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ایدهآلهای اول چسبیدۀ آخرین مدول ناصفرکوهمولوژی موضعی
Attached primes of top local cohomology modules
جبر
alg
مقاله مستقل
Original Manuscript
<span dir="LTR"> </span>فرض کنیم <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image001.png" > یک ایدهآل از حلقۀ نوتری <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image002.png" > و <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image003.png" > و <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image004.png" > دو <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image002.png" > - مدول متناهیمولد باشند. در این مقاله<br>
ایدهآلهای اول چسبیدۀ ­­­­­­­­­آخرین مدول ناصفر کوهمولوژی موضعی را بررسی میکنیم. در ابتدا ثابت میکنیم اگر <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image003.png" > یک<br>
<img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image005.png" > مدول متناهیمولد ناصفر باشد و <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image006.png" > آنگاه<br>
<img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image007.png" ><br>
در ادامه ثابت میکنیم اگر <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image008.png" > <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image003.png" > یک <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image002.png" > - مدول متناهیمولد ناصفر با بعد تصویری <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image009.png" > و <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image010.png" > یک <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image002.png" > - مدول متناهیمولد ناصفر با<br>
بعد همولوژیک <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image011.png" > باشد، <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image008.png" > آنگاه <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image012.png" > <strong>و</strong><br>
<img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image013.png" ><br>
همچنین نشان میدهیم تساوی <img alt="" chromakey="white" src="file:///C:UsersrezaeiAppDataLocalTempmsohtmlclip11clip_image014.png" > تحت شرایطی خاص برقرار است.
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:14.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">Introduction</span></span></span></b></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"> <span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> Let </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> be an ideal of Noetherian ring </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> and </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>M</m:r><m:r>,</m:r><m:r>N</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> be finitely generated </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">-modules. Recall that, for each </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>i</m:r><m:r>≥0</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">, i-th generalized local cohomology module </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>M</m:r><m:r>, </m:r><m:r>N</m:r><m:r> </m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">with respect to </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> is defined by</span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><m:omathpara><m:omath><m:ssubsup><m:ssubsuppr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:sub><m:sup><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>M</m:r><m:r>,</m:r><m:r>N</m:r></span></span></span></i></m:e></m:d><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>≔</m:r></span></span></span></i><m:func><m:funcpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:funcpr><m:fname><m:limlow><m:limlowpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:limlowpr><m:e><m:limlow><m:limlowpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:limlowpr><m:e><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>lim</m:r></span></span></span></m:e><m:lim><m:box><m:boxpr><m:opemu m:val="on"><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:opemu></m:boxpr><m:e><m:acc><m:accpr><m:chr m:val="⃗"><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:chr></m:accpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></span></i></m:e></m:acc></m:e></m:box></m:lim></m:limlow></m:e><m:lim><m:box><m:boxpr><m:opemu m:val="on"><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:opemu></m:boxpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r> </m:r></span></span></span></i></m:e></m:box></m:lim></m:limlow><m:ssubsup><m:ssubsuppr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:sub><m:sup><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></span></i></m:sup></m:ssubsup><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>(</m:r></span></span></span></i></m:fname><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>M</m:r><m:r>/</m:r></span></span></span></i><m:ssup><m:ssuppr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:e><m:sup><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></span></i></m:sup></m:ssup><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>M</m:r><m:r>,</m:r><m:r>N</m:r><m:r>).</m:r></span></span></span></i></m:e></m:func></m:omath></m:omathpara><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"> </span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">Also, recall that </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>cd</m:r></span></span></span></i><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r><m:r>,</m:r><m:r>M</m:r><m:r>,</m:r><m:r>N</m:r></span></span></span></i></m:e></m:d><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>, </m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> the cohomological dimension of R-modules M and N with respect to an ideal </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> of a commutative Noetherian ring </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> is </span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><m:omathpara><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>sup</m:r></span></span></span></i><m:d><m:dpr><m:begchr m:val="{"><m:endchr m:val="}"><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>i</m:r><m:r>∈</m:r></span></span></span></i><m:ssub><m:ssubpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>N</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>0</m:r></span></span></span></i></m:sub></m:ssub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>: </m:r></span></span></span></i><m:ssubsup><m:ssubsuppr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:sub><m:sup><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r> </m:r><m:r>M</m:r><m:r>,</m:r><m:r>N</m:r></span></span></span></i></m:e></m:d><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>≠0</m:r></span></span></span></i></m:e></m:d><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>.