\u86c7\u5f62\u5f15\u7406\u662f\u540c\u8c03\u4ee3\u6570\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u5f15\u7406, \u5728\u6784\u9020\u6b63\u5408\u540c\u8c03\u5217\u4e2d\u6709\u8d77\u5230\u91cd\u8981\u7684\u4f5c\u7528. <\/p>\n
<\/p>\n
\u8bb0\u5f97\u521d\u89c1\u8fd9\u4e2a\u5f15\u7406\u4e4b\u65f6\u89c9\u5f97\u62bd\u8c61\u65e0\u6bd4, \u65e0\u6cd5\u7406\u89e3. \u611f\u89c9\u5b8c\u5168\u4e3a\u4e86\u62bd\u8c61\u800c\u62bd\u8c61\u6ca1\u6709\u4ec0\u4e48\u610f\u4e49.<\/p>\n
\u7136\u800c\u968f\u7740\u5b66\u4e60\u4ee3\u6570\u62d3\u6251\u7684\u6df1\u5165, \u9010\u6e10\u7406\u89e3\u6b63\u5408\u5217\u7684\u5f3a\u5927, \u4ee5\u53ca\u8fd9\u4e2a\u5f15\u7406\u7684\u7528\u9014.<\/p>\n
\u63a5\u4e0b\u6765\u672c\u6587\u5c06\u4f7f\u7528\u6309\u56fe\u7d22\u9aa5\u6cd5\u8bc1\u660e\u86c7\u5f62\u5f15\u7406.<\/p>\n
Lemma. (Snake)<\/strong> $R$ \u662f\u4e00\u4e2a\u73af, $A,B,C,D,E,F$ \u662f $R$ – \u6a21, \u4e0b\u56fe\u4e2d\u7684\u6620\u5c04\u90fd\u662f $R$ – \u6a21\u540c\u6001, \u4e14\u4e0b\u56fe\u4ea4\u6362\u4e14\u884c\u662f\u6b63\u5408\u7684: \u5b9a\u4e49 $d:z\\mapsto j^{-1}\\partial p^{-1}z+\\operatorname{im}\\partial'$.<\/p>\n (\u53ef\u4ee5\u6ce8\u610f\u5230 $p$ \u4e0d\u4e00\u5b9a\u662f\u5355\u5c04, \u6240\u4ee5 $p^{-1}z$ \u53ef\u80fd\u6709\u591a\u4e2a\u5143\u7d20, \u8fd9\u91cc\u7684\u5b9a\u4e49\u8bf4\u7684\u662f\u53ef\u4ee5\u53d6\u5176\u4e2d\u4efb\u610f\u4e00\u4e2a\u5143\u7d20)<\/p>\n \u5148\u8bc1 $d$ \u662f\u826f\u5b9a\u4e49\u7684, \u8fd9\u91cc\u4efb\u53d6 $z\\in \\operatorname{ker} \\partial''$.<\/p>\n \u8fd9\u4e2a\u8bc1\u660e\u5206\u4e3a\u4e24\u90e8\u5206, \u4e00\u90e8\u5206\u662f\u8bc1\u660e $\\partial p^{-1}z\\in \\operatorname{im} j$, \u800c\u53e6\u4e00\u90e8\u5206\u662f\u8bc1\u660e\u53d6 $p^{-1}z$ \u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u5143\u7d20\u5f97\u5230\u7684\u7ed3\u679c\u90fd\u662f\u76f8\u540c\u7684.<\/p>\n \u7531\u4ea4\u6362\u56fe\u77e5 $q\\partial s=\\partial'' ps=\\partial''z=0$, \u56e0\u6b64\u518d\u7ed3\u5408\u6b63\u5408\u6027\u77e5 $\\partial s\\in \\operatorname{ker} q=\\operatorname {im} j$.<\/p>\n \u6240\u4ee5 $\\exists a\\in A,ia=s-s'$. \u7531\u6b64\u53ef\u77e5 $d$ \u662f\u826f\u5b9a\u4e49\u7684.<\/p>\n<\/li>\n \u518d\u8bc1\u660e\u6b63\u5408\u6027.<\/p>\n \u9996\u5148\u660e\u786e\u6bcf\u4e24\u4e2a\u96c6\u5408\u95f4\u7684\u6620\u5c04:<\/p>\n $i_\\# :\\operatorname{ker}{\\partial'}\\to\\operatorname{ker}{\\partial},z\\mapsto iz$;<\/p>\n $p_\\#:\\operatorname{ker}{\\partial}\\to\\operatorname{ker}{\\partial''},z\\mapsto pz$;<\/p>\n $j_\\# :\\operatorname{coker}{\\partial'}\\to\\operatorname{coker}{\\partial},z+\\operatorname{im}\\partial'\\mapsto jz+\\operatorname{im}\\partial$;<\/p>\n $q_\\#:\\operatorname{coker}{\\partial}\\to\\operatorname{coker}{\\partial''},z+\\operatorname{im}\\partial\\mapsto pz+\\operatorname{im}\\partial''$.