Geodesic
On almost free torus actions and Horrocks conjecture
Dalʹnevostočnyj matematičeskij žurnal, Tome 12 (2012) no. 1, pp. 98-107.

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

We consider a model for cohomology groups of a space X with an action of torus, representing Koszul complex of its equivariant cohomology. Studying homological properties of modules over polynomial ring we derive new estimates on homological rank (total dimension of rational cohomology) of X. In particular, we obtain simple proof of toral rank conjecture in the case of torus dimension ≤4. 
@article{DVMG_2012_12_1_a8,
     author = {Yu. M. Ustinovskii},
     title = {On almost free torus actions and {Horrocks} conjecture},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {98--107},
     publisher = {mathdoc},
     volume = {12},
     number = {1},
     year = {2012},
     language = {ru},
     url = {https://geodesic-test.mathdoc.fr/item/DVMG_2012_12_1_a8/}
}
TY  - JOUR
AU  - Yu. M. Ustinovskii
TI  - On almost free torus actions and Horrocks conjecture
JO  - Dalʹnevostočnyj matematičeskij žurnal
PY  - 2012
SP  - 98
EP  - 107
VL  - 12
IS  - 1
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/DVMG_2012_12_1_a8/
LA  - ru
ID  - DVMG_2012_12_1_a8
ER  - 
%0 Journal Article
%A Yu. M. Ustinovskii
%T On almost free torus actions and Horrocks conjecture
%J Dalʹnevostočnyj matematičeskij žurnal
%D 2012
%P 98-107
%V 12
%N 1
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/DVMG_2012_12_1_a8/
%G ru
%F DVMG_2012_12_1_a8
Yu. M. Ustinovskii. On almost free torus actions and Horrocks conjecture. Dalʹnevostočnyj matematičeskij žurnal, Tome 12 (2012) no. 1, pp. 98-107. https://geodesic-test.mathdoc.fr/item/DVMG_2012_12_1_a8/

[1] {D. Buchsbaum and D. Eisenbud}, “{Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3}”, Amer. J. Math., 99:3 (1977), 447–485 | DOI | MR | Zbl

[2] V. M. Bukhshtaber, T. E. Panov, {Toricheskie deistviya v topologii i kombinatorike}, MTsNMO, Moskva, 2004

[3] X. Cao and Z. Lu, {Möbius transform, moment-angle complexes and Halperin-Carlsson conjecture}, 2009, arXiv: 0908.3174 | MR

[4] G. Carlsson, “{Free $(\mathbb Z/2)^k$-actions and a problem in commutative algebra}”, Lecture Notes in Math., 1217 (1986), 79–83, Springer, Berlin | DOI | MR

[5] P. E. Conner, “On the action of a finite group on $S^n\times S^n$”, Ann. of Math, 66 (1957), 586–588 | DOI | MR | Zbl

[6] M. W. Davis and T. Januszkiewicz, “{Convex polytopes, Coxter orbifolds and torus actions}”, Duke Math. J, 62:2 (1991), 417–451 | DOI | MR | Zbl

[7] E. G. Evans and P. Griffith, “Binomial behavior of Betti numbers for modules of finite length”, Pacific J. Math, 133 (1988), 267–276 | DOI | MR | Zbl

[8] D. Erman, “A special case of the Buchsbaum-Eisenbud-Horrocks rank conjecture”, Math. Res. Lett., 17:06 (2010), 1079–1089 | DOI | MR | Zbl

[9] Y. Félix, J. Oprea, D. Tanré, {Algebraic Models in Geometry}, Oxford University Press, 2008 | MR | Zbl

[10] Y. Félix, S. Halperin, J. Thomas, {Rational Homotopy Theory}, Springer-Verlag, 2001 | MR

[11] S. Halperin, “{Rational homotopy and torus actions}”, Aspects of topology, 93 (1985), 293–306, Cambridge Univ. Press, Cambridge | DOI | MR

[12] R. Hartshorne, {Algebraic Geometry}, Springer-Verlag, 1977 | MR | Zbl

[13] R. Hartshorne, “{Algebraic vector bundles on projective spaces: a problem list}”, Topology, 18:2 (1979), 117–128 | DOI | MR | Zbl

[14] T. Höfer, “{Remarks on torus principal bundles}”, {J.Math. Kyoto Univ}, 33:1 (1993), 227–259 | MR | Zbl

[15] V. Puppe, “Multiplicative aspects of the Halperin-Carlsson conjecture”, Georgian Mathematical Journal, 16:2 (2009), 369–379, arXiv: 0811.3517 | MR | Zbl

[16] E. Spener, Algebraicheskaya topologiya, Mir, Moskva, 1971 | MR

[17] Yu. M. Ustinovskii, “Gipoteza o toricheskom range dlya moment-ugol kompleksov”, Matem. zametki, 90:2 (2011), 300–305, arXiv: 0909.1053 | DOI | MR | Zbl