Ext Functor - Ring Structure and Module Structure On Specific Exts

Ring Structure and Module Structure On Specific Exts

One more very useful way to view the Ext functor is this: when an element of Extn
R(A, B) = 0 is considered as an equivalence class of maps f: P_n → B for a projective resolution P_* of A ; so, then we can pick a long exact sequence Q_* ending with B and lift the map f using the projectivity of the modules P_m to a chain map f_*: P_* → Q_* of degree -n. It turns out that homotopy classes of such chain maps correspond precisely to the equivalence classes in the definition of Ext above.

Under sufficiently nice circumstances, such as