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Non-linear Extension of Linear Modulesover Exterior AlgebrasCandidate Jiao FuSupervisor and Rank Prof. Jinyun GuoCollege Mathematics and Computational ScienceProgram Pure MathematicsSpecialization Representation Theory of AlgebrasDegree Master of ScienceUniversity Xiangtan UniversityDate June 1th, 2010BIAZCLBSBSB9A6BCC8CJBOAMA9AICAB6BNCYB1A5AFBBBWC7CPBAAKBCDDAICAAQC4D1C7BDC4BQCGDICDCHCSCNDFC8AFC2C6C7CSCNBAABA1BLA5BCBKAMATBKD9ARBVDFA1C7BBCFB2A4AIAKBCAYAFCBALBPAGDACAB6BBAQD8CJCQASB6BYCAACC7BAABC8BLA1CLAIBCC7CSCNC7BKBNCYDHBUC7DACAAKBBAQA4CUD8AQBCBKD9B1C6CVD8ARB1A1AICAB3C5DCD5C5AICYB1C7CRAIARABA3AICABFC0A1C8AZBTB2A5 CDBOA5 BF AI CDBSB9A6BCCAAHAQC4AVAHAWAICPBAAKBCC8AZB3C5A5C7CPC6A4A3ACABA0D6A1CPBAAKBCC7A7CEA4AUDCCPC6ACABAXC2AABJA4A3B0ATB6B8DIA9BWAKBCC7D3DGBSAKCCC1A9A4AMCMAKBCAHB5AKAKC9AKA1AICADHC4BYAJBXCPD0D9BUAICPBAAKBCC7C5B0B6B0CZBBCFAOCHA4A3A1CRD3CDCHBO?A4D0D9B2A1DIDGA0ADDGB6CLAYC8D3BIDFCKACBUAKB3AOAICPBAAKBCA1CSAVAKBCA7CPC6A7CEBNDBA1C8AZBTB2A5 CDBOA5 BF AI CDC4D1BTB2A5 CDBOA5 BF AI CD? A3CB BX........................................................IAbstract.....................................................IICU1CF C3 BV.............................................1CU2CF C7CCCNAP.........................................7CU3CF A0CIADCSCEASDECK. ............................16CU4CF A0CIADCSB3D0. ................................. 23CU5CF CUDBD4CDB6. ....................................29CHDGBDBG.................................................. 30CR BM...................................................... 33CA BWB2BYA1DDCEDDAQD2DAC2A4D2BN V COC7D2DACYB7BNCYC7BYA1, B2BYA1BCBPCOC7B4CSA4ANBVC7DHA1AGCK. CFBFD8, CLB2BYA1BCBPCOC7B4A4D2BOA6C7CSCN, CNB4C7D7AXBEAOCLA9B4C7C6DIC7CSCNDDD2DAB7AICNA8CYB7A4C3C7BEAO.AIBCBUASD9CPB3CSA4CGBQCGD8?????????????????ab ab a... ...c ... ...... ... ...c b a?????????????????nn,??????????????ab a0 b a... ... ... ...0 0 ··· ... a0 0 ··· ··· b??????????????m1mC7BWCIB4CZATC0C7D3AOCIB8 1C7CQAXB8CIB8 nC7a,b,cCFt?BWCIB4, BPBKtDDcBKBTAQCAD2A6BKC7CHARAKD3AOCIB8 2C7CQAXB8CIB8 mC7a,bCFBAC5BWCIB4.AIBCBRCYCSCND3AOCIB8 1C7CQAXB8CIAYAUC7 a,b,cCFt?B6s?BWCIB4C7CYBWCID7AXBEAO, D9BCD3AOCIB8 2C7CQAXB8CIB8 mC7 a,bCFBAC5BWCIB4ABD3AOCIB81C7CQAXB8CIB8 nC7a,b,cCFt?BWCIB4C7CYBWCID7AXBEAO.BCAWCCB2A1ASD9CPB3C7CVCR. BFAUDGBRBFABCYBWCID7AXB4C7ASD9CPB3, C8ARALAKBPAUDIC7ASBS. CEDB 3.1B7B1A5D3AOCIB8 