<\/p>\n \u7531\u4ea4\u6362\u56fe\u5bb9\u6613\u9a8c\u8bc1\u4e0a\u9762\u7684\u6620\u5c04\u90fd\u662f\u826f\u5b9a\u4e49.<\/p>\n \u6b63\u5408\u6027\u7684\u8bc1\u660e\u5206\u4e3a\u56db\u4e2a\u90e8\u5206, \u5206\u522b\u8bc1\u660e\u5728\u6bcf\u4e2a $R$ – \u6a21\u5904\u6b63\u5408.<\/p>\n \u4e0b\u8bc1 $\\operatorname {im}i_\\#=\\operatorname {ker}p_\\#$.<\/p>\n<\/li>\n \u4efb\u53d6 $z\\in \\operatorname{ker}{\\partial'}$, \u6709 $p_\\#i_\\#z=piz=0\\Rightarrow i_\\#z\\in \\operatorname{ker} p_\\#$, \u6240\u4ee5 $\\operatorname {im}i_\\#\\subset\\operatorname {ker}p_\\#$.<\/p>\n<\/li>\n \u4efb\u53d6 $z\\in \\operatorname {ker} p_\\#$, \u6709 $z\\in\\ker p=\\operatorname{im} i$, \u56e0\u6b64 $\\exists s\\in A, is=z$.<\/p>\n \u53c8\u7531\u4e8e $j\\partial' s=\\partial is=\\partial z=0$ \u4e14 $j$ \u662f\u5355\u5c04, \u56e0\u6b64\u6709 $\\partial's=0$.<\/p>\n \u6240\u4ee5 $s\\in \\operatorname{ker} \\partial'$ \u4e14 $i_\\#s=z\\Rightarrow z\\in\\operatorname {im}i_\\#$.<\/p>\n<\/li>\n \u4e0b\u8bc1 $\\operatorname {im}p_\\#=\\operatorname {ker} d$.<\/p>\n<\/li>\n \u4efb\u53d6 $z\\in \\operatorname{ker}{\\partial}$, \u6709 $dp_\\#z=dpz=j^{-1}\\partial p^{-1}pz+\\operatorname{im}\\partial'=j^{-1}(\\partial z)+\\operatorname{im}\\partial'=0$.<\/p>\n<\/li>\n \u4efb\u53d6 $z\\in \\operatorname{ker} d$, \u6709 $j^{-1}\\partial p^{-1}z \\in \\operatorname{im}\\partial'$, \u6240\u4ee5 $\\exists s\\in A,\\partial' a=j^{-1}\\partial p^{-1}z$.<\/p>\n \u5219\u6709 $\\partial ia=j\\partial' a=\\partial p^{-1}z\\Rightarrow p^{-1}z-ia\\in \\operatorname{ker}\\partial$.<\/p>\n \u5219\u6709 $p_\\#(p^{-1}z-ia)=pz-pia=z$, \u56e0\u6b64 $z\\in \\operatorname {im}p_\\#$.<\/p>\n<\/li>\n \u4e0b\u8bc1 $\\operatorname {im}d=\\operatorname {ker} j_\\#$.<\/p>\n<\/li>\n \u4efb\u53d6 $z\\in \\operatorname{ker}{\\partial''}$, \u6709 $j_\\#dz=jj^{-1}\\partial p^{-1}z+\\operatorname{im}\\partial=0$.<\/p>\n<\/li>\n \u4efb\u53d6 $z+\\operatorname{im}\\partial'\\in \\operatorname{ker} j_\\#$, \u5219\u6709 $jz\\in \\operatorname{im}\\partial$.<\/p>\n \u5219 $\\exists s\\in B,\\partial s=jz$.<\/p>\n \u6240\u4ee5, $\\partial'' ps=q\\partial s=qjz=0$.<\/p>\n \u56e0\u6b64, $ps\\in \\operatorname{ker}\\partial''$, \u4e14 $dps=j^{-1}\\partial p^{-1}ps+\\operatorname{im}\\partial'=j^{-1}jz+\\operatorname{im}\\partial'=z+\\operatorname{im}\\partial'\\Rightarrow z+\\operatorname{im}\\partial'\\in \\operatorname{im}d$.<\/p>\n<\/li>\n \u4e0b\u8bc1 $\\operatorname{im} j_\\#=\\operatorname{ker} q_\\#$.<\/p>\n<\/li>\n \u4efb\u53d6 $z+\\operatorname{im}\\partial'\\in \\operatorname{coker}{\\partial'}$, \u6709 $q_\\#j_\\#(z+\\operatorname{im}\\partial')=qjz+\\operatorname{im}\\partial''=0$.<\/p>\n<\/li>\n \u4efb\u53d6 $z+\\operatorname{im}\\partial\\in \\operatorname{ker} q_\\#$, \u6709 $qz\\in \\operatorname{im}\\partial''$.