“Kamran Divaani-Aazar”

Tel:  (+98)(21)2310 ext. 2118
Fax:  (+98)(21)22290648
Email: 

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]
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