1C7CQAXB8CIAYAUC7 a,b,cCFt?B6s?BWCIB4BEA4BMCSC7CYBWCID7AX. CEDB 3.2 AKCEDB 4.1 CZATB7B1A5BCD3AOCIB8 2C7CQAXB8CIB8 mC7a,bCFBAC5BWCIB4ABD3AOCIB8 1 C7CQAXB8CIB8 nC7a,b,cCFt?BWCIB4C7CYBWCID7AXB4C7ASD9CPB3AKAUDIC7ASBS.D1D7CKB2BYA1,BWCIB4, CYBWCID7AX, ASD9CPB3, AUDIIAbstractExterior algebras, which are defined in a vector space V, are a class of veryimportant algebra. Exterior algebras and their modules, with strong applicationbackground. There is a series of study on exterior algebras and their modules re-cently, however, the extension of two modules is an elementary and very interestingpart of the study of modules.In this paper, we called such linear modulestheir representation matrix hasthe following ?????????????????ab ab a... ...c ... ...... ... ...c b a?????????????????nn,??????????????ab a0 b a... ... ... ...0 0 ··· ... a0 0 ··· ··· b??????????????m1mt-linear module of type a,b,c and of complexity one with cyclic length m,in which, t is row mark of the first column, in which c appears; and minimal linearmodule of type a,b and of complexity two with cyclic length n respectively.In this paper, we make efforts to study on the non-linear extension of two t?or s? linear modules of complexity one and of type a,b,c with different cycliclength, and the non-linear extension of two linear modules –one is minimal linearmodule of complexity two and of type a,b with cyclic length m, another is t?linear module of complexity one and of type a,b,cwith cyclic length n.We still apply the of representation matrix in this paper. Firstly,we calculate the representation matrix of non-linear extension modules, and thusdiscuss terms of isomorphism. Theorem 3.1 prove that there is only trivial non-linear extension between t?ors?linear modules of typea,b,c and of complexityone with different cyclic length respectively. Theorem 3.1 and theorem 4.1 proveIIthat representation matrix and isomorphism terms of nonlinear extension modulesbetween t? linear module of complexity one and of type a,b,c with cyclic lengthn.Keywords Exterior algebra; Linear module; Non-linear extension; Represen-tation matrix; IsomorphismIIICTBZCE C2 BUB2BYA1DD GrassmannAQ19DBBH 40BFBYCQBTC7CEDDAQD2DAC2A4D2BN VCOC7BYA1,D0B9B8BWBVBYA1. CUVDDACkCOC7C2A4D2BN, TV k⊕V ⊕V ?V⊕···DDVC7AXA4BYA1, C2A4D2BNVCOC7B2BYA1CEDDB8 Λ ΛV TV/I,BPBKI B8A3{x?x|x ∈ V}AFCZBAC7DBBZ.B2BYA1DDD2DAA4BVD3AOCIC7C2CHCR Koszul BYA1. 