Geodesic
Classifying finite localizations of quasi-coherent sheaves
Algebra i analiz, Tome 21 (2009) no. 3, pp. 93-129.

Voir la notice de l'article provenant de la source Math-Net.Ru

Given a quasicompact, quasiseparated scheme $X$, a bijection between the tensor localizing subcategories of finite type in $\operatorname{Qcoh}(X)$ and the set of all subsets $Y\subseteq X$ of the form $Y=\bigcup_{i\in\Omega}Y_i$, with $X\setminus Y_i$ quasicompact and open for all $i\in\Omega$, is established. As an application, an isomorphism of ringed spaces $$ (X,\mathcal{O}_X)\overset{\cong}{\longrightarrow}(\sf{spec}(\operatorname{Qcoh}(X)),\mathcal{O}_{\operatorname{Qcoh}(X)}) $$ is constructed, where $(\sf{spec}(\operatorname{Qcoh}(X)),\mathcal{O}_{\operatorname{Qcoh}(X)})$ is a ringed space associated with the lattice of tensor localizing subcategories of finite type. Also, a bijective correspondence between the tensor thick subcategories of perfect complexes $\mathcal{D}_{\operatorname{per}}(X)$ and the tensor localizing subcategories of finite type in $\operatorname{Qcoh}(X)$ is established. 
@article{AA_2009_21_3_a3,
     author = {G. A. Garkusha},
     title = {Classifying finite localizations of quasi-coherent sheaves},
     journal = {Algebra i analiz},
     pages = {93--129},
     publisher = {mathdoc},
     volume = {21},
     number = {3},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AA_2009_21_3_a3/}
}
TY  - JOUR
AU  - G. A. Garkusha
TI  - Classifying finite localizations of quasi-coherent sheaves
JO  - Algebra i analiz
PY  - 2009
SP  - 93
EP  - 129
VL  - 21
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2009_21_3_a3/
LA  - ru
ID  - AA_2009_21_3_a3
ER  - 
%0 Journal Article
%A G. A. Garkusha
%T Classifying finite localizations of quasi-coherent sheaves
%J Algebra i analiz
%D 2009
%P 93-129
%V 21
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2009_21_3_a3/
%G ru
%F AA_2009_21_3_a3
G. A. Garkusha. Classifying finite localizations of quasi-coherent sheaves. Algebra i analiz, Tome 21 (2009) no. 3, pp. 93-129. http://geodesic.mathdoc.fr/item/AA_2009_21_3_a3/

[1] Gabriel P., “Des catégories abéliennes”, Bull. Soc. Math. France, 90 (1962), 323–448 | MR | Zbl

[2] Buan A. B., Krause H., Solberg Ø., “Support varieties: an ideal approach”, Homology, Homotopy Appl., 9 (2007), 45–74 | MR | Zbl

[3] Garkusha G., Prest M., “Classifying Serre subcategories of finitely presented modules”, Proc. Amer. Math. Soc., 136 (2008), 761–770 | DOI | MR | Zbl

[4] Garkusha G., Prest M., “Reconstructing projective schemes from Serre subcategories”, J. Algebra, 319 (2008), 1132–1153 | DOI | MR | Zbl

[5] Garkusha G., Prest M., “Torsion classes of finite type and spectra”, $K$-Theory and Noncomm. Geometry, European Math. Soc. Publ. House, 2008, 393–412 | MR | Zbl

[6] Thomason R. W., “The classification of triangulated subcategories”, Compositio Math., 105 (1997), 1–27 | DOI | MR | Zbl

[7] Hopkins M. J., “Global methods in homotopy theory”, Homotopy Theory (Durham, 1985), London Math. Soc. Lecture Note Ser., 117, Cambridge Univ. Press, Cambridge, 1987, 73–96 | MR

[8] Neeman A., “The chromatic tower for $D(R)$”, Topology, 31 (1992), 519–532 | DOI | MR | Zbl

[9] Balmer P., “Presheaves of triangulated categories and reconstruction of schemes”, Math. Ann., 324 (2002), 557–580 | DOI | MR | Zbl

[10] Hovey M., “Classifying subcategories of modules”, Trans. Amer. Math. Soc., 353 (2001), 3181–3191 | DOI | MR | Zbl

[11] Rosenberg A. L., “The spectrum of abelian categories and reconstruction of schemes”, Rings, Hopf Algebras, and Brauer Groups (Antwerp/Brussels, 1996), Lecture Notes in Pure and Appl. Math., 197, Marcel Dekker, New York, 1998, 257–274 | MR | Zbl

[12] Herzog I., “The Ziegler spectrum of a locally coherent Grothendieck category”, Proc. London Math. Soc. (3), 74 (1997), 503–558 | DOI | MR | Zbl

[13] Krause H., “The spectrum of a locally coherent category”, J. Pure Appl. Algebra, 114 (1997), 259–271 | DOI | MR | Zbl

[14] Enochs E., Estrada S., “Relative homological algebra in the category of quasi-coherent sheaves”, Adv. Math., 194 (2005), 284–295 | DOI | MR | Zbl

[15] Grothendieck A., Dieudonné J. A., Eléments de géométrie algébrique. I, Grundlehren Math. Wiss., 166, Springer-Verlag, Berlin etc., 1971 | Zbl

[16] Khartskhorn R., Algebraicheskaya geometriya, Mir, M., 1981 | MR

[17] Garkusha G. A., “Kategorii Grotendika”, Algebra i analiz, 13:2 (2001), 1–68 | MR | Zbl

[18] Murfet D., Modules over a scheme, dostupen v seti: http://therisingsea.org

[19] Ziegler M., “Model theory of modules”, Ann. Pure Appl. Logic, 26 (1984), 149–213 | DOI | MR | Zbl

[20] Kacivara M., Shapira P., Puchki na mnogoobraziyakh, Mir, M., 1997

[21] Lipman J., Notes on derived categories and derived functors, dostupen v seti: http://www.math.purdue.edu/~lipman

[22] Bondal A., van den Bergh M., “Generators and representability of functors in commutative and noncommutative geometry”, Moscow Math. J., 3:1 (2003), 1–36 | MR | Zbl

[23] Hochster M., “Prime ideal structure in commutative rings”, Trans. Amer. Math. Soc., 142 (1969), 43–60 | DOI | MR | Zbl

[24] Balmer P., “The spectrum of prime ideals in tensor triangulated categories”, J. Reine Angew. Math., 588 (2005), 149–168 | MR | Zbl

[25] Prest M., The Zariski spectrum of the category of finitely presented modules, Eprint, dostupen v seti: http://www.maths.man.ac.uk/~mprest

[26] Johnstone P. T., Stone spaces, Cambridge Stud. in Adv. Math., 3, Cambridge Univ. Press, Cambridge, 1982 | MR | Zbl

[27] Soublin J.-P., “Anneaux et modules cohérents”, J. Algebra, 15 (1970), 455–472 | DOI | MR | Zbl