## Algebra & Number Theory

### Thick tensor ideals of right bounded derived categories

#### Abstract

Let $R$ be a commutative noetherian ring. Denote by $D−(R)$ the derived category of cochain complexes $X$ of finitely generated $R$-modules with $Hi(X) = 0$ for $i ≫ 0$. Then $D−(R)$ has the structure of a tensor triangulated category with tensor product $⋅⊗RL⋅$ and unit object $R$. In this paper, we study thick tensor ideals of $D−(R)$, i.e., thick subcategories closed under the tensor action by each object in $D−(R)$, and investigate the Balmer spectrum $Spc D − ( R )$ of $D−(R)$, i.e., the set of prime thick tensor ideals of $D−(R)$. First, we give a complete classification of the thick tensor ideals of $D−(R)$ generated by bounded complexes, establishing a generalized version of the Hopkins–Neeman smash nilpotence theorem. Then, we define a pair of maps between the Balmer spectrum $SpcD−(R)$ and the Zariski spectrum $SpecR$, and study their topological properties. After that, we compare several classes of thick tensor ideals of $D−(R)$, relating them to specialization-closed subsets of $SpecR$ and Thomason subsets of $SpcD−(R)$, and construct a counterexample to a conjecture of Balmer. Finally, we explore thick tensor ideals of $D−(R)$ in the case where $R$ is a discrete valuation ring.

#### Article information

Source
Algebra Number Theory, Volume 11, Number 7 (2017), 1677-1738.

Dates
Revised: 9 June 2017
Accepted: 16 July 2017
First available in Project Euclid: 12 December 2017

https://projecteuclid.org/euclid.ant/1513096743

Digital Object Identifier
doi:10.2140/ant.2017.11.1677

Mathematical Reviews number (MathSciNet)
MR3697152

Zentralblatt MATH identifier
06775557

#### Citation

Matsui, Hiroki; Takahashi, Ryo. Thick tensor ideals of right bounded derived categories. Algebra Number Theory 11 (2017), no. 7, 1677--1738. doi:10.2140/ant.2017.11.1677. https://projecteuclid.org/euclid.ant/1513096743

#### References

• L. L. Avramov and H.-B. Foxby, “Homological dimensions of unbounded complexes”, J. Pure Appl. Algebra 71:2–3 (1991), 129–155.
• P. Balmer, “Presheaves of triangulated categories and reconstruction of schemes”, Math. Ann. 324:3 (2002), 557–580.
• P. Balmer, “The spectrum of prime ideals in tensor triangulated categories”, J. Reine Angew. Math. 588 (2005), 149–168.
• P. Balmer, “Spectra, spectra, spectra - tensor triangular spectra versus Zariski spectra of endomorphism rings”, Algebr. Geom. Topol. 10:3 (2010), 1521–1563.
• P. Balmer, “Tensor triangular geometry”, pp. 85–112 in Proceedings of the International Congress of Mathematicians (Hyderabad, India, 2010), vol. II, edited by R. Bhatia et al., Hindustan, 2010.
• P. Balmer, “Separable extensions in tensor-triangular geometry and generalized Quillen stratification”, Ann. Sci. Éc. Norm. Supér. $(4)$ 49:4 (2016), 907–925.
• P. Balmer and B. Sanders, “The spectrum of the equivariant stable homotopy category of a finite group”, Invent. Math. 208:1 (2017), 283–326.
• D. J. Benson, J. F. Carlson, and J. Rickard, “Thick subcategories of the stable module category”, Fund. Math. 153:1 (1997), 59–80.
• D. J. Benson, S. B. Iyengar, and H. Krause, “Stratifying modular representations of finite groups”, Ann. of Math. $(2)$ 174:3 (2011), 1643–1684.
• W. Bruns and J. Herzog, Cohen–Macaulay rings, 2nd ed., Cambridge Studies in Advanced Mathematics 39, 1998.
• L. W. Christensen, Gorenstein dimensions, Lecture Notes in Mathematics 1747, Springer, 2000.
• I. Dell'Ambrogio and G. Tabuada, “Tensor triangular geometry of non-commutative motives”, Adv. Math. 229:2 (2012), 1329–1357.
• E. S. Devinatz, M. J. Hopkins, and J. H. Smith, “Nilpotence and stable homotopy theory, I”, Ann. of Math. $(2)$ 128:2 (1988), 207–241.
• E. M. Friedlander and J. Pevtsova, “$\Pi$-supports for modules for finite group schemes”, Duke Math. J. 139:2 (2007), 317–368.
• M. J. Hopkins, “Global methods in homotopy theory”, pp. 73–96 in Homotopy theory (Durham, England, 1985), edited by E. Rees and J. D. S. Jones, London Mathematical Society Lecture Note Series 117, Cambridge University, 1987.
• M. J. Hopkins and J. H. Smith, “Nilpotence and stable homotopy theory, II”, Ann. of Math. $(2)$ 148:1 (1998), 1–49.
• M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homotopy theory, Memoirs of the American Mathematical Society 610, 1997.
• H. Krause, “Derived categories, resolutions, and Brown representability”, pp. 101–139 in Interactions between homotopy theory and algebra, edited by L. L. Avramov et al., Contemporary Mathematics 436, American Mathematical Society, 2007.
• A. Neeman, “The chromatic tower for $D(R)$”, Topology 31:3 (1992), 519–532.
• T. J. Peter, “Prime ideals of mixed Artin–Tate motives”, J. K-Theory 11:2 (2013), 331–349.
• G. Stevenson, “Duality for bounded derived categories of complete intersections”, Bull. Lond. Math. Soc. 46:2 (2014), 245–257.
• G. Stevenson, “Subcategories of singularity categories via tensor actions”, Compos. Math. 150:2 (2014), 229–272.
• R. Takahashi, “Classifying thick subcategories of the stable category of Cohen–Macaulay modules”, Adv. Math. 225:4 (2010), 2076–2116.
• R. W. Thomason, “The classification of triangulated subcategories”, Compos. Math. 105:1 (1997), 1–27.