“Kamran DivaaniAazar”
IPM Positions 

Senior Associate Researcher, School of Mathematics
(2011  2017 ) 

Past IPM Positions 

Resident Researcher, School of Mathematics
(2007  2011) Senior Associate Researcher, School of Mathematics (2005  2006) Associate Researcher, School of Mathematics (2004  2005) Associate Researcher, School of Mathematics (2002  2003) Senior Associate Researcher, School of Mathematics (2001  2002) Associate Researcher, School of Mathematics (1999  2001) 

Non IPM Affiliations 

Associate Professor of Azzahra University, (Tel: +98 21 8044051, ext. 413)  
Research Interests 

Commutative algebra and algebraic geometry  
Research Activities 

This project has two parts. In the first part, we examine the theory of
tight closure.
The theory tight of closure was introduced by M. Hochster and C. Huneke in 1980s. There are many applications for this notion in both commutative algebra and algebraic geometry. However, there are many basic open questions concerning tight closure. One of the essential questions is whether tight closure commutes with localization. In a joint paper, with M. Tousi, we show that tight closure commutes with localization in any two dimensional Noetherian ring of prime characteristic. In this project, we study the theory tight closure further. In particular, we interested in using local cohomology technics in our investigations . A generalization of local cohomology functors has been given by J. Herzog. Let \fraka denote an ideal of a commutative Noetherian ring R. For each i ? 0, the functor H^{i}_{\fraka}(.,.) defined by H^{i}_{\fraka}(M,N)=\undersetn\varinjlim\operatornameExt_{R}^{i}(M/\fraka^{n}M,N), for all Rmodules M and N. Clearly, this notion is a generalization of the usual local cohomology functor. In the second part of this project, we focus on studying the finiteness properties of generalized local cohomology. Mainly, we try to extend the known cofiniteness results about local cohomology to generalized local cohomology. 