</m:r></span></span></span></i></m:omath></m:omathpara><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"> </span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">An important problem concerning local cohomology is determining the set of attached prime ideals of the top local cohomology modules. This problem has been studied by several authors. In this paper, we study attached prime ideals of top local cohomology modules.</span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"> <b><span style="font-size:14.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">Material and methods </span></span></span></b><span style="font-size:14.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> In this paper, we first obtain some subsets of the set of attached prime ideals of top local cohomology module. By using these, we obtain a result about finiteness of top local cohomology modules.</span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><span style="font-size:12.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:14.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">Results and discussion</span></span></span></b><span style="font-size:14.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><b> </b><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">Let </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> be a Noetherian ring and </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> be an ideal of </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r> </m:r><m:r>R</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">. Let </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> and </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>N</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> be non-zero finitely generated </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">-modules. Assume that </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>pd</m:r></span></span></span></i><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></span></i></m:e></m:d><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r>d</m:r><m:r><∞</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">, </span></span></span><m:omath><m:func><m:funcpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:funcpr><m:fname><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>cd</m:r></span></span></span></i><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r><m:r>,</m:r><m:r>N</m:r></span></span></span></i></m:e></m:d></m:fname><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r>c</m:r><m:r><∞</m:r></span></span></span></i></m:e></m:func><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r> . </m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">We will prove that </span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><m:omathpara><m:omathparapr><m:jc m:val="left"></m:jc></m:omathparapr><m:omath><m:ssubsup><m:ssubsuppr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><m:ssub><m:ssubpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>i</m:r><m:r>) </m:r></span></span></span></i><m:d><m:dpr><m:begchr m:val="{"><m:endchr m:val="}"><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>p</m:r><m:r>∈</m:r></span></span></span></i><m:ssub><m:ssubpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>Supp</m:r></span></span></span></i></m:e><m:sub><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:sub></m:ssub><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>N</m:r><m:r>: </m:r><m:r>cd</m:r></span></span></span></i><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r><m:r>,</m:r><m:r>R</m:r><m:r>/</m:r><m:r>p</m:r></span></span></span></i></m:e></m:d><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r>c</m:r><m:r>, </m:r></span></span></span></i><m:func><m:funcpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:funcpr><m:fname><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr> dim</m:r></span></span></span></m:fname><m:e><m:f><m:fpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:fpr><m:num><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:num><m:den><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></span></i></m:den></m:f><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r>c</m:r></span></span></span></i></m:e></m:func></m:e></m:d><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>⊆</m:r><m:r>Att</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:sub></m:ssub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>H</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:sub><m:sup><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>c</m:r></span></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>N</m:r><m:r>)</m:r></span></span></span></i></m:e></m:d><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>,</m:r></span></span></span></i></m:omath></m:omathpara><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"> </span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><m:omathpara><m:omathparapr><m:jc m:val="left"></m:jc></m:omathparapr><m:omath><m:ssubsup><m:ssubsuppr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r> </m:r><m:r>ii</m:r><m:r>) </m:r></span></span></span></i><m:ssub><m:ssubpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r> </m:r><m:r>Att</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:sub></m:ssub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>H</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:sub><m:sup><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>d</m:r><m:r>+</m:r><m:r>c</m:r></span></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>M</m:r><m:r>,</m:r><m:r>N</m:r><m:r>)</m:r></span></span></span></i></m:e></m:d><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>⊆</m:r></span></span></span></i><m:d><m:dpr><m:begchr m:val="{"><m:endchr m:val="}"><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>p</m:r><m:r>∈</m:r></span></span></span></i><m:ssub><m:ssubpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>Supp</m:r></span></span></span></i></m:e><m:sub><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:sub></m:ssub><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>N</m:r><m:r>: </m:r><m:r>cd</m:r></span></span></span></i><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r><m:r>,</m:r><m:r>M</m:r><m:r>,</m:r><m:r>R</m:r><m:r>/</m:r><m:r>p</m:r></span></span></span></i></m:e></m:d><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r>d</m:r><m:r>+</m:r><m:r>c</m:r></span></span></span></i></m:e></m:d><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>,</m:r></span></span></span></i></m:omath></m:omathpara><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"> </span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><m:omathpara><m:omathparapr><m:jc m:val="left"></m:jc></m:omathparapr><m:omath><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>iii</m:r><m:r>) </m:r></span></span></span></i><m:d><m:dpr><m:begchr m:val="{"><m:endchr