Finite determinacy of matrices over local rings. tangent modules to the miniversal deformation for \(R\)-linear group actions. (English) Zbl 1406.58022
The results of the previous authors’ paper [C. R. Math. Acad. Sci., Soc. R. Can. 38, No. 4, 113–153 (2016; Zbl 1359.58020)] reduce the study of (in)finite determinacy to the computation of the support of \(T^1_{(\Sigma,G,A)}\), i.e. the annihilator ideal \(\mathrm{ann}(T^1_{(\Sigma,G,A)})\). Here \(\mathrm{Mat}_{m\times n}(R)\) denotes the \(R\)-module of \(m\times n\) matrices with entries in a local ring \(R\), \(G\) is a group acting on \(\mathrm{Mat}_{m\times n}(R)\), and \(\Sigma \subset \mathrm{Mat}_{m\times n}(R)\) is a subset of allowed deformations.
In the current paper, the authors study the module \(T^1_{(\Sigma,G,A)}\) and its support for the \(R\)-linear actions of the group \(G\) on \(\mathrm{Mat}_{m\times n}(R)\) described in Example 1.1. Some ready-to-use criteria of determinacy for deformations of (embedded) modules, (skew-)symmetric forms, filtered modules, filtered morphisms of filtered modules, chains of modules and others, are obtained.
In the current paper, the authors study the module \(T^1_{(\Sigma,G,A)}\) and its support for the \(R\)-linear actions of the group \(G\) on \(\mathrm{Mat}_{m\times n}(R)\) described in Example 1.1. Some ready-to-use criteria of determinacy for deformations of (embedded) modules, (skew-)symmetric forms, filtered modules, filtered morphisms of filtered modules, chains of modules and others, are obtained.
Reviewer: Dorin Andrica (Riyadh)
MSC:
| 58K40 | Classification; finite determinacy of map germs |
| 58K50 | Normal forms on manifolds |
| 32A19 | Normal families of holomorphic functions, mappings of several complex variables, and related topics (taut manifolds etc.) |
| 14B07 | Deformations of singularities |
| 15A21 | Canonical forms, reductions, classification |
Citations:
Zbl 1359.58020References:
| [1] | Ahmed, I.; Soares-Ruas, M. A., Determinacy of determinantal varieties · Zbl 1423.32025 |
| [2] | Arnol’d, V. I.; Goryunov, V. V.; Lyashko, O. V.; Vasil’ev, V. A., Singularity theory. I, (1998), Springer-Verlag Berlin, iv+245 pp., Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [Dynamical systems. VI, Encyclopaedia Math. Sci., 6, Springer, Berlin, 1993] · Zbl 0901.58001 |
| [3] | Beasley, L. B.; Laffey, T. J., Linear operators on matrices: the invariance of rank-k matrices, Linear Algebra Appl., 133, 175-184, (1990) · Zbl 0721.15006 |
| [4] | Belitski, G.; Kerner, D., Group actions on filtered modules and finite determinacy. finding large submodules in the orbit by linearization, C. R. Math. Rep. Acad. Sci. Can., 38, 4, 113-155, (2016) · Zbl 1359.58020 |
| [5] | Belitski, G.; Kerner, D., Finite determinacy of matrices over local rings. II. tangent modules to the miniversal deformations for group-actions involving the ring automorphisms |
| [6] | G. Belitski, D. Kerner, in preparation.; G. Belitski, D. Kerner, in preparation. |
| [7] | Mac Lane, S.; Birkhoff, G., Algebra, (1988), Chelsea Publishing Co. New York, xx+626 pp · Zbl 0641.12001 |
| [8] | Bruce, J. W., On families of symmetric matrices. dedicated to vladimir I. Arnold on the occasion of his 65th birthday, Mosc. Math. J., 3, 2, 335-360, (2003), 741 · Zbl 1054.15012 |
| [9] | Bruce, J. W.; Goryunov, V. V.; Zakalyukin, V. M., Sectional singularities and geometry of families of planar quadratic forms, (Trends in Singularities, Trends Math., (2002), Birkhäuser Basel), 83-97 · Zbl 1018.32026 |
