(Journal of Pure and Applied Algebra 39 (1986) 197-250 korth-Holland 197 AN EXACT SEQUENCE IN THE FIRST VARIABLE FOR TORSOR COHOMOLOGY: THE 2-DIMENSIONAL THEORY OF OBSTRUCTIONS Antonio M. CEGARRA and Enrique R. AZNAR Departamento de Algebra y Fundamentos, Facultad de Ciencias, Universidad de Granada, Granada, Spain Communicated by F.W. Lawvere Received April 1983 Revised July 1984 Introduction In [9] J.W. Duskin gets an inner interpretation for the ‘cotriple’ cohomology of ia monadic category in terms of n-dimensional torsors, generalizing to any dimen- ision Beck’s interpretation of dimension 1. Later, in [15], P. Glenn, using a slightly ,different n-torsor concept, developes a cohomology theory for an exact category in the sense of Barr [4] without requiring free resolutions, which is applicable in both algebraic and topological settings and coincides with the most important cohom- ology theories in the known examples. Clearly this general context is an appropriate one in which to formulate and solve the classical problems considered in these theories of relating them in low dimensions to ‘obstructions and extensions’. This is the object of our paper. More precisely, with each of the classical cohomology theories in Algebra has been associated a theory relating Hi (Zf2 as classically numbered) to the classification of non-singular extensions and W2 (H3 as classically numbered) to obstructions to the existence of such extensions. This problem has been studied using cocycle calcula- tions in several algebraic contexts (e.g. [5], [12], [163, [17], [22], [23], [24]) and with- out these ([13], [26]). In this paper, we interpret the two-dimensional torsors E. under an abelian group A (whose connected components [E.] E Tors’(R, A) define H2(R,A) in the Glenn theory) as obstructions to the existence of (l-dimensional) torsors under the groupoid G(E.) associated to E.. Specifically, we have Theorem 5.3. TorsiR(R, G(E,)) #0 if and only if [E.] =0 in Tors2(R,A). Moreover, we have Theorem 5.4. If [EJ = 0, then the abelian group To_rs’(R, A) acts on the set of iso- morphism classes Torsr,(R, G(E.)) as a principal homogeneous representation. 0022-4049/86/$3.50 0 1986, Elsevier Science Publishers B.V. (North-Holland)