. Commun. Math. Phys. 304, 411–432 (2011)CrossRefMATH
57.
Metadata
Title
Introduction and Survey
Authors
David Eisenbud
Irena Peeva
Copyright Year
2016
DOI
https://doi.org/10.1007/978-3-319-26437-0_1

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