| [10] | Bruce, J. W.; Tari, F., On families of square matrices, Proc. Lond. Math. Soc. (3), 89, 3, 738-762, (2004) · Zbl 1070.58032 |
| [11] | Buchsbaum, D. A.; Eisenbud, D., Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Am. J. Math., 99, 3, 447-485, (1977) · Zbl 0373.13006 |
| [12] | Cutkosky, S. D.; Srinivasan, H., Equivalence and finite determinancy of mappings, J. Algebra, 188, 1, 16-57, (1997) · Zbl 0955.13010 |
| [13] | Damon, J., The unfolding and determinacy theorems for subgroups of A and K, Mem. Am. Math. Soc., 50, 306, (1984), x+88 pp. · Zbl 0545.58010 |
| [14] | Damon, J.; Pike, B., Solvable groups, free divisors and nonisolated matrix singularities I: towers of free divisors, Ann. Inst. Fourier, 65, 3, 1251-1300, (2015) · Zbl 1371.14056 |
| [15] | Damon, J.; Pike, B., Solvable groups, free divisors and nonisolated matrix singularities II: vanishing topology, Geom. Topol., 18, 2, 911-962, (2014) · Zbl 1301.32017 |
| [16] | Dieudonné, J., Sur une généralisation du groupe orthogonal à quatre variables, Arch. Math., 1, 282-287, (1949) · Zbl 0032.10601 |
| [17] | de Jong, T.; van Straten, D., A deformation theory for nonisolated singularities, Abh. Math. Semin. Univ. Hamb., 60, 177-208, (1990) · Zbl 0721.32014 |
| [18] | du Plessis, A., On the genericity of topologically finitely-determined map-germs, Topology, 21, 2, 131-156, (1982) · Zbl 0499.58007 |
| [19] | du Plessis, A., Genericity and smooth finite determinacy, (Singularities, Part 1, Arcata, Calif., 1981, Proc. Sympos. Pure Math., vol. 40, (1983), Amer. Math. Soc. Providence, RI), 295-312 · Zbl 0507.58014 |
| [20] | Eisenbud, D., Commutative algebra. with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, (1995), Springer-Verlag New York, xvi+785 pp. · Zbl 0819.13001 |
| [21] | Elkik, R., Solutions d’équations à coefficients dans un anneau hensélien, Ann. Sci. Éc. Norm. Supér. (4), 6, 553-603, (1973) · Zbl 0327.14001 |
| [22] | Frühbis-Krüger, A.; Zach, M., On the vanishing topology of isolated Cohen-Macaulay codimension 2 singularities · Zbl 1483.32015 |
| [23] | Goryunov, V.; Mond, D., Tjurina and Milnor numbers of matrix singularities, J. Lond. Math. Soc. (2), 72, 1, 205-224, (2005) · Zbl 1106.32021 |
| [24] | Goryunov, V. V.; Zakalyukin, V. M., Simple symmetric matrix singularities and the subgroups of Weyl groups \(A_\mu\), \(D_\mu\), \(E_\mu\), Mosc. Math. J., 3, 2, 507-530, (2003), 743-744 · Zbl 1040.58018 |
| [25] | Grandjean, V., Finite determinacy relative to closed and finitely generated ideals, Manuscr. Math., 103, 3, 313-328, (2000) · Zbl 0971.58023 |
| [26] | Grandjean, V., Infinite relative determinacy of smooth function germs with transverse isolated singularities and relative łojasiewicz conditions, J. Lond. Math. Soc. (2), 69, 2, 518-530, (2004) · Zbl 1054.58030 |
| [27] | Greub, W., Multilinear algebra, Universitext, (1978), Springer-Verlag New York-Heidelberg, vii+294 pp. · Zbl 0387.15001 |
| [28] | Greuel, G.-M.; Pham, T. H., Mather-Yau theorem in positive characteristic, J. Algebraic Geom., 26, 2, 347-355, (2017) · Zbl 1357.14010 |
| [29] | Haslinger, G.-J., Families of skew-symmetric matrices, (2001), The University of Liverpool United Kingdom, Ph.D. Thesis |
| [30] | Heymans, P., Pfaffians and skew-symmetric matrices, Proc. Lond. Math. Soc. (3), 19, 730-768, (1969) · Zbl 0182.05102 |
| [31] | Huneke, C.; Swanson, I., Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, (2006), Cambridge University Press Cambridge, xiv+431 pp. · Zbl 1117.13001 |
| [32] | Jiang, G., Functions with non-isolated singularities on singular spaces, (1997), Universiteit Utrecht Utrecht, Netherlands, Ph.D. Thesis |
| [33] | Kucharz, W., Power series and smooth functions equivalent to a polynomial, Proc. Am. Math. Soc., 98, 3, 527-533, (1986) · Zbl 0626.58005 |
| [34] | Mather, J. N., Stability of \(C^\infty\) mappings. VI. the Nice dimensions, (Proceedings of Liverpool Singularities—Symposium, I, 1969/1970, Lecture Notes in Math., vol. 192, (1971), Springer Berlin), 4, 207-253, (1970) · Zbl 0211.56105 |
| [35] | Li, C.-K.; Pierce, S., Linear preserver problems, Am. Math. Mon., 108, 7, 591-605, (2001) · Zbl 0991.15001 |
| [36] | Molnár, L., Selected preserver problems on algebraic structures of linear operators and on function spaces, Lecture Notes in Mathematics, vol. 1895, (2007), Springer-Verlag Berlin · Zbl 1119.47001 |
| [37] | Pellikaan, R., Finite determinacy of functions with nonisolated singularities, Proc. Lond. Math. Soc. (3), 57, 2, 357-382, (1988) · Zbl 0621.32019 |
| [38] | Pellikaan, R., Deformations of hypersurfaces with a one-dimensional singular locus, J. Pure Appl. Algebra, 67, 1, 49-71, (1990) · Zbl 0733.32026 |
| [39] | Pereira, M. S., Variedades determinantais e singularidades de matrizes, (2010), Universidade de São Paulo, USP Brasil, PhD thesis |
| [40] | Pereira, M. S., Properties of G-equivalence of matrices |
| [41] | Pham, T. H., On finite determinacy of hypersurface singularities and matrices in arbitrary characteristic, (2016), Technische Universität Kaiserslautern, PhD thesis |
| [42] | Rudin, W., Real and complex analysis, (1987), McGraw-Hill Book Co. New York, xiv+416 pp. · Zbl 0925.00005 |
| [43] | Siersma, D., Isolated line singularities, (Singularities, Part 2, Arcata, Calif., 1981, Proc. Sympos. Pure Math., vol. 40, (1983), Amer. Math. Soc. Providence, RI), 485-496 · Zbl 0514.32007 |
| [44] | Siersma, D., The vanishing topology of non isolated singularities, (New Developments in Singularity Theory, Cambridge, 2000, NATO Sci. Ser. II Math. Phys. Chem., vol. 21, (2001), Kluwer Acad. Publ. Dordrecht), 447-472 · Zbl 1011.32021 |
| [45] | Teissier, B., Variétés polaires. II. multiplicités polaires, sections planes, et conditions de Whitney, (Algebraic Geometry, La Rábida, 1981, Lecture Notes in Math., vol. 961, (1982), Springer Berlin), 314-491 · Zbl 0585.14008 |
| [46] | Thilliez, V., Infinite determinacy on a closed set for smooth germs with non-isolated singularities, Proc. Am. Math. Soc., 134, 1527-1536, (2006) · Zbl 1090.58022 |
| [47] | Tougeron, J. C., Idéaux de fonctions différentiables. I, Ann. Inst. Fourier (Grenoble), 18, 1, 177-240, (1968) · Zbl 0188.45102 |
| [48] | Wall, C. T.C., Are maps finitely determined in general?, Bull. Lond. Math. Soc., 11, 2, 151-154, (1979) · Zbl 0431.58008 |
| [49] | Wall, C. T.C., Finite determinacy of smooth map-germs, Bull. Lond. Math. Soc., 13, 6, 481-539, (1981) · Zbl 0451.58009 |
| [50] | Wilson, L.-C., Infinitely determined map germs, Can. J. Math., 33, 3, 671-684, (1981) · Zbl 0476.58005 |
| [51] | Wilson, L.-C., Map-germs infinitely determined with respect to right-left equivalence, Pac. J. Math., 102, 1, 235-245, (1982) · Zbl 0454.58005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
