Mathematical Researches

fa تابع‌گون‌های توسیع مدول‌های کوهمولوژی موضعی تعمیم‌یافته Extension Functors of Generalized Local Cohomology Modules جبر alg مقاله مستقل Original Manuscript فرض کنیم <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.png" >  یک حلقه نوتری جابه‌جایی با واحد ناصفر، <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.png" >  ایده‌آلی از حلقه <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.png" > ، <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image003.png" >  یک <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.png" > مدول متناهی‌مولد و <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image005.png" >  یک <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.png" > مدول دل‌خواه باشد. در این مقاله، برای اعداد صحیح و نامنفی <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.png" > ، <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image007.png" >  و <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.png" > مدول متناهی‌مولد <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.png" > ، متعلق بودن <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image009.png" >  را در زیررسته‌های سر از رسته <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.png" > مدول‌ها بررسی می‌کنیم و کران‌های بالایی برای بعد انژکتیو و اعداد باس <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.png" >  ارایه می‌کنیم. هم‌چنین برخی نتایج در مورد هم‌متناهی بودن و مینیماکس بودن <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.png" >  و متناهی بودن <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image011.png" >  به‌دست می‌آوریم.     Introduction Throughout this paper, <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" > is a commutative Noetherian ring with non-zero identity, <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="7" > is an ideal of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" >, <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="12" > is a finitely generated <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" >-module, &lrm;and <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="8" > is an arbitrary <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" >-module which is not necessarily finitely generated.      Let L be a finitely generated R-module. Grothendieck, in [11], conjectured that <img alt="" height="28" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="125" > is finitely generated for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="4" >. In [12], &lrm;Hartshorne gave a counter-example and raised the question whether <img alt="" height="28" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image014.gif" width="116" > is finitely generated for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="4" > and <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="4" >. The <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="4" >th generalized local cohomology module of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="12" > and <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="8" > with respect to <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="7" >, <img alt="" height="34" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="203" > was introduced by Herzog in [14]. It is clear that <img alt="" height="23" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image020.gif" width="51" > is just the ordinary local cohomology module <img alt="" height="23" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image022.gif" width="36" > of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="8" > with respect to <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="7" >. As a generalization of Hartshorne's question, we have the following question for generalized local cohomology modules (see [25, Question 2.7]). Question. When is <img alt="" height="28" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image024.gif" width="134" > finitely generated for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="4" > and <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="4" >?      In this paper, we study <img alt="" height="28" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image026.gif" width="117" > in general for a finitely generated <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" >-module <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image028.gif" width="10" > and an arbitrary <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" >-module <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="8" >. Material and methods The main tool used in the proofs of the main results of this paper is the spectral sequences. Results and discussion We present some technical results (Lemma 2.1 and Theorems 2.2, 2.9, and 2.14) which show that, in certain situation, for non-negative integers <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image030.gif" width="6" >, <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image032.gif" width="5" >, <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image034.gif" width="10" >, and <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image036.gif" width="9" > with <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image038.gif" width="85" >, <img alt="" height="21" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image040.gif" width="227" > and the <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" >-modules <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image042.gif" width="103" > and <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image044.gif" width="101" > are in a Serre subcategory of the category of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" >-modules (i.e. the class of   <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" >-modules which is closed under taking submodules, quotients, and extensions). Conclusion We apply the main results of this paper to some Serre subcategories (e.g. the class of zero        <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" >-modules and the class of finitely generated <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" >-modules) and deduce some properties of generalized local cohomology modules. In Corollaries 3.1-3.3, we present some upper bounds for the injective dimension and the Bass numbers of generalized local cohomology modules. We study cofiniteness and minimaxness of generalized local cohomology modules in Corollaries 3.4-3.8. Recall that, an <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" >-module <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="8" > is said to be <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="7" >-cofinite (resp. minimax) if <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image046.gif" width="120" > and <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image048.gif" width="82" > is finitely generated for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="4" > [12] (resp. there is a finitely generated submodule <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image050.gif" width="12" > of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="8" > such that <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image052.gif" width="28" > is Artinian [26]) where <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image054.gif" width="185" >. We show that if <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image048.gif" width="82" > is finitely generated for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image056.gif" width="28" > and <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image058.gif" width="56" > is minimax for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image060.gif" width="29" >, then <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image058.gif" width="56" > is <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="7" >-cofinite for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image060.gif" width="29" > and <img alt="" height="21" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image062.gif" width="140" > is finitely generated (Corollary 3.5). We prove that if <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image048.gif" width="82" > is finitely generated for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image064.gif" width="63" >, where <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image066.gif" width="39" > is the arithmetic rank of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="7" >, and <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image058.gif" width="56" > is <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="7" >-cofinite for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image068.gif" width="28" >, then <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image070.gif" width="56" > is also an <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="7" >-cofinite <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" >-module (Corollary 3.6). We show that if <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" > is local, <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image072.gif" width="87" >, and <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image048.gif" width="82" > is finitely generated for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image074.gif" width="54" >, then <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image058.gif" width="56" > is <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="7" >-cofinite for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image060.gif" width="29" > if and only if <img alt="" height="28" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image076.gif" width="143" > is finitely generated for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image056.gif" width="28" > (Corollary 3.7). We also prove that if <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" > is local, <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image072.gif" width="87" >, <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image048.gif" width="82" > is finitely generated for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="4" >, and <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image078.gif" width="59" > (or <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image080.gif" width="73" >) is <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="7" >-cofinite for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="4" >, then <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image058.gif" width="56" > is <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="7" >-cofinite for all <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="4" > (Corollary 3.8). In Corollary 3.9, we state the weakest possible conditions which yield the finiteness of associated prime ideals of generalized local cohomology modules. Note that, one can apply the main results of this paper to other Serre subcategories to deduce more properties of generalized local cohomology modules.<a href="./files/site1/files/71/15.pdf">./files/site1/files/71/15.pdf</a>   اعداد باس, ایده‌آل‌های اول وابسته, بعدهای انژکتیو, تابع‌گون‌های توسیع, مدول‌های کوهمولوژی موضعی تعمیم‌یافته, مدول‌های مینیماکس, مدول‌های هم‌متناهی. Associated prime ideals, Bass numbers, Cofinite modules, Extension functors, Generalized local cohomology modules, Injective dimensions, Minimax modules. 151 164 http://mmr.khu.ac.ir/browse.php?a_code=A-10-453-2&slc_lang=fa&sid=1 Alireza Vahidi علی‌رضا وحیدی vahidi.ar@pnu.ac.ir 10031947532846004463 10031947532846004463 Yes Payame Noor University دانشگاه پیام نور، گروه ریاضی، تهران Faisal Hassani فیصل حسنی f_hasani@pnu.ac.ir 10031947532846004464 10031947532846004464 No Payame Noor University دانشگاه پیام نور، گروه ریاضی، تهران Elham Hoseinzade الهام حسین‌زاده el.hosseinzade.phd@gmail.com 10031947532846004465 10031947532846004465 No Payame Noor University دانشگاه پیام نور، گروه ریاضی، تهران