Mathematical Researches

fa عدد نظم توان دوم ایده آلهای یالی Regularity of second power of edge ideals جبر alg علمی پژوهشی بنیادی S فرض کنید G یک گراف با ایده آل یالی I(G) باشد. بنرجی<a href="#_ftn1" name="_ftnref1" title="">[1]</a> و نِوو<a href="#_ftn2" name="_ftnref2" title="">[2]</a> ثابت کردند که برای هر گراف G، نامساوی reg(I(G)2)≤reg(I(G))+2 برقرار است. در این مقاله اثبات دیکری برای این مظلب ارائه می کنیم. <div>  <hr align="left" size="1" width="33%" > <div id="ftn1"><a href="#_ftnref1" name="_ftn1" title="">[1]</a> Banerjee</div> <div id="ftn2"><a href="#_ftnref2" name="_ftn2" title="">[2]</a> Nevo</div> </div> Introduction &lrm;&lrm; The study of the minimal free resolution of homogenous ideals and their powers is an interesting and active area of research in commutative algebra. Two invariants which measure the complexity of the minimal free resolutions are the so-called “projective dimension” and “Castelnuovo-Mumford regularity” (or simply, regularity) of the given ideal. Projective dimension determines the length of the minimal free resolution, while regularity is defined in terms of the degree of the entries of the matrices defining the differentials of the resolution. The focus of this paper is on the regularity of powers of ideals. One of the main results in this area is obtained by Cutkosky, Herzog, Trung [7], and independently Kodiyalam [8]. They proved that for a homogenous ideal <m:omath><m:r>I</m:r></m:omath> in a polynomial ring, the regularity of powers of <m:omath><m:r>I</m:r></m:omath> is asymptotically linear. In other words, there exist integers <m:omath><m:r>a</m:r><m:r>(</m:r><m:r>I</m:r><m:r>)</m:r></m:omath> and <m:omath><m:r>b</m:r><m:r>(</m:r><m:r>I</m:r><m:r>)</m:r></m:omath> such that <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>reg</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>I</m:r></m:e><m:sup><m:r>s</m:r></m:sup></m:ssup></m:e></m:d><m:r>=</m:r><m:r>a</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>I</m:r></m:e></m:d><m:r>s</m:r><m:r>+</m:r><m:r>b</m:r><m:r>(</m:r><m:r>I</m:r><m:r>)</m:r></m:omath> for every integer <m:omath><m:r>s</m:r><m:r>≫0</m:r></m:omath>. It is known that <m:omath><m:r>a</m:r><m:r>(</m:r><m:r>I</m:r><m:r>)</m:r></m:omath> is bounded above by the maximum degree of generators of <m:omath><m:r>I</m:r></m:omath>. Moreover, if <m:omath><m:r>I</m:r></m:omath> is generated in a single degree <m:omath><m:r>d</m:r></m:omath>, then <m:omath><m:r>a</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>I</m:r></m:e></m:d><m:r>=</m:r><m:r>d</m:r><m:r>.</m:r></m:omath> But in general, it is not so much known about <m:omath><m:r>b</m:r><m:r>(</m:r><m:r>I</m:r><m:r>)</m:r></m:omath> even if <m:omath><m:r>I</m:r></m:omath> is monomial ideal. However, when <m:omath><m:r>I</m:r></m:omath> is a quadratic squarefree monomial ideal, Alilooee, Banerjee, Beyarslan and Ha [9] conjectured that <m:omath><m:r>b</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>I</m:r></m:e></m:d><m:r>≤</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>reg</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>I</m:r></m:e></m:d><m:r>-</m:r><m:r>2</m:r></m:omath>. In fact, they conjectured that the inequality <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>reg</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>I</m:r></m:e><m:sup><m:r>s</m:r></m:sup></m:ssup></m:e></m:d><m:r>≤2</m:r><m:r>s</m:r><m:r>+</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>reg</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>I</m:r></m:e></m:d><m:r>-</m:r><m:r>2</m:r></m:omath> holds for any integer <m:omath><m:r>s</m:r><m:r>≥1</m:r></m:omath>, when <m:omath><m:r>I</m:r></m:omath> is quadratic squarefree monomial ideal. Recently, Benerjee and Nevo [10] proved this conjecture for <m:omath><m:r>s</m:r><m:r>=2</m:r></m:omath>. In this paper, we provide an alternative proof for their result. While the proof in [10] is based on topological arguments and using the Hochster’s formula, our proof is purely algebraic.  