<\/p>\n \u5219 $\\exists s\\in C,\\partial'' s=qz$. \u7531\u4e8e $p$ \u662f\u6ee1\u5c04, \u6240\u4ee5 $\\exists b\\in B,pb=s$.<\/p>\n \u5219\u6709 $q\\partial b=\\partial''pb=qz$, \u6240\u4ee5 $q(z-\\partial b)=0$, \u56e0\u6b64 $z-\\partial b\\in \\operatorname{ker} q=\\operatorname{im}j$.<\/p>\n \u5219\u6709 $j_\\#(j^{-1}(z-\\partial b)+\\operatorname{im}\\partial')=z-\\partial b+\\operatorname{im}\\partial=z+\\operatorname{im}\\partial\\Rightarrow z+\\operatorname{im}\\partial\\in \\operatorname{im} j_\\#$.<\/p>\n<\/li>\n<\/ul>\n \u7531\u6b64\u53ef\u77e5, \u6b63\u5408\u6027\u6210\u7acb. $\\square$<\/p>\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":" \u86c7\u5f62\u5f15\u7406\u662f\u540c\u8c03\u4ee3\u6570\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u5f15\u7406, \u5728\u6784\u9020\u6b63\u5408\u540c\u8c03\u5217\u4e2d\u6709\u8d77\u5230\u91cd\u8981\u7684\u4f5c\u7528.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[11,8],"class_list":["post-426","post","type-post","status-publish","format-standard","hentry","category-6","tag-11","tag-8"],"_links":{"self":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/426","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=426"}],"version-history":[{"count":3,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/426\/revisions"}],"predecessor-version":[{"id":432,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/426\/revisions\/432"}],"wp:attachment":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=426"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=426"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=426"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n$$% https:\/\/darknmt.github.io\/res\/xypic-editor\/#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
\n\\xymatrix{
\n & A \\ar@{->}[r]^{i} \\ar@{->}[d]^{\\partial'} & B \\ar@{->}[r]^{p} \\ar@{->}[d]^{\\partial} & C \\ar@{->}[r] \\ar@{->}[d]^{\\partial''} & 0 \\\\
\n0 \\ar@{->}[r] & D \\ar@{->}[r]_{j} & E \\ar@{->}[r]_{q} & F &
\n}
\n$$
\n\u5219\u5b58\u5728 $d:\\operatorname{ker}\\partial''\\to\\operatorname{coker}\\partial'$, \u4f7f\u5f97\u4ee5\u4e0b\u5217\u6b63\u5408:
\n$$\\operatorname{ker}{\\partial'}\\to\\operatorname{ker}{\\partial}\\to \\operatorname{ker}{\\partial''}\\xrightarrow{d}\\operatorname{coker}\\partial'\\to\\operatorname{coker}\\partial\\to\\operatorname{coker}\\partial''
\n$$
\nProof.<\/p>\n\n
\n
\n
\n$$\\begin{align}
\nia&=s-s'\\\\
\nj^{-1}\\partial ia&=j^{-1}\\partial(s-s')\\\\
\nj^{-1}j\\partial' a&=j^{-1}\\partial(s-s')\\ \\ \\ (j\\partial'=\\partial i)\\\\
\n\\partial' a&=j^{-1}\\partial(s-s')
\n\\end{align}
\n$$
\n\u56e0\u6b64 $j^{-1}\\partial(s-s')\\in \\operatorname{im}\\partial'\\Rightarrow j^{-1}\\partial s+\\operatorname{im}\\partial'=j^{-1}\\partial s'+\\operatorname{im}\\partial'$.<\/p>\n\n