Balance in Generalized Tate Cohomology

<div class="line" id="line-19"> We consider two preenveloping classes of left R-modules &imagline;, &expectation; such that Inj&nbsp;&sub;&nbsp;&imagline;&nbsp;&sub;&nbsp;&expectation;, and a left R-module N. For any left R-module M and n&nbsp;&ge;&nbsp;1 we define the relative extension modules <img src="http://www.tandfonline.com/na101/home/literatum/publisher/tandf/journals/content/lagb20/2005/lagb20.v033.i06/agb-200063353/20170907/images/lagb_a_10362644_o_ilm0001.gif"/> (M, N) and prove the existence of an exact sequence connecting these modules and the modules <img src="http://www.tandfonline.com/na101/home/literatum/publisher/tandf/journals/content/lagb20/2005/lagb20.v033.i06/agb-200063353/20170907/images/lagb_a_10362644_o_ilm0046.gif"/> (M, N) and <img src="http://www.tandfonline.com/na101/home/literatum/publisher/tandf/journals/content/lagb20/2005/lagb20.v033.i06/agb-200063353/20170907/images/lagb_a_10362644_o_ilm0047.gif"/> (M, N). We show that there is a long exact sequence of <img src="http://www.tandfonline.com/na101/home/literatum/publisher/tandf/journals/content/lagb20/2005/lagb20.v033.i06/agb-200063353/20170907/images/lagb_a_10362644_o_ilm0002.gif"/> (M, &minus;) associated with a Hom(&minus;, &expectation;) exact sequence 0&nbsp;&rarr;&nbsp;N&prime;&nbsp;&rarr;&nbsp;N&nbsp;&rarr;&nbsp;N&prime;&prime;&nbsp;&rarr;&nbsp;0 and a long exact sequence of <img src="http://www.tandfonline.com/na101/home/literatum/publisher/tandf/journals/content/lagb20/2005/lagb20.v033.i06/agb-200063353/20170907/images/lagb_a_10362644_o_ilm0003.gif"/> (&minus;, N) associated with a Hom(&minus;, &expectation;) exact sequence 0&nbsp;&rarr;&nbsp;M&prime;&nbsp;&rarr;&nbsp;M&nbsp;&rarr;&nbsp;M&prime;&prime;&nbsp;&rarr;&nbsp;0. Using these properties we prove that for two complete hereditary cotorsion theories (&Cscr;, &lagran;), (&lagran;, &expectation;) we have <img src="http://www.tandfonline.com/na101/home/literatum/publisher/tandf/journals/content/lagb20/2005/lagb20.v033.i06/agb-200063353/20170907/images/lagb_a_10362644_o_ilm0004.gif"/> (M, N) for any left R modules M, N and for any n&nbsp;&ge;&nbsp;1, where <img src="http://www.tandfonline.com/na101/home/literatum/publisher/tandf/journals/content/lagb20/2005/lagb20.v033.i06/agb-200063353/20170907/images/lagb_a_10362644_o_ilm0005.gif"/> (M, N) are the generalized Tate cohomology modules (see Section 1 for the definition). So in this case we have an occurrence of balance, i.e. the generalized Tate cohomology can be computed either using a left &Cscr;-resolution and a projective resolution of M or using a right &expectation;-resolution and an injective resolution of N.</div>