Material and methods To every simple graph <m:omath><m:r>G</m:r></m:omath> one associates a quadratic squarefree monomial ideal, called its edge ideal, whose generators are the quadratic squarefree monomials corresponding to the edges of <m:omath><m:r>G</m:r></m:omath>. This association is a strong tool in the study of squarefree monomial ideals, as one can use the combinatorial properties of <m:omath><m:r>G</m:r></m:omath> to obtain information about the algebraic and homological properties of it, s edge ideal. One of the main results for bounding the regularity of powers of edge ideals is obtained by Benerjee [1]. He proved that the regularity of the <m:omath><m:r>s</m:r></m:omath>th power of an edge ideal <m:omath><m:r>I</m:r><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:omath> has an upper bound which is defined in terms of the regularity of its  <m:omath><m:r>(</m:r><m:r>s</m:r><m:r>-</m:r><m:r>1)</m:r></m:omath>th power and the regularity of the edge ideal of some graphs which are explicitly determined by the structure of the <m:omath><m:r>G</m:r></m:omath>. This result has an essential role in our proof. Results and discussion The main result of this paper states that for every graph <m:omath><m:r>G</m:r></m:omath>, with edge ideal <m:omath><m:r>I</m:r><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:omath>, we have <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>reg</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>I</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d></m:e><m:sup><m:r>2</m:r></m:sup></m:ssup></m:e></m:d><m:r>≤</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>reg</m:r><m:r>(</m:r><m:r>I</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d><m:r>)+2</m:r></m:omath>. In order to prove this inequality, using the aforementioned result of Benerjee, we must prove that the regularity of certain colon ideals are at most <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>reg</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>I</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d></m:e></m:d></m:omath>. To achieve this goal, we use a short exact sequence argument which allows us to estimate the regularity of the colon ideas in terms of the regularity of edge ideal of some graphs which are strictly smaller than <m:omath><m:r>G</m:r></m:omath>. Conclusion The following conclusions were drawn from this research. <ul> <li class="CxSpMiddle" style="text-align:justify">The conjectured inequality of Alilooee, Banerjee, Beyarslan and Ha [9] is true for the case of <m:omath><m:r>s</m:r><m:r>=2</m:r></m:omath>.</li> <li class="CxSpMiddle" style="text-align:justify">It is known that for every graph <m:omath><m:r>G</m:r></m:omath> with edge ideal <m:omath><m:r>I</m:r><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:omath> and induced matching number <m:omath><m:r>ν</m:r><m:r>(</m:r><m:r>G</m:r><m:r>)</m:r></m:omath>, we have <m:omath><m:r>2</m:r><m:r>s</m:r><m:r>+</m:r><m:r>ν</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d><m:r>-</m:r><m:r>1≤</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>reg</m:r><m:r>(</m:r><m:r>I</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d></m:e><m:sup><m:r>s</m:r></m:sup></m:ssup><m:r>)</m:r></m:omath>, for every integer <m:omath><m:r>s</m:r><m:r>≥1</m:r></m:omath>. Thus, our result implies that if <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>reg</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>I</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d></m:e></m:d><m:r>=</m:r><m:r>ν</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d><m:r>+1</m:r></m:omath>, then <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>reg</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>I</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d></m:e><m:sup><m:r>2</m:r></m:sup></m:ssup></m:e></m:d><m:r>=</m:r><m:r>ν</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>G</m:r></m:e></m:d><m:r>+3</m:r></m:omath>.</li> </ul> The short exact sequence argument is a common technique in the study of regularity of monomial ideals. So, it would be interesting if one can prove the above-mentioned conjecture, using this method, even in the case of <m:r>s</m:r><m:r>=3</m:r> . ایده آل یالی, عدد نظم کاستلنوو-مامفورد Edge ideal, Castelnuovo-Mumford regularity 106 116 http://mmr.khu.ac.ir/browse.php?a_code=A-10-987-1&slc_lang=fa&sid=1 Seyed Amin Seyed Fakhari سید امین سید فخاری aminfakhari@ut.ac.ir 10031947532846005467 10031947532846005467 Yes دانشگاه تهران