Balance of Tate cohomology in triangulated categories

Let C$\mathcal {C}$ be a triangulated category and E$\mathcal {E}$ a proper class of triangles. We show that Tate cohomology in triangulated category is balanced, i.e. there is an isomorphism Ext̂Pi(A,B)≅Ext̂Ii(A,B)$\widehat {\mathcal {E}\text {xt}}^{i}_{\mathcal {P}}(A, B)\cong \widehat {\mathcal {E}\text {xt}}^{i}_{\mathcal {I}}(A, B)$ for any integer i∈ℤ$i\in \mathbb {Z}$, where the first cohomology group is computed by complete E$\mathcal {E}$-projective resolution for A∈C$A\in \mathcal {C}$ and the second one is computed by complete E$\mathcal {E}$-injective coresolution for B∈C$B\in \mathcal {C}$. This improves the theorem proposed by J. Asadollahi and Sh. Salarian [J. Algebra 299, 480-502 (2006)].

[1]  A. Neeman.  Chapter 1. Definition and elementary properties of triangulated categories , 2001 .

[2]  Overtoun M. G. Jenda,et al.  8 Relative Homological Algebra and Balance , 2000 .

[3]  M. Bridger,et al.  Stable Module Theory , 1969 .

[4]  David A. Jorgensen,et al.  Tate (co)homology via pinched complexes , 2011, 1105.2286.

[5]  Alex Martsinkovsky,et al.  Absolute, Relative, and Tate Cohomology of Modules of Finite Gorenstein Dimension , 2002 .

[6]  J. Asadollahi,et al.  Gorenstein objects in triangulated categories , 2004 .

[7]  Henrik Holm,et al.  Gorenstein homological dimensions , 2004 .

[8]  A. Iacob.  Balance in Generalized Tate Cohomology , 2005 .

[9]  C. Weibel,et al.  AN INTRODUCTION TO HOMOLOGICAL ALGEBRA , 1996 .

[10]  E. Enochs,et al.  Balance with unbounded complexes , 2011, 1108.1100.

[11]  J. Asadollahi,et al.  Cohomology theories based on Gorenstein injective modules , 2005 .

[12]  Pu Zhang,et al.  Gorenstein derived categories , 2010 .

[13]  I. Emmanouil.  Balance in complete cohomology , 2014 .

[14]  Zhongkui Liu,et al.  Gorenstein homological dimensions for triangulated categories , 2014 .

[15]  Oana Veliche.  Gorenstein projective dimension for complexes , 2004 .

[16]  J. Asadollahi,et al.  Tate cohomology and Gorensteinness for triangulated categories , 2006 .

[17]  J. Rotman.  An Introduction to Homological Algebra , 1979 .

[18]  Apostolos Beligiannis.  Relative Homological Algebra and Purity in Triangulated Categories , 2000 .

[19]  J. Verdier.  Categories Derivees Quelques résultats (Etat 0) , 1977 .