Ballard, Matthew; Farman, Blake Kernels for noncommutative projective schemes. (English) Zbl 1493.14003 J. Noncommut. Geom. 15, No. 4, 1129-1180 (2021). Summary: We give a noncommutative geometric description of the internal Hom dg-category in the homotopy category of dg-categories between two noncommutative projective schemes in the style of M. Artin and J. J. Zhang [Adv. Math. 109, No. 2, 228–287 (1994; Zbl 0833.14002)]. As an immediate application, we give a noncommutative projective derived Morita statement along the lines of Rickard and Orlov. MSC: 14A22 Noncommutative algebraic geometry 14A30 Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.) 14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry 16E35 Derived categories and associative algebras Keywords:noncommutative algebra; noncommutative projective schemes; derived categories; Fourier-Mukai transforms Citations:Zbl 0833.14002 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] M. Artin, Some problems on three-dimensional graded domains. In Representation Theory and Algebraic Geometry (Waltham, MA, 1995), pp. 1-19, London Math. Soc. Lecture Note Ser. 238, Cambridge Univ. Press, Cambridge, 1997 Zbl 0888.16025 MR 1477464 · Zbl 0888.16025 [2] M. Artin and W. F. Schelter, Graded algebras of global dimension 3. Adv. in Math. 66 (1987), no. 2, 171-216 Zbl 0633.16001 MR 917738 · Zbl 0633.16001 [3] M. Artin, J. Tate, and M. van den Bergh, Some algebras associated to automorphisms of ellip-tic curves. In The Grothendieck Festschrift, Vol. I, pp. 33-85, Progr. Math. 86, Birkhäuser, Boston, MA, 1990 Zbl 0744.14024 MR 1086882 · Zbl 0744.14024 [4] M. Artin and J. J. Zhang, Noncommutative projective schemes. Adv. Math. 109 (1994), no. 2, 228-287 Zbl 0833.14002 MR 1304753 · Zbl 0833.14002 [5] D. Auroux, L. Katzarkov, and D. Orlov, Mirror symmetry for weighted projective planes and their noncommutative deformations. Ann. of Math. (2) 167 (2008), no. 3, 867-943 Zbl 1175.14030 MR 2415388 · Zbl 1175.14030 [6] M. Ballard, D. Favero, and L. Katzarkov, A category of kernels for equivariant factorizations and its implications for Hodge theory. Publ. Math. Inst. Hautes Études Sci. 120 (2014), 1-111 Zbl 1401.14086 MR 3270588 · Zbl 1401.14086 [7] A. Bayer and E. Macrì, MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations. Invent. Math. 198 (2014), no. 3, 505-590 Zbl 1308.14011 MR 3279532 · Zbl 1308.14011 [8] A. A. Beilinson, Coherent sheaves on P n and problems of linear algebra. Funct. Anal. Appl. 12 (1978), 214-216 Zbl 0424.14003 · Zbl 0424.14003 [9] D. Ben-Zvi, J. Francis, and D. Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry. J. Amer. Math. Soc. 23 (2010), no. 4, 909-966 Zbl 1202.14015 MR 2669705 · Zbl 1202.14015 [10] A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties. 1995, arXiv:alg-geom/9506012 [11] A. Bondal and M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3 (2003), no. 1, 1-36 Zbl 1135.18302 MR 1996800 · Zbl 1135.18302 [12] A. I. Bondal and A. E. Polishchuk, Homological properties of associative algebras: the method of helices. Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 2, 3-50 Zbl 0847.16010 MR 1230966 · Zbl 0847.16010 [13] T. Bridgeland, Flops and derived categories. Invent. Math. 147 (2002), no. 3, 613-632 Zbl 1085.14017 MR 1893007 · Zbl 1085.14017 [14] T. Bridgeland, Stability conditions on triangulated categories. Ann. of Math. (2) 166 (2007), no. 2, 317-345 Zbl 1137.18008 MR 2373143 · Zbl 1137.18008 [15] A. Canonaco and P. Stellari, Internal Homs via extensions of dg functors. Adv. Math. 277 (2015), 100-123 Zbl 1355.14012 MR 3336084 · Zbl 1355.14012 [16] T. Dyckerhoff, Compact generators in categories of matrix factorizations. Duke Math. J. 159 (2011), no. 2, 223-274 Zbl 1252.18026 MR 2824483 · Zbl 1252.18026 [17] P. Gabriel, Des catégories abéliennes. Bull. Soc. Math. France 90 (1962), 323-448 Zbl 0201.35602 MR 232821 · Zbl 0201.35602 [18] D. Happel, Triangulated Categories in the Representation Theory of Finite-Dimensional Alge-bras. London Math. Soc. Lecture Note Ser. 119, Cambridge Univ. Press, Cambridge, 1988 · Zbl 0635.16017 [19] Kernels for noncommutative projective schemes 1179 [20] A. Harder and L. Katzarkov, Perverse sheaves of categories and some applications. Adv. Math. 352 (2019), 1155-1205 Zbl 1440.14084 MR 3975708 · Zbl 1440.14084 [21] M. Hovey, Model category structures on chain complexes of sheaves. Trans. Amer. Math. Soc. 353 (2001), no. 6, 2441-2457 Zbl 0969.18010 MR 1814077 · Zbl 0969.18010 [22] M. Hovey, Cotorsion pairs and model categories. In Interactions Between Homotopy Theory and Algebra, pp. 277-296, Contemp. Math. 436, Amer. Math. Soc., Providence, RI, 2007 Zbl 1129.18004 MR 2355778 · Zbl 1129.18004 [23] A. Kanazawa, Non-commutative projective Calabi-Yau schemes. J. Pure Appl. Algebra 219 (2015), no. 7, 2771-2780 Zbl 1342.14003 MR 3313507 · Zbl 1342.14003 [24] B. Keller, Deriving DG categories. Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63-102 Zbl 0799.18007 MR 1258406 · Zbl 0799.18007 [25] M. Kontsevich, Homological algebra of mirror symmetry. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pp. 120-139, Birkhäuser, Basel, 1995 Zbl 0846.53021 MR 1403918 · Zbl 0846.53021 [26] C. Li and X. Zhao, The minimal model program for deformations of Hilbert schemes of points on the projective plane. Algebr. Geom. 5 (2018), no. 3, 328-358 Zbl 1432.14011 MR 3800356 · Zbl 1432.14011 [27] V. A. Lunts and D. Orlov, Uniqueness of enhancement for triangulated categories. J. Amer. Math. Soc. 23 (2010), no. 3, 853-908 Zbl 1197.14014 MR 2629991 · Zbl 1197.14014 [28] J. N. Mather and S. S. T. Yau, Classification of isolated hypersurface singularities by their moduli algebras. Invent. Math. 69 (1982), no. 2, 243-251 Zbl 0499.32008 MR 674404 · Zbl 0499.32008 [29] S. Mukai, Duality between D.X/ and D. O X/ with its application to Picard sheaves. Nagoya Math. J. 81 (1981), 153-175 Zbl 0417.14036 MR 607081 · Zbl 0417.14036 [30] S. Mukai, On the moduli space of bundles on K3 surfaces. I. In Vector Bundles on Algebraic Varieties (Bombay, 1984), pp. 341-413, Tata Inst. Fund. Res. Stud. Math. 11, Tata Inst. Fund. Res., Bombay, 1987 Zbl 0674.14023 MR 893604 · Zbl 0674.14023 [31] D. Orlov, Equivalences of derived categories and K3 surfaces. J. Math. Sci. (New York) 84 (1997), no. 5, 1361-1381 Zbl 0938.14019 MR 1465519 · Zbl 0938.14019 [32] 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, pp. 503-531, Progr. Math. 270, Birkhäuser, Boston, MA, 2009 Zbl 1200.18007 MR 2641200 · Zbl 1200.18007 [33] A. Polishchuk and A. Vaintrob, Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations. Duke Math. J. 161 (2012), no. 10, 1863-1926 Zbl 1249.14001 MR 2954619 · Zbl 1249.14001 [34] P. Seidel, A 1 -subalgebras and natural transformations. Homology Homotopy Appl. 10 (2008), no. 2, 83-114 Zbl 1215.53079 MR 2426130 · Zbl 1215.53079 [35] N. Sheridan, Homological mirror symmetry for Calabi-Yau hypersurfaces in projective space. Invent. Math. 199 (2015), no. 1, 1-186 Zbl 1344.53073 MR 3294958 · Zbl 1344.53073 [36] B. Stenström, Rings of Quotients: An Introduction to Methods of Ring Theory. Grundlehren Math. Wiss. 217, Springer, New York, 1975 Zbl 0296.16001 MR 0389953 · Zbl 0296.16001 [37] D. R. Stephenson, Artin-Schelter regular algebras of global dimension three. J. Algebra 183 (1996), no. 1, 55-73 Zbl 0868.16027 MR 1397387 · Zbl 0868.16027 [38] D. R. Stephenson, Algebras associated to elliptic curves. Trans. Amer. Math. Soc. 349 (1997), no. 6, 2317-2340 Zbl 0868.16028 MR 1390046 · Zbl 0868.16028 [39] B. Toën, The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167 (2007), no. 3, 615-667 Zbl 1118.18010 MR 2276263 · Zbl 1118.18010 [40] B. Toën and M. Vaquié, Moduli of objects in dg-categories. Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 3, 387-444 Zbl 1140.18005 MR 2493386 · Zbl 1140.18005 [41] M. van den Bergh, Existence theorems for dualizing complexes over non-commutative graded and filtered rings. J. Algebra 195 (1997), no. 2, 662-679 Zbl 0894.16020 MR 1469646 · Zbl 0894.16020 [42] M. van den Bergh, Noncommutative quadrics. Int. Math. Res. Not. IMRN 2011 (2011), no. 17, 3983-4026 Zbl 1311.14003 MR 2836401 · Zbl 1311.14003 [43] K. van Rompay, Segre product of Artin-Schelter regular algebras of dimension 2 and embed-dings in quantum P 3 ’s. J. Algebra 180 (1996), no. 2, 483-512 Zbl 0853.17012 MR 1378541 · Zbl 0853.17012 [44] J. J. Zhang, Twisted graded algebras and equivalences of graded categories. Proc. London Math. Soc. (3) 72 (1996), no. 2, 281-311 Zbl 0852.16005 MR 1367080 · Zbl 0852.16005 This reference list is based on information provided by the publisher or from digital mathematics libraries. 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