m:val="}"><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>p</m:r><m:r>∈</m:r><m:r>Sup</m:r></span></span></span></i><m:ssub><m:ssubpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:sub></m:ssub><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>N</m:r><m:r> :</m:r><m:r>cd</m:r></span></span></span></i><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r><m:r>,</m:r><m:r>M</m:r><m:r>,</m:r></span></span></span></i><m:f><m:fpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:fpr><m:num><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:num><m:den><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></span></i></m:den></m:f></m:e></m:d><i><span lang="EN-GB" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r>d</m:r><m:r>+</m:r><m:r>c</m:r></span></span></span></i><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>,</m:r></span></span></span></i><m:func><m:funcpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:funcpr><m:fname><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>dim</m:r></span></span></span></m:fname><m:e><m:f><m:fpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:fpr><m:num><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:num><m:den><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>p</m:r></span></span></span></i></m:den></m:f><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r>c</m:r></span></span></span></i></m:e></m:func><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r> </m:r></span></span></span></i></m:e></m:d><span dir="RTL" lang="FA" style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>⊆</m:r></span></span></span><m:ssubsup><m:ssubsuppr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><m:ssub><m:ssubpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>Att</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:sub></m:ssub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>H</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:sub><m:sup><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>d</m:r><m:r>+</m:r><m:r>c</m:r></span></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>M</m:r><m:r>,</m:r><m:r>N</m:r><m:r>)</m:r></span></span></span></i></m:e></m:d><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>.</m:r></span></span></span></i></m:omath></m:omathpara><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"> </span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:14.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">Conclusion </span></span></span></b></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> Let </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> and </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>N</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> be non-zero finitely generated </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">-modules. Assume that </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>pd</m:r></span></span></span></i><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>M</m:r></span></span></span></i></m:e></m:d><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r>d</m:r><m:r><∞</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">, </span></span></span><m:omath><m:func><m:funcpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:funcpr><m:fname><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>cd</m:r></span></span></span></i><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r><m:r>,</m:r><m:r>N</m:r></span></span></span></i></m:e></m:d></m:fname><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r>c</m:r><m:r><∞</m:r></span></span></span></i></m:e></m:func><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>. </m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">The following conclusions were drawn from this research. </span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> • If </span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>R</m:r><m:r>,</m:r><m:r>m</m:r><m:r>)</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> is a Noetherian local ring such that</span></span></span><m:omath><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r> </m:r></span></span></span></i><m:ssubsup><m:ssubsuppr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:sub><m:sup><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>d</m:r><m:r>+</m:r><m:r>c</m:r></span></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>M</m:r><m:r>,</m:r><m:r>N</m:r></span></span></span></i></m:e></m:d><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>≠0</m:r></span></span></span></i></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> then </span></span></span><m:omath><m:ssubsup><m:ssubsuppr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>H</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:sub><m:sup><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>d</m:r><m:r>+</m:r><m:r>c</m:r></span></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>M</m:r><m:r>,</m:r><m:r>N</m:r></span></span></span></i></m:e></m:d></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> is not of finite length.</span></span></span> </span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times=""> • If R is a Noetherian domain, then under certain conditions we have </span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><m:omath><m:ssubsup><m:ssubsuppr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><m:ssub><m:ssubpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>Att</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:sub></m:ssub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>H</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:sub><m:sup><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>d</m:r><m:r>+</m:r><m:r>c</m:r></span></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>M</m:r><m:r>,</m:r><m:r>N</m:r><m:r>)</m:r></span></span></span></i></m:e></m:d><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>=</m:r></span></span></span></i><m:ssubsup><m:ssubsuppr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><m:ssub><m:ssubpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>Att</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>R</m:r></span></span></span></i></m:sub></m:ssub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>(</m:r><m:r>H</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>a</m:r></span></span></span></i></m:sub><m:sup><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>c</m:r></span></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:12.0pt"><span style="line-height:107%"><span cambria="" math="" style="font-family:"><m:r>N</m:r><m:r>)</m:r></span></span></span></i></m:e></m:d></m:omath><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.0pt"> </span></span></span></span></span><span style="font-size:12.0pt"><span style="line-height:107%"><span new="" roman="" style="font-family:" times="">.</span></span></span></span></span></span><br>
مدولهای کوهمولوژی موضعی , ایدهآلهای اول چسبیده.
local cohomology modules, attached primes
136
146
http://mmr.khu.ac.ir/browse.php?a_code=A-14-634-5&slc_lang=fa&sid=1
Shahram
Rezaei
شهرام
رضایی
sha.Rezaei@gmail.com
10031947532846006181
10031947532846006181
Yes
PNU
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