Grothendieck, A., Dieudonne, J.A.: Elements de geometrie algebrique,

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1 References [Be70] [Bo61] [Bru85] [D-C70] [Del71] [Do72] [G-D64] [G-D71] [Gr60] [H-082] [Ha66] [Ha77] [He80] [He80'] [He82] [He83] [He90] [He91] [He92] [Hi70] [Ja90] [Ja93] Bennett, B. M.: On the characteristic functions of a local ring, Ann. Math. 91 (1970) Bourbaki, N.: Algebre commutative, Hermann, Paris Brundu, M.: On tangential flatness, Commun. Algebra 13 (1985) Dieudonne, J.A., Carrel, T.: Invariant theory, old and new, Advances in Math. 4 (1970) Deligne, P.: Theorie de Hodge II, Pub!. Math. IRES 40 (1971), 5-58 liojiraueb, 11. B. :A6cTpaKTHaR anrefipaauecxaa reoxrerpa.a, 11TOrl1 nayxn, Anrefip a Feowe-rpna TOIIOJIOrl1R, T.10 (1972) CTp Grothendieck, A., Dieudonne, J.A.: Elements de geornetrie algebrique V t-, Pub!. Math. IRES No. 20, Paris 1964 Grothendieck, A., Dieudonne, J.A.: Elements de geometrie algebrique, Springer-Verlag, Berlin 1971 Grothendieck, A.: Techniques de construction et theorernes d' existence en geometrie algebrique IV: Les schemes de Hilbert, Seminaire Bourbaki 221 (1960/61) Herrmann, M., Orbanz, U.: Two notes on flatness, Manuscripta Math. 40 (1982) Hartshorne, R.: Connectedness of the Hilbert scheme, Pub!. Math. IRES 29 (1966) 5-48 Hartshorne, Algebraic geometry, Springer-Verlag, New York 1977 Herzog, B.: Die Wirkung lokaler Homomorphismen auf die Hilbert function, Math. Nachr. 97 (1980) Herzog, B.: On the Macaulayfication of local rings, J. Algebra 67 (1980) Herzog, B.: On a relation between the Hilbert functions belonging to a local homomorphism, J. London Math. Soc. 25 (1982), Herzog, B.: A criterion for tangential flatness, Manuscripta Math. 43 (1983), Herzog, B.: Hironaka-Lech inequalities for flat couples of locals rings, Manuscripta Math. 68 (1990), Herzog, B.: Local singularities such that all deformations are tangentially flat, Trans. Amer. Math. 324 (1991), Herzog, B.: Tangential flatness for filtered modules over local rings, Reports, Department of Mathematics, University of Stockholm 4 (1991) vi+113 pp Hironaka, H.: Certain numerical characters of singularities, J. Math. Kyoto Univ. 10 (1970), J ahnel, J.: Tangentiale Flachheit und Lech-Hironaka-Ungleichungen, Promotionsschrift Jena 1990 J ahnel, J.: Lech's conjecture on deformations of singularities and second Harrison cohomology, to appear in J. London Math. Soc. 171

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3 Index F-distinguished basis 59 FA-filtration 9 FA-minimal 62 with respect to (b, d) 62 I-adic filtration 3, 10 q-adic filtration 4 P-distinguished basis 59 Artin-Rees filtration 10 algebra generators, homogeneous 115 associated graded ring 2 base change 145 change homomorphism 145 basic exact sequence 42 closure 14 cofinite 3 compatible filtrations 9 complete 13 completion 13 contained in a filtration 4 critical 163 definable over a subalgebra 111 defined over a subalgebra linear map 111 module 111 defining equations of the deformation space 153 of the fibre 150 defining exact sequence for F7:r/\ 14 for cl(n) 14 for in(i) 9 defining ideal 103 submodule 111 deformation x embedded germ 145 germ ix space, defining equations 153 deformations embedded of first order 157 functor of embedded 157 degree 1 degreewise lexicographic order 148 derivations 103 direct image filtration 7, 9 distinguished basis 59 embedded deformation functor 157 deformation germ 145 deformations 157 first order deformations 157 Kodaira-Spencer map 106 embedding of a permissible graded homomorphism 106 exact sequence associated with a family 33 basic 42 defining for F7:r/\ 14 defining for cl(n) 14 defining for in(i) 9 exponential Hilbert series 133 family associated exact sequence 33 generating a filtration 7 of structure constants 118 filtered module 8 module homomorphism 8 ring 2 filtration 2, 8 q-adic 4 contained in another one 4 direct image of 7, 9 generated by a family 7 ideal-adic 3, 10 induced on a factor ring 7 induced on a subring 6 inverse image of 6, 9 minimal 62 quasi-homogeneous 4 weight 4 finite Hilbert function 76 flatifying filtration 91 prefiltration 91 formal degree 163 generating a filtration 7 graded module 1 homomorphism 2 173

