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 Hi\fraka(.,.) defined by Hi\fraka(M,N)=\undersetn\varinjlim\operatornameExtRi(M/\frakanM,N), for all R-modules 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. Divaani-Aazar (Joint with H. Faridian and M. Tousi) Stable under specialization sets and cofiniteness J. Algebra Appl. (2019), DOI: 10.1142/S0219498819500154  [abstract]

2.	K. Divaani-Aazar (Joint with H. Faridian and M. Tousi) A New outlook on cofiniteness Kyoto J. Math. (Accepted) [abstract]

3.	K. Divaani-Aazar (Joint with H. Faridian and M. Tousi) Local homology, Koszul homology and Serre classes Rocky Mountain J. Math. (Accepted) [abstract]

4.	K. Divaani-Aazar (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. Divaani-Aazar Weakly Laskerian rings versus Noetherian rings Rend. Sem. Math. Univ. Padova 138 (2017), 239-256  [abstract]

6.	K. Divaani-Aazar and M. Tousi (Joint with M. Nikkhah) A criterion for dualizing modules Tokyo J. Math. 38 (2015), 133-143  [abstract]

7.	M. Asgharzadeh and K. Divaani-Aazar (Joint with F. Zareh-Khoshchehreh) Gorenstein homology, relative pure homology and virtually Gorenstein rings J. Pure Appl. Algebra 218 (2014), 2356-2366  [abstract]

8.	K. Divaani-Aazar and A. Soleyman Jahan (Joint with S. Bandari) Almost complete intersections and Stanley's conjecture Kodai Mathematical Journal 37 (2014), 396-404  [abstract]

9.	K. Divaani-Aazar and M. Tousi (Joint with F. Mohammadi Aghjeh Mashhad) On the existence of certain modules of finite Gorenstein homological dimensions Comm. Algebra 42 (2014), 1630-1643  [abstract]

10.	K. Divaani-Aazar and A. Soleyman Jahan (Joint with S. Bandari) Pretty cleanness and filter-regular sequences Czechoslovak Math. J. (Accepted) [abstract]

11.	K. Divaani-Aazar (Joint with M. Nikkhah Babaei) Gorenstein injective envelopes of artinian modules Comm. Algebra (Accepted) [abstract]

12.	K. Divaani-Aazar (Joint with F. Zareh-Khoshchehreh) The existence of relative pure injective envelopes Colloq. Math. 130 (2013), 251-264  [abstract]

13.	K. Divaani-Aazar (Joint with M. Hatamkhani) On the vanishing of local homology modules Glasgow Math. J. (2013), DOI: 10.1017/S0017089512000717  [abstract]

14.	K. Divaani-Aazar (Joint with M. Hatamkhani) The derived category analogue of the Hartshorne-Lichtenbaum vanishing theorem Tokyo J. Math. 36 (2013), 11 pages  [abstract]

15.	K. Divaani-Aazar (Joint with F. Mohammadi Aghjeh Mashhad) Local homology and Gorenstein flat modules J. Algebra Appl. (Accepted) [abstract]

16.	M. Asgharzadeh and K. Divaani-Aazar A unified approach to formal local cohomology and local Tate cohomology Rocky Mountain J. Math. (Accepted) [abstract]

17.	M. Asgharzadeh and K. Divaani-Aazar Finiteness properties of formal local cohomology modules and Cohen-Macaulayness Comm. Algebra 39 (2011), 1082-1103  [abstract]

18.	K. Divaani-Aazar (Joint with A. Hajikarimi) Generalized local cohomology modules and homological Gorenstein dimensions Comm. Algebra 39 (2011), 2051-2067  [abstract]

19.	M. Asgharzadeh, K. Divaani-Aazar and M. Tousi The finiteness dimension of local cohomology modules and its dual notion J. Pure Appl. Algebra 213 (2009), 321-328  [abstract]

20.	K. Divaani-Azar, M. A. Esmkhani and M. Tousi Some criteria of cyclically pure injective modules J. Algebra 304 (2006), 367-381  [abstract]

21.	K. Divaani-Aazar (Joint with M.A. Esmkhani, and M. Tousi) Two characterizations of pure injective modules Proc. Amer. Math. Soc. 134 (2006), 2817-2822  [abstract]

22.	K. Divaani-Aazar, R. Sazeedeh and M. Tousi On vanishing of generalized local cohomology modules Algebra Colloq. 12 (2005), 213-218  [abstract]

23.	K. Divaani-Aazar and M.A. Esmkhani Associated prime submodules of finitely generated modules Comm. Algebra 33 (2005), 4259 - 4266   [abstract]

24.	K. Divaani-Aazar and A. Mafi A new characterization of commutative Artinian rings Vietnam J. Math. 32 (2004), 319-322  [abstract]

25.	R. Naghipour, K. Divaani-Aazar and M. Tousi The Lichtenbaum-Hartshorne theorem for generalized local cohomology and connectedness Comm. Algebra 30 (2002), 3687-3702  [abstract]

26.	K. Divaani-Aazar, R. Naghipour and M. Tousi Cohomological dimension of certain algebraic varieties Proc. Amer. Math. Soc. 130 (2002), 3537-3544  [abstract]

27.	K. Divaani-Aazar and R. Naghipour Integral closure and ideal topologies in modules Comm. Algebra 29 (2001), 5239-5250  [abstract]

28.	K. Divaani-Aazar On associated and attached prime ideals of certain modules Colloq. Math. 89 (2001), 147-157  [abstract]

29.	K. Divaani-Aazar and P. Schenzel Ideal topologies, local cohomology and connectedness Math. Proc. Cambridge Philos. Soc. 131 (2001), 211-226  [abstract]

30.	K. Divaani-Aazar and M. Tousi A new definition of associated prime ideals Ital. J. pure Appl. Math. 8 (2000), 99-113  [abstract]

31.	M. T. Dibaei and K. Divaani-Aazar The theory of local homology for Artinian modules Southeast Asian Bull. Math. 24 (2000), 31-40  [abstract]

32.	K. Divaani-Aazar and M. Tousi Some remarks on coassociated primes J. Korean Math. Soc. 36 (1999), 847-853  [abstract]

33.	K. Divaani-Aazar and M. Tousi Asymptotic attached prime ideals related to flat modules Algebra Colloq. 6 (1999), 193-204  [abstract]

34.	K. Divaani-Aazar and M. Tousi Asymptotic prime ideals related to exact functors Comm. Algebra 27 (1999), 3949-3968  [abstract]

35.	K. Divaani-Aazar and Sh. Payrovi Reduced atomic chains of Artinian modules Southeast Asian Bull. Math. 23 (1999), 221-230  [abstract]

36.	K. Divaani-Aazar and M. Tousi Asymptotic associated and attached prime ideals related to projective modules Comm. Algebra 25 (1997), 1129-1142  [abstract]