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CRDID2BNCOC7BICQB4CTBIDDBYA1BEALC7BNCYCSCNCLC3, BICQB4C7C4BKCTBIABAFA5BTC7B3B4BEALCEBM [5].1978BFBerstein?Gel′fand?Gel′fandCOB7B1A5CRDID2BNCOBICQB4C7A4C8C4BKCTBIDbcohPm?1ABC2CHCRBYA1COC7A4BVCZBACZBRB4C7BDCECTBIgrmod∧V C7C8BM,BBC1BUB2BYA1ABBYA1BEALC7CSCNA0BOBRD8 [6]. Eisenbud, FloystadAKSchreyerAQBCBU [7]BKA1BGGCLDHBUB2BYA1COC7CZBRB4ABCRDID2BNCOC7B4A0BOBRD8, BSCNDCBKA5CRDID2BNCOB4C7 Beilinson monadC7CSAQDIAR,D9BCB4C7COAUCDC7BFABC7A4C7CVCR, AXBYA1B0BMCVCRCSCNA5B2BYA1COC7BAC5C2A3CZC7. Eisenbud AK FloystadCSCNA5CRDID2BNCOC7BICQB4COC7A4C8D3CGABBP Koszul CLBJ - B2BYA1COC7BAC5C7BGBVC2A3A2CZC7BABNC7 BGG CLDHC7CSAQCGD8. B7A9B0C8CSCND0C4BKA5CRDID2BNCOB4C7COAUCDBFABC7A4C7CVCR, CNBYB7D4A5BCBU [7] BKCSBCC5C7CRDID2BNCOB4C7BeilinsonmonadC7BUAQCIABCSAQC7DIARCVCR.1B2BYA1BCBPCOC7B4C7CSCNAQBTBYA1CPC7DBDACZB8CG A1CPBIDBA0B7CZBEALAKB1BNCPD9BCDFBDBIDBA0DJCPC8CMCMA9ACCFC6C5A5A6CUCNCWCHC7DHA1, CGBSB2C7de.Rham C7COAUCDDBAK, B7CZCGD8C7BFABAKCSCN, AXA4CZBK, CPB3C7CHA6D8C7BFABC8C8 [8,9,10]. B2BYA1D0AQBYA1BEALA0A4C1C7C8BUCTC7A1CPA9ACCQB0BNCYC7C8A1[11,12,13,14]. CFBFB2BYA1C7DHA1CXAXAQDBDAA9AC CG [15]BKA9BOAOC9BTB2BYA1C7CVCRDCBKA5BRAYAQA4ASBWD8C7BOAVCGD8C7B0C4, BCBRAYAQA4C4A1C7BBC1B7B1; AnAK Tu AQ [16] BKA1B2BYA1D8CSCNADCGCOC7AQCZBEAO. B0C8DFC8CFASB1B2BYA1DDD2DACYB7BNCYC7BYA1.B2BYA1DBAKCOC7CSCND0B8B5C8A1CPBCBIDBDBAKANDGA5D2BMCCDJA0BPC5C7BFABCVCR. AQBCBU [17] BK, B5DGB0A1B2BYA1C7CVCRD8BNDBCHA6D8DBAK, B0AJD9C8BYA1BKBOAVC7BNDBCVCRDEBPC5. CNAQB2APC8AQBCBU [18] BK, A1B2BYA1C7CVCRBNDBA5Laplace CEDBC8D2C8BEAO. DCADA4AQBCBU [19] BKA1B2BYA1C7CVCRDCBKA5D2BMCCC7CCAGB6ALCZBKABCR. B0BMC7CRBKCVA5C4AFC7AZCM, AMAWA5COBAAF,AYB2B6CZARCH,BSCNC4D0A5BFABB8C7BUBUA1A4. B0C8CFASB1A4ALCYCLB2BYA1BCBPCOC7B4C8BOAVC7CSCN,BSCNB8B5C8A9ACC8D1BJ.ACC8B2BYA1BCBPB4AQDBDAA9ACDHA1A6CU, C2BPASD9C7CSCNAXAYCMBQ, D0BCDDDEB8C3D2BNB9A1 ≥ 3D3, B0DABYA1DDAFBBC7A2CZA3ASD9CFBYA1 [1]. CFBFD8, Eisenbud[20] AKA9C8CA [21] CLB2BYA1COBPBOB4CDCHA5CSCN, A9C8CABRCYDDBSD3AOCIB8 1 C7KoszulB4C7BXCICDCHCSCN, B0A6A5 tameCFD6BOBYA1C7 tubeDBAK.B0C8DFC8C4BGCLD3AOCIB8 1 AK 2 C7B4C7D2BOA6C7CSCN, D2DAC2C8C7BEAODDCGALATACB4C7D7AXBEAOD8CSCNB2BYA1C7B4. C5CF,A9CAAKCVBVDGA1B2BYA1COASD9CPB3BXCLBPC1C7BWCIB4CSCNA5BPD7AXBEAO [22,23].AQAIBCBK, BFAUBLCE kDDD2DABYA1AKAC, VDDkCOC7 3B9C2A4D2BN, Λ ∧VDDBDVCOC7B2BYA1.AIBCCCB2A1ASD9CPB3C7CSCNCVCR. ASD9CPB3C8B8D2BMCSCNCVCR, CLA9C6CEB4C7C6DIDDD2CZBNCYC7. CJΛDDD2DABYA1, ΛCOAWCRB4C7AICRCTBIBGB8 