Table of contents for Threading homology through algebra : selected patterns / Giandomenico Boffi, David A. Buchsbaum.


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I   Recollections and Perspectives                                     1
1.1  Factorization                                                 1
1.1.1  Factorization domains
I1.2   Polynomial and power series rings                      6
1.2  Linear algebra                                                8
1.2.1  Free modules                                           8
1.2.2  Projective modules                                    13
1.2.3  Projective resolutons                                 17
1.3  Mutilinear algebra                                           21
1.31   R[KXI, .. , Xt] as a symmetric algebra                22
1.3.2  The divided power algebra                             28
1.3.3  The exterior algebra                                  30
fI Local Ring Theory                                                  37
11.1 Koszul complexes                                             38
11.2 Local rings                                                  43
11.3 HIilbert- Samuel polynomils                                  46
S1.4 Codimension and finitistie global dimension                  50
11.5 Regular local rings                                          54
11 6 Unique factorization                                         56
II.7 Multiplicity                                                 59
11.8 Intersection multiplicity and the homological conjectures    64
III Generalized Koszul Complexes                                      69
l1.l A few standard complexes                                     69
II. 11 The graded Koszul complex and its "derivatives'       70
II. 1.2 Definitions of the hooks and their explicit bases    72
IIL 2 General setup                                               80
111.2.1 The fat complexes                                    82
111.2.2 Slimming down                                        83
III.3 Families of complexes                                       85
111.3.1 The "homothety homotopy"                             88
111.3.2 Conmparison of the fat and s!im complexes            91
III.4 Depth-sensitivity of T(q; f)                                94
III.5 Another kind of multiplicity                                99
IV Structure Theorems for Finite Free Resolutions                    103
IV 1 Some criteria for exactness                                 104
IV.2 The first structure theorem                                 110
IV.3 Proof of the first structure theorem                      115
SV.3.1 Part (a)                                           115
IV.3.2 Part (b)                                           118
IV.4 The second structure theorem                              119
V    Exactness Criteria at Work                                     127
V.1   Pfafian ideals                                           128
V. 1  Pfaffians                                          128
V ..2 Resolution of a certain pfafian ideal              131
V.1.3 Algebra structures on resolutions                   132
V1 4 Proof of Part 2 of Theorem V.1.8                    134
V.2   Powers of pfaffian ideals                                136
V,2.1 Intrinsic description of thematrix X                137
V.2.2 Hooks again                                         138
V.2.3 Some reprsentation theory                           139
V.2.4 A counting argunent                                140
V.2.5 Description of the resolutions                      143
V.2 6 Proof of Theorem V.2.4 1                              5
VI Weyl and Schur Modules                                           149
VI,  Shape matrices and tableaux                              149
VI. 1.1 Shape matrices                                   149
V1I.12 rableaux                                           153
V .2 Weyl and Schur modules associated to shape matricns       154
1VI.  Letter-place algebra                                     156
V1.3.1 Positive places and the divided power algebra      156
V1.3.2 Negative places and the exterior aebra             159
VI.3 3 The symnmeric algebra (or negative letters and plces)  164
VI.3.4 Putting it all together                            li1
VI.   Placw polarization rmaps and C(pelIi identtities         t5
VI.5 Weyl and Schur maps revsited                              167
VI 6 Some kernel elements of Weyl and Schur maps               169
VI.7 Tahbleax, straightening, and the straight basis theorem   17
V17.1 'ableaux for -Weyl and Schur modules                174
V1 7.2 Straightening tableaux                            176
VI.7 3 T  lor-mnade tableaux, or a straightflling algorithm  81
VI,7.4 Proof of linear independence of straight tableaux  18
VI.7.5 Modifications for Schur modules                    1(6
VI ..6 Duality                                            187
1V.8  Weyl-Schur complexes                                     187
VII Some Applications of Weyl and Schur Modules                     193
VII. 1 The fundamental exact sequence                           93
VII.2 Direct surs and iltrations for skew-shapes               197
V11.3 Resolution of determinantal ideals                       199
VII3.1 The Lascoux resolutions                             200
VII.3.2 The submaximal minors                              202
VII.3.3 Z-forms                                            203
VII.4 Arithmetic considerations                                 206
VII.4.1 Intertwining numbers                               206
V11.4,2 Z-forms again                                      208
VII.5 Resolutions revisited; the Hashimoto counterexample       209
VII.6 Resolutions of  eyl modules                               211
VII.6.1 The bar complex                                    212
VII.6.2 The two-rowed case                                 215
VII.6.3 A three-rowed example                              217
VII.6.4 Resolutions of ske-hooks                           225
VII .65 Comparison with the Lascoux resolutions            227



Library of Congress subject headings for this publication: Algebra, Homological