Quadratic forms of projective spaces over rings

V. M. Levchuk   ; O. A. Starikova

Sbornik. Mathematics, Tome 197 (2006) no. 6, pp. 887-899

Résumé

In the passage from fields to rings of coefficients quadratic forms with invertible matrices lose their decisive role. It turns out that if all quadratic forms over a ring are diagonalizable, then in effect this is always a local principal ideal ring $R$ with $2\in R^*$. The problem of the construction of a ‘normal’ diagonal form of a quadratic form over a ring $R$ faces obstacles in the case of indices $|R^*:R^{*2}|$ greater than 1. In the case of index 2 this problem has a solution given in Theorem 2.1 for $1+R^{*2}\subseteq R^{*2}$ (an extension of the law of inertia for real quadratic forms) and in Theorem 2.2 for $1+R^2$ containing an invertible non-square. Under the same conditions on a ring $R$ with nilpotent maximal ideal the number of classes of projectively congruent quadratic forms of the projective space associated with a free $R$-module of rank $n$ is explicitly calculated (Proposition 3.2). Up to projectivities, the list of forms is presented for the projective plane over $R$ and also (Theorem 3.3) over the local ring $F[[x,y]]/\langle x^{2},xy,y^{2}\rangle$ with non-principal maximal ideal, where $F=2F$ is a field with an invertible non-square in $1+F^{2}$ and $|F^{*}:F^{*2}|=2$. In the latter case the number of classes of non-diagonalizable quadratic forms of rank 0 depends on one's choice of the field $F$ and is not even always finite; all the other forms make up 21 classes. Bibliography: 28 titles.

Bibliographie

[1] E. Artin, Geometricheskaya algebra, Nauka, Fizmatlit, M., 1969 | MR | Zbl

[2] A. I. Maltsev, Osnovy lineinoi algebry, Nauka, Fizmatlit, M., 1970 | MR | MR | Zbl

[3] W. Scharlau, Quadratic and Hermitian forms, Grundlehren Math. Wiss., 270, Springer-Verlag, Berlin, 1985 | MR | Zbl

[4] T. Y. Lam, The algebraic theory of quadratic forms, Benjamin/Cummings Publ., Reading, MA, 1980 | MR | Zbl

[5] R. Baeza, “The norm theorem for quadratic forms over a field of characteristic 2”, Comm. Algebra, 18:5 (1990), 1337–1348 | DOI | MR | Zbl

[6] Z. I. Borevich, I. R. Shafarevich, Teoriya chisel, Nauka, Fizmatlit, M., 1972 | MR | Zbl

[7] B. A. Venkov, Izbrannye trudy. Issledovaniya po teorii chisel, Nauka, L., 1981 | MR | Zbl

[8] E. B. Vinberg, “Ob unimodulyarnykh tselochislennykh kvadratichnykh formakh”, Funkts. analiz i ego prilozh., 6:2 (1972), 24–31 | MR | Zbl

[9] D. Kassels, Ratsionalnye kvadratichnye formy, Mir, M., 1982 | MR | Zbl

[10] W. Benz, Vorlesungen über Geometrie der Algebren, Springer-Verlag, Berlin, 1973 | MR | Zbl

[11] V. V. Vishnevskii, B. A. Rozenfeld, A. P. Shirokov, “O razvitii geometrii prostranstv nad algebrami”, Izv. vuzov. Ser. matem., 1984, no. 7, 38–44 | MR | Zbl

[12] V. V. Vishnevskii, A. P. Shirokov, V. V. Shurygin, Prostranstva nad algebrami, Izd-vo Kazanskogo un-ta, Kazan, 1985 | MR | Zbl

[13] M. A. Marshall, Bilinear forms and orderings on commutative rings, Queen's Papers in Pure and Appl. Math., 71, Queen's Univ., Kingston, ON, 1985 | MR | Zbl

[14] Yu. I. Merzlyakov (red.), Avtomorfizmy klassicheskikh grupp, Sb. per., Mir, M., 1976 | MR

[15] M. Ojanguren, R. Sridharan, “A note on the fundamental theorem of projective geometry”, Comment Math. Helv., 44 (1969), 310–315 | DOI | MR | Zbl

[16] R. Baeza, Quadratic forms over semilocal rings, Lecture Notes in Math., 655, Springer-Verlag, Berlin, 1978 | MR | Zbl

[17] R. Baeza, “On the classification of quadratic forms over semilocal rings”, Colloque sur les Formes Quadratiques, 2 (Montpellier, 1977), Bull. Soc. Math. France Mém., 59, 1979, 7–10 | MR | Zbl

[18] J.-L. Colliot-Thélène, “Formes quadratiques sur les anneaux semi-locaux réguliers”, Colloque sur les Formes Quadratiques, 2 (Montpellier, 1977), Bull. Soc. Math. France Mém., 59, 1979, 13–31 | MR | Zbl

[19] J. L. Yucas, “A classification theorem for quadratic forms over semilocal rings”, Ann. Math. Sil., 2:14 (1986), 7–12 | MR | Zbl

[20] D. R. Estes, R. M. Guralnick, “A stable range for quadratic forms over commutative rings”, J. Pure Appl. Algebra, 120:3 (1997), 255–280 | DOI | MR | Zbl

[21] Dzh. Milnor, D. Khyuzmoller, Simmetricheskie bilineinye formy, Nauka, Fizmatlit, M., 1986 | MR | MR | Zbl

[22] O. A. Starikova, “Perechislenie kvadrik proektivnykh ploskostei i prostranstv nad lokalnymi koltsami glavnykh idealov”, Algebra i teoriya modelei, no. 4, NGTU, Novosibirsk, 2003, 110–115

[23] V. M. Levchuk, O. A. Starikova, “Kvadriki i simmetrichnye formy modulei nad lokalnym koltsom glavnykh idealov”, Tezisy dokladov “Mezhdunarodnaya algebraicheskaya konferentsiya”, MGU, M., 2004, 86–88

[24] O. A. Starikova, Perechislenie kvadrik i simmetrichnykh form modulei nad lokalnymi koltsami, Dis. ... kand. fiz.-matem. nauk, Severnyi mezhdunarodnyi universitet, Magadan, 2004

[25] M. Kholl, Kombinatorika, Mir, M., 1970 | MR | MR | Zbl

[26] B. L. Van der Varden, Algebra, Nauka, Fizmatlit, M., 1979 | MR | Zbl

[27] W. Brown, Matrices over commutative rings, Monogr. Textbooks Pure Appl. Math., 169, Marcel Dekker, New York, 1993 | MR | Zbl

[28] E. A. Hornix, “Totally indefinite quadratic forms over formally real fields”, Nederl. Akad. Wetensch. Indag. Math., 47:3 (1985), 305–312 | MR | Zbl