On adjunctions for Fourier-Mukai transforms. (English) Zbl 1316.14033
Summary: We show that the adjunction counits of a Fourier-Mukai transform \(\varPhi :D(X_{1})\to D(X_{2})\) arise from maps of the kernels of the corresponding Fourier-Mukai transforms. In a very general setting of proper separable schemes of finite type over a field we write down these maps of kernels explicitly – facilitating the computation of the twist (the cone of an adjunction counit) of \(\varPhi \). We also give another description of these maps, better suited to computing cones if the kernel of \(\varPhi \) is a pushforward from a closed subscheme \(Z\subset X_{1}\times X_{2}\). Moreover, we show that we can replace the condition of properness of the ambient spaces \(X_{1}\) and \(X_{2}\) by that of \(Z\) being proper over them and still have this description apply as is. This can be used, for instance, to compute spherical twists on non-proper varieties directly and in full generality.
MSC:
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
Keywords:
Fourier-Mukai transforms; derived categories; algebraic geometry; spherical twists; adjunction; Künneth mapReferences:
| [1] | Rina Anno, Timothy Logvinenko, Braiding criteria for spherical fibrations (in preparation).; Rina Anno, Timothy Logvinenko, Braiding criteria for spherical fibrations (in preparation). · Zbl 1435.14019 |
| [2] | Rina Anno, Spherical functors, 2007. arXiv:0711.4409; Rina Anno, Spherical functors, 2007. arXiv:0711.4409 · Zbl 1374.14015 |
| [3] | Luchezar L. Avramov, Srikanth B. Iyengar, Joseph Lipman, Reflexivity and rigidity for complexes, II: schemes, 2010. arXiv:1001.3450; Luchezar L. Avramov, Srikanth B. Iyengar, Joseph Lipman, Reflexivity and rigidity for complexes, II: schemes, 2010. arXiv:1001.3450 · Zbl 1246.14005 |
| [4] | Alexei Bondal, Dmitri Orlov, Semi-orthogonal decompositions for algebraic varieties, 1995. arXiv:alg-geom/9506012; Alexei Bondal, Dmitri Orlov, Semi-orthogonal decompositions for algebraic varieties, 1995. arXiv:alg-geom/9506012 |
| [5] | Grothendieck, Alexander; Dieudonné, Jean, Éléments de géométrie algébrique I: Le langage des schémas, Publ. math. l’I.H.É.S., 4, 5-228 (1960) |
| [6] | Grothendieck, Alexander; Dieudonné, Jean, Éléments de géométrie algébrique IV: Étude locale des schémas et des morphismes de schémas. Première partie, Publ. math. l’I.H.É.S., 20, 5-259 (1964) |
| [7] | Hartshorne, R., Residues and Duality (1966), Springer-Verlag |
| [8] | Illusie, Luc, Conditions de finitude relatives, (Théorie des Intersections et Théorème de Riemann-Roch (SGA 6). Théorie des Intersections et Théorème de Riemann-Roch (SGA 6), Lecture Notes in Math., vol. 225 (1971), Springer-Verlag), 222-273 · Zbl 0229.14010 |
| [9] | Illusie, Luc, Généralités sur les conditions de finitude dans les catégories dérivées, (Théorie des Intersections et Théorème de Riemann-Roch (SGA 6). Théorie des Intersections et Théorème de Riemann-Roch (SGA 6), Lecture Notes in Math., vol. 225 (1971), Springer-Verlag), 78-159 · Zbl 0229.14009 |
| [10] | Kashiwara, Masaki; Schapira, Pierre, Categories and sheaves, (Grundlehren der mathematischen Wissenschaften, vol. 332 (2006), Springer) · Zbl 1118.18001 |
| [11] | Khovanov, Mikhail; Thomas, Richard, Braid cobordisms, triangulated categories, and flag varieties, Homology, Homotopy Appl., 9, 2, 19-94 (2007) · Zbl 1119.18008 |
| [12] | Kuznetsov, Alexander G., Hyperplane sections and derived categories, Izv. RAN. Ser. Mat., 70, 3, 23-128 (2006) · Zbl 1133.14016 |
| [13] | Lipman, Joseph, Notes on derived functors and Grothendieck duality, (Foundations of Grothendieck Duality for Diagrams of Schemes. Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Math., vol. 1960 (2009), Springer: Springer Berlin), 1-259 · Zbl 1467.14052 |
| [14] | MacLane, Saunders, Categoriesfortheworkingmathematician (1998), Springer-Verlag: Springer-Verlag New York |
| [15] | Mukai, Shigeru, Duality between \(D(X)\) and \(D(\hat{X})\) and its application to Picard sheaves, Nagoya Math. J., 81, 153-175 (1981) · Zbl 0417.14036 |
| [16] | Nagata, M., Imbedding of an abstract variety in a complete variety, J. Math. Kyoto Uni., 2, 1 (1962) · Zbl 0109.39503 |
| [17] | Namikawa, Yoshinori, Mukai flops and derived categories, J. Reine Angew. Math., 560, 65-76 (2003) · Zbl 1033.18008 |
| [18] | Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown’s representability, J. Amer. Math. Soc., 9, 205-236 (1996) · Zbl 0864.14008 |
| [19] | Neeman, Amnon, Triangulated categories, (Annals of Mathematics Studies, vol. 148 (2001), Princeton University Press) · Zbl 0974.18008 |
| [20] | Spaltenstein, Nicolas, Resolutions of unbounded complexes, Compos. Math., 65, 2, 121-154 (1988) · Zbl 0636.18006 |
| [21] | Seidel, Paul; Thomas, Richard, Braid group actions on derived categories of coherent sheaves, Duke Math. J., 108, 1, 37-108 (2001) · Zbl 1092.14025 |
| [22] | Verdier, Jean-Louis, Base change for twisted inverse image of coherent sheaves, (Algebraic geometry (Bombay Colloquium, 1968) (1969), Oxford University Press), 393-408 · Zbl 0202.19902 |
| [23] | Paul Vojta, Nagata’s embedding theorem, 2007. arXiv:0706.1907; Paul Vojta, Nagata’s embedding theorem, 2007. arXiv:0706.1907 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