Present Research Project at IPM 

Cofiniteness of Generalized Local Cohomology and Some Problems in Tight Closure  
Related Papers 
1.  K. DivaaniAazar (Joint with H. Faridian and M. Tousi) Stable under specialization sets and cofiniteness J. Algebra Appl. (2019), DOI: 10.1142/S0219498819500154 [abstract] 
2.  K. DivaaniAazar (Joint with H. Faridian and M. Tousi) A New outlook on cofiniteness Kyoto J. Math. (Accepted) [abstract] 
3.  K. DivaaniAazar (Joint with H. Faridian and M. Tousi) Local homology, Koszul homology and Serre classes Rocky Mountain J. Math. (Accepted) [abstract] 
4.  K. DivaaniAazar (Joint with H. Farhadian and M. Tousi) Local homology, finiteness of Tor modules and cofiniteness J. Algebra Appl. 16 (2017), # 10 Pages [abstract] 
5.  K. Bahmanpour and K. DivaaniAazar Weakly Laskerian rings versus Noetherian rings Rend. Sem. Math. Univ. Padova 138 (2017), 239256 [abstract] 
6.  K. DivaaniAazar and M. Tousi (Joint with M. Nikkhah) A criterion for dualizing modules Tokyo J. Math. 38 (2015), 133143 [abstract] 
7.  M. Asgharzadeh and K. DivaaniAazar (Joint with F. ZarehKhoshchehreh) Gorenstein homology, relative pure homology and virtually Gorenstein rings J. Pure Appl. Algebra 218 (2014), 23562366 [abstract] 
8.  K. DivaaniAazar and A. Soleyman Jahan (Joint with S. Bandari) Almost complete intersections and Stanley's conjecture Kodai Mathematical Journal 37 (2014), 396404 [abstract] 
9.  K. DivaaniAazar and M. Tousi (Joint with F. Mohammadi Aghjeh Mashhad) On the existence of certain modules of finite Gorenstein homological dimensions Comm. Algebra 42 (2014), 16301643 [abstract] 
10.  K. DivaaniAazar and A. Soleyman Jahan (Joint with S. Bandari) Pretty cleanness and filterregular sequences Czechoslovak Math. J. (Accepted) [abstract] 
11.  K. DivaaniAazar (Joint with M. Nikkhah Babaei) Gorenstein injective envelopes of artinian modules Comm. Algebra (Accepted) [abstract] 
12.  K. DivaaniAazar (Joint with F. ZarehKhoshchehreh) The existence of relative pure injective envelopes Colloq. Math. 130 (2013), 251264 [abstract] 
13.  K. DivaaniAazar (Joint with M. Hatamkhani) On the vanishing of local homology modules Glasgow Math. J. (2013), DOI: 10.1017/S0017089512000717 [abstract] 
14.  K. DivaaniAazar (Joint with M. Hatamkhani) The derived category analogue of the HartshorneLichtenbaum vanishing theorem Tokyo J. Math. 36 (2013), 11 pages [abstract] 
15.  K. DivaaniAazar (Joint with F. Mohammadi Aghjeh Mashhad) Local homology and Gorenstein flat modules J. Algebra Appl. (Accepted) [abstract] 
16.  M. Asgharzadeh and K. DivaaniAazar A unified approach to formal local cohomology and local Tate cohomology Rocky Mountain J. Math. (Accepted) [abstract] 
17.  M. Asgharzadeh and K. DivaaniAazar Finiteness properties of formal local cohomology modules and CohenMacaulayness Comm. Algebra 39 (2011), 10821103 [abstract] 
18.  K. DivaaniAazar (Joint with A. Hajikarimi) Generalized local cohomology modules and homological Gorenstein dimensions Comm. Algebra 39 (2011), 20512067 [abstract] 
19.  M. Asgharzadeh, K. DivaaniAazar and M. Tousi The finiteness dimension of local cohomology modules and its dual notion J. Pure Appl. Algebra 213 (2009), 321328 [abstract] 
20.  K. DivaaniAzar, M. A. Esmkhani and M. Tousi Some criteria of cyclically pure injective modules J. Algebra 304 (2006), 367381 [abstract] 
21.  K. DivaaniAazar (Joint with M.A. Esmkhani, and M. Tousi) Two characterizations of pure injective modules Proc. Amer. Math. Soc. 134 (2006), 28172822 [abstract] 
22.  K. DivaaniAazar, R. Sazeedeh and M. Tousi On vanishing of generalized local cohomology modules Algebra Colloq. 12 (2005), 213218 [abstract] 
23.  K. DivaaniAazar and M.A. Esmkhani Associated prime submodules of finitely generated modules Comm. Algebra 33 (2005), 4259  4266 [abstract] 
24.  K. DivaaniAazar and A. Mafi A new characterization of commutative Artinian rings Vietnam J. Math. 32 (2004), 319322 [abstract] 
25.  R. Naghipour, K. DivaaniAazar and M. Tousi The LichtenbaumHartshorne theorem for generalized local cohomology and connectedness Comm. Algebra 30 (2002), 36873702 [abstract] 
26.  K. DivaaniAazar, R. Naghipour and M. Tousi Cohomological dimension of certain algebraic varieties Proc. Amer. Math. Soc. 130 (2002), 35373544 [abstract] 
27.  K. DivaaniAazar and R. Naghipour Integral closure and ideal topologies in modules Comm. Algebra 29 (2001), 52395250 [abstract] 
28.  K. DivaaniAazar On associated and attached prime ideals of certain modules Colloq. Math. 89 (2001), 147157 [abstract] 
29.  K. DivaaniAazar and P. Schenzel Ideal topologies, local cohomology and connectedness Math. Proc. Cambridge Philos. Soc. 131 (2001), 211226 [abstract] 
30.  K. DivaaniAazar and M. Tousi A new definition of associated prime ideals Ital. J. pure Appl. Math. 8 (2000), 99113 [abstract] 
31.  M. T. Dibaei and K. DivaaniAazar The theory of local homology for Artinian modules Southeast Asian Bull. Math. 24 (2000), 3140 [abstract] 
32.  K. DivaaniAazar and M. Tousi Some remarks on coassociated primes J. Korean Math. Soc. 36 (1999), 847853 [abstract] 
33.  K. DivaaniAazar and M. Tousi Asymptotic attached prime ideals related to flat modules Algebra Colloq. 6 (1999), 193204 [abstract] 
34.  K. DivaaniAazar and M. Tousi Asymptotic prime ideals related to exact functors Comm. Algebra 27 (1999), 39493968 [abstract] 
35.  K. DivaaniAazar and Sh. Payrovi Reduced atomic chains of Artinian modules Southeast Asian Bull. Math. 23 (1999), 221230 [abstract] 
36.  K. DivaaniAazar and M. Tousi Asymptotic associated and attached prime ideals related to projective modules Comm. Algebra 25 (1997), 11291142 [abstract] 
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