Geodesic
Construction of a linear filtration for bundles of rank $2$ on $\mathbf{P}^1_{\mathbb Z}$
Sbornik. Mathematics, Tome 208 (2017) no. 4, pp. 568-584 Cet article a éte moissonné depuis la source Math-Net.Ru

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We obtain an algorithm for the construction of a filtration with linear factors for vector bundles of rank 2 over the surface $\mathbf{P}^1_A$, where $A$ is a Euclidean domain. In other words, we produce an algorithm that, for an invertible $2$-matrix $\sigma$ over the ring $A[x,x^{-1}]$, constructs matrices $\lambda$ over $A[x]$ and $\rho$ over $A[x^{-1}]$ for which $\lambda\sigma\rho$ is an upper triangular matrix. Bibliography: 13 titles.
Keywords: vector bundle, arithmetic surface, projective line, reduction.
Mots-clés : filtration 
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A. L. Smirnov; S. S. Yakovenko. Construction of a linear filtration for bundles of rank $2$ on $\mathbf{P}^1_{\mathbb Z}$. Sbornik. Mathematics, Tome 208 (2017) no. 4, pp. 568-584. http://geodesic.mathdoc.fr/item/SM_2017_208_4_a5/

[1] Ch. C. Hanna, “Subbundles of vector bundles on the projective line”, J. Algebra, 52:2 (1978), 322–327 | DOI | MR | Zbl

[2] A. L. Smirnov, “On filtrations of vector bundles over $\operatorname{\mathbb P}^1_\mathbb Z$”, Arithmetic and geometry, London Math. Soc. Lecture Note Ser., 420, Cambridge Univ. Press, Cambridge, 2015, 436–457 | DOI | MR | Zbl

[3] A. N. Parshin, “Higher Bruhat–Tits buildings and vector bundles on an algebraic surface”, Algebra and number theory (Essen, 1992), de Gruyter, Berlin, 1994, 165–192 ; Preprint, Institut für Experimentelle Mathematik, Essen, 1993, 34 pp. | MR | Zbl

[4] J.-P. Serre, Trees, Transl. from the French, Springer-Verlag, Berlin–New York, 1980, ix+142 pp. | MR | Zbl

[5] R. Hartshorne, Algebraic geometry, Grad. Texts in Math., 52, Springer-Verlag, New York–Heidelberg, 1977, xvi+496 pp. | MR | MR | Zbl | Zbl

[6] Ch. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces, Progr. Math., 3, Birkhäuser, Boston, Mass., 1980, vii+389 pp. | MR | MR | Zbl

[7] A. Grothendieck, “Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes”, Inst. Hautes Études Sci. Publ. Math., 8:1 (1961), 5–205 | DOI | MR | Zbl

[8] G. Horrocks, “Projective modules over an extension of a local ring”, Proc. London Math. Soc. (3), 14:4 (1964), 714–718 | DOI | MR | Zbl

[9] B. L. van der Varden, Algebra, Mir, M., 1976, 648 pp. ; B. L. van der Waerden, Algebra, v. I, Heidelberger Taschenbücher, 12, 8. Aufl., Springer-Verlag, Berlin–New York, 1971, ix+272 pp. ; v. II, Heidelberger Taschenbücher, 23, 5. Aufl., 1967, x+300 pp. | MR | Zbl | Zbl | MR | Zbl

[10] C. S. Seshadri, “Triviality of vector bundles over the affine space $K^2$”, Proc. Nat. Acad. Sci. U.S.A., 44 (1958), 456–458 | DOI | MR | Zbl

[11] D. Quillen, “Projective modules over polynomial rings”, Invent. Math., 36 (1976), 167–171 | DOI | MR | Zbl

[12] A. A. Suslin, “Projective modules over a polynomial ring are free”, Soviet Math. Dokl., 17:4 (1976), 1160–1164 | MR | Zbl

[13] A. L. Smirnov, “Vektornye rassloeniya na $\mathbf P^1_\mathbb Z$ s prostymi podskokami”, Voprosy teorii predstavlenii algebr i grupp. 30, Zap. nauch. sem. POMI, 452, POMI, SPb., 2016, 202–217