4 with finite Hilbert function 76 graded ring 1 homomorphism 2 graduationally flat x Hausdorff 3 Hilbert scheme 143 point functor 143 Hilbert series exponential 133 of a filtered ring 76 of a graded module 76 of a homomorphism 77 Hilbert-Samuel series of a filtered ring 77 of a graded module 76 of a homomorphism 77 homogeneous 1 algebra generators 115 syzygy 31 homomorphism of embedded deformation germs 145 of filtered modules 8 of filtered rings 3 of the associated graded rings 3 ideal-adic filtration of a module 10 of a ring 3 inclusion of filtrations 4 induced 6, 7 filtration on a factor ring 7 filtration on a subring 6 initial element 9 form 9, 148 ideal 9 intersection of filtrations 4 inverse image 9 image filtration 6, 9 irrelevant entry 163 ideal 1 Kodaira-Spencer map 108 map, embedded 106 Krull dimension 77 Lech's problem 78 liftable to a syzygy 31 local singularity x minimal 115, 116 flatifying prefiltration 93 over a filtration 62 w.r.t. a distinguished basis 62 module filtration 8 monomial lift 118 module generators 116 multiplicity 77 natural filtration 3 topology 3 nice distinguished basis 117 non-standard degree 151 normal module 103 normally flat 101 order degreewise lexicographic 148 in a filtered module 9 in a filtered ring 2 total 76 permissible graded algebra 101 homomorphism 101 homomorphism, embedding of 106 prefiltration 39 quasi-homogeneous filtration 4 Russian Conjecture 141 refinement 14, 21 residually rational xvi separable xvi ring filtration 2 Schlessinger's T separated 3 shifting degrees 2 special fibre x, 7 standard basis 30 monomial 148 strict 174

5 homomorphism 32 system of generators 30 stronger 21 structure constants 66, 117, 150 sum of filtrations 5 transform 76, 77 syzygy 31 tangent space 157, 158 tangentially flat x, 3 total order 76 underlying lifts 117 system of algebra generators 117 system of module generators 117 vector space basis 117 universal germ 147 weaker 21 weakly graded 102 weight filtration 4 Zariski tangent space 157,

6 Formula Index rl(a) xvi Ass 153 AnF B 6 MA(M) xvi < a,b > xvi < a,b > 31 < a, b > 153 B(A) 31 <C xvi c 152 cl(n) xvi, o, 103 dass 153 deg 1 nf 3 F'iJ xvi FrS(X, y) 153 FA 2 Fi(X, y) 153 nf M xvi, 10 prs(x) 150 Pi(X) 150 i' ---> 1" 145 i.f M 9 f*(f B ) 6 f*f M, 8 G(d) 1 G+ 1 G(A) 2 G(f) 3 G(M) xvi, 10 G(M)(d) 10 G(M)(2 e) 10 G(M)(::::; e) 10 GF A (M) 10 GF(A) 2 GF(M) xvi Gr(M) Hilbx/s 143 lhi xvi, 144 HM xvi HM xvi i, 103 in(m) 9 im 13 KE; 159 LA(M) xvi M(d) xvi, 1, 10 M(? e) xvi M(::::; e) xvi M[e] 2 M(A) xvi M/\ xvi M F 127 m(a) xvi (N I mod N 2 ) xvi s, 103 Nxvi nsdeg 151 ord9 ordm(x) xvi lp'(g) 116 xvi Q xvi lr xvi RB(X) xvi, 31 TC,Bo 106 Tb 103 1I.J ---> lhi xvi : 1I.J := x K lhi ---> lhi 144 1I.Jo 144 x 153 y 153 Z xvi 176

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