MorphPΛ,BPCLC3B8AWCRB4BABNC7AICR f P1 → P2, A3DACLC3 f C5 f′ C7AICRB8D6C6BQAYBWAZC7AUAICL g1,g2,2P1 f→ P2↓ g1 ↓ g2P′1 f′→ P′2BPBK gi Pi → P′i,i 1,2 DD Λ? B4AUAI, A9DD Coker MorphPΛ →modΛ,f → Cokerf CEDDA5D2DAAGC1. CU C DDD2DA Λ B4, C C7BAC5AWCRASD9C7BUA3C0B8 P1f→ P0 → C → 0,ATCoker ~ C C4BKD2DAC8BM G MorphPΛ/R →modΛ,BPBKRDDMorphPΛCOC7D2DAA3BO [24]C7CAIVAWC7B3AO 1.2. C2CEC2A3B4C7B7, ATAICR f CLDHA9D2DAAFAADJA9 ΛC7CPB3 [30]. B7A9BQ, CSCNBYA1COC7B4D0D9BXAWB8CSCNBPAFCLDHC7ASD9CPB3. B0DACPB3DBD0CWBADDAWCR?C7D8BKAWCRB4C7CAD2DAA0CRCLDHC7CPB3.ABBQBXA3C7D2DABEAODDA3DAB4ALD3DDAUDIC7.CGABBWCIB4M,LC7BAC5?AWCRCZC7CZATB8···PtM ft→ ··· → P1M f1→ P0M f0→ M → 0···PtN ft→ ··· → P1N f1→ P0N f0→ N → 0N1,N2 DDM ABLC7CYBWCID7AXB4. C3N1 ~ N2 D3,ATA4BQAYBWAZ··· ?12N→ P11McircleplustextP11L ?11N→ P10McircleplustextP10L ?10N→ N1 → 0↓ h2 ↓ h1 ↓ g··· ?22N→ P21McircleplustextP21L ?21N→ P20McircleplustextP20L ?20N→ N2 → 0B0D3h1,h2 DBCLDHΛCOC7D0BECPB3 H1,H2.B4C7D7AXBEAODDD2DAA4C3C7BEAO, AHABAUCDBYA1, A6DDC4C1C8A4C0AVBXC7A0BO[24,25]. CLA9tameCFD6BOBYA1, DCCEC7CLDDCLDHD2DA P1 BTAYD0CZC7B4, BPB7B1COAGA9CL Kronecker BYA1COC7C1B4D7AX.B4BUAPAQ [26]BKCSCNA5D9 a,b,cB8D2C4B7C7 3B9D2BNCOD3AOCIB8 1C7AYD0CZC7BWCIB4C7ASD9CPB3. BPBKA4D2DAB4C7ASD9CPB3CSA4CGBQCGD83???????????????????????ab a0 b a... ... ... ...0 ... ... ... ...c 0 ... ... ... ...0 c ... ... ... ... ...... ... ... 0 ··· 0 b a0 ··· 0 c 0 ··· 0 b a???????????????????????nnB0CXC7BWCIB4BFAUC0C7CQAXB8CIB8 nC7D3AOCIB8 1C7a,b,cCFt?BWCIB4, BPBKtDDcBKBTAQCAD2A6BKC7CHAR.A9AQ[3]BKDFCHA5D3AOCIB8 2C7BAC5BWCIB4, BPASD9CPB3CSA4BQA6CGD8??????????????ab a0 b a... ... ... ...0 0 ··· ... a0 0 ··· ··· b??????????????m1mB0CXC7BWCIB4BFAUC0C7CQAXB8CIB8 mC7D3AOCIB8 2C7a,bCFBAC5BWCIB4.CGAB 0 → M → N → L → 0B5AMBY N DDBWCIB4, B9N DDM C9BTA9 LC7D2DAD7AXBWCIB4. D0ATB9BAB8D2DACYBWCID7AX.A3AMAPDFDBBFAUD0D9A8B4C7D7AXBEAOA1ASD9CPB3BKD9AYA0. CGAB 0 → M →N → L → 0 DDB5AMA6, A,B CZATDD M,L C7ASD9CPB3, AT N C7ASD9CPB3CSA4?? A OD B??C7CGBZ,CLD C7CSCND0D9D1AVD7AXB4 N.AIBCCSCNA5A3DABEAO D2DADDD3AOCIB8 1C7CQAXB8CIAYAUC7 a,b,cCFt?B6s? BWCIB4C7CYBWCID7AXBEAO; D2DADDD3AOCIB8 2 C7CQAXB8CIB8 m C7 a,b CFBAC5BWCIB4ABD3AOCIB8 1 C7CQAXB8CIB8 n C7 a,b,c CF t?BWCIB4C7CYBWCID7AXAKAUDIBEAO.CJCU N DDA3DABWCIB4C7CYBWCID7AXB4, AT N C7ASD9CPB3B8?? A OD B??.BF4
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