## Geometric phantom categories.(English)Zbl 1285.14018

For a smooth, projective variety $$X$$ one can define a quasiphantom to be an admissible subcategory of the bounded derived category of coherent sheaves $$\mathrm{D}^b(X)$$, with trivial Hochschild homology (and finite abelian Grothendieck group). For a quasiphantom subcategory to be a phantom subcategory one requires that it also has a trivial Grothendieck group.
The authors construct phantom triangulated categories by considering the tensor product of quasiphantoms for which the orders of their Grothendieck groups are coprime. They not only prove that such products are phantoms, but that they are also universal phantoms (which they show to be equivalent to the vanishing of their $$K$$-motive). Last, but not least, they prove that, over $$\mathbb C$$, the vanishing of $$K_0$$-groups of an admissible subcategory imply the vanishing of its Hochschild homology.
In the paper, they first consider Chow motives of surfaces of general type with $$q=p_g=0$$ which satisfy Bloch’s conjecture for $$0$$-cycles (i.e. $$\mathrm{CH}^2(S)\cong\mathbb Z$$) and prove that for such an $$S$$, there exists a decomposition of its Chow motive $$M(S)\cong 1\oplus\mathbb L^{\oplus r}\oplus\mathbb L^2\oplus M$$, where $$1$$ is the unit object, $$\mathbb L$$ is the Lefschetz motive and M is a Chow motive over $$\mathbb C$$. Then, they go on to show that, for S and S’ surfaces as above, the tensor product of the corresponding $$M$$ and $$M'$$ is trivial, whereas the external product map $$\mathrm{CH}^\bullet(S)\otimes\mathrm{CH}^\bullet(S')\rightarrow\mathrm{CH}^\bullet(S\times S')$$ is an isomorphism. This allows them to prove that, assuming the derived categories $$\mathrm{D}^b(S)$$ and $$\mathrm{D}^b(S')$$ have exceptional collections of maximal length and the orders of $$\mathrm{Pic}(S)_{\mathrm{tors}}$$ and $$\mathrm{Pic}(S)_{\mathrm{tors}}$$ are coprime, the tensor product of the orthogonals to these exceptional collections is a phantom. Applying this to the Burniat surface with $$c_1^2=6$$ and the classical Godeaux surface (or a Beauville surface), they construct a phantom category.
The second part of the paper deals with $$K$$-motives and the proof that the above constructed phantoms are actually universal phantoms. We should remark here that a universal phantom is indeed a phantom, however the proofs that the corresponding tensor product is a phantom (respectively a universal phantom) are different and independently interesting.

### MathOverflow Questions:

Vanishing of Hochschild homology of a category

### MSC:

 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 18E30 Derived categories, triangulated categories (MSC2010) 18F30 Grothendieck groups (category-theoretic aspects) 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 14F42 Motivic cohomology; motivic homotopy theory
Full Text:

### References:

 [1] V. Alexeev and D. Orlov, Derived categories of Burniat surfaces and exceptional collections, preprint (2012), arXiv:1208.4348 . · Zbl 1282.14030 [2] P. Berthelot, A. Grothendieck, and L. Illusie, Théorie des intersections et théorème de Riemann–Roch, in Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Lecture Notes in Mathematics, vol. 225, 1971. [3] S. Bloch, Lectures on Algebraic Cycles, Duke Univ. Math. Series IV, 1980. · Zbl 0436.14003 [4] Ch. Böhning, H.-Ch. Graf von Bothmer, and P. Sosna, On the derived category of the classical Godeaux surface, preprint (2012), arXiv:1206.1830 . · Zbl 1299.14015 [5] Ch. Böhning, H.-Ch. Graf von Bothmer, L. Katzarkov, and P. Sosna, Determinantal Barlow surfaces and phantom categories, preprint (2012), arXiv:1210.0343 . · Zbl 1323.14014 [6] A. Bondal and M. Kapranov, Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR, Ser. Mat., 53 (1989), 1183–1205, 1337. · Zbl 0703.14011 [7] A. Bondal and M. Kapranov, Enhanced triangulated categories, Mat. Sb., 181 (1990), 669–683. [8] A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties, preprint MPIM 95/15 (1995), arXiv:math.AG/9506012 . [9] W. Fulton, Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer, Berlin, 1984. · Zbl 0541.14005 [10] S. Galkin and E. Shinder, Exceptional collection on the Beauville surface, preprint (2012), arXiv:1210.3339 . · Zbl 1408.14068 [11] S. Gorchinskiy and V. Guletskii, Motives and representability of algebraic cycles on threefolds over a field, J. Algebr. Geom., 21 (2012), 347–373. · Zbl 1256.14007 [12] H. Inose and M. Mizukami, Rational equivalence of 0-cycles on some surfaces of general type with p g =0, Math. Ann., 244 (1979), 205–217. · Zbl 0444.14006 [13] B. Kahn, J. P. Murre, and C. Pedrini, On the transcendental part of the motive of a surface, in London Math. Soc. Lecture Notes Ser., vol. 344, pp. 143–202, Cambridge University Press, Cambridge, 2004. · Zbl 1130.14008 [14] B. Keller, Invariance and localization for cyclic homology of DG algebras, J. Pure Appl. Algebra, 123 (1998), 223–273. · Zbl 0890.18007 [15] B. Keller, On differential graded categories, in International Congress of Mathematicians, vol. II, pp. 151–190, Eur. Math. Soc., Zürich, 2006. · Zbl 1140.18008 [16] M. Kontsevich and Y. Soibelman, Notes on A algebras, A categories and non-commutative geometry, in Homological Mirror Symmetry, Lecture Notes in Phys., vol. 757, pp. 153–219, Springer, Berlin, 2009. · Zbl 1202.81120 [17] A. Kuznetsov, Hochschild homology and semiorthogonal decompositions, preprint (2009), arXiv:0904.4330 . [18] A. Kuznetsov, Height of exceptional collections and Hochschild cohomology of quasiphantom categories, preprint (2012), arXiv:1211.4693 . · Zbl 1331.14024 [19] V. Lunts and D. Orlov, Uniqueness of enhancement for triangulated categories, J. Am. Math. Soc., 23 (2010), 853–908. · Zbl 1197.14014 [20] V. A. Lunts and O. M. Schnürer, Smoothness of equivariant derived categories, preprint (2012), arXiv:1205.3132 . [21] Yu. I. Manin, Correspondences, motifs and monoidal transformations, Math. USSR Sb., 6 (1968), 439–470. [22] M. Marcolli and G. Tabuada, From exceptional collections to motivic decompositions via noncommutative motives, preprint (2012), arXiv:1202.6297 . · Zbl 1349.14021 [23] A. S. Merkurjev and A. A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Math. USSR, Izv., 21 (1983), 307–340. · Zbl 0525.18008 [24] D. Orlov, Derived categories of coherent sheaves and equivalences between them, Russ. Math. Surv., 58 (2003), 511–591. · Zbl 1050.01509 [25] D. Quillen, Algebraic K-theory I, Lect. Notes Math., 341 (1973), 85–147. · Zbl 0292.18004 [26] A. Scholl, Classical motives, in Proc. Sympos. Pure Math., vol. 55, pp. 163–187, Am. Math. Soc., Providence, 1994. · Zbl 0814.14001 [27] G. Tabuada, Invariants additifs de dg-catégories, Int. Math. Res. Not., 53 (2005), 3309–3339. · Zbl 1094.18006 [28] B. Toën, The homotopy theory of dg-categories and derived Morita theory, Invent. Math., 167 (2007), 615–667. · Zbl 1118.18010 [29] B. Toën and M. Vaquié, Moduli of objects in dg-categories, Ann. Sci. École Norm. Sup. (4), 40 (2007), 387–444. · Zbl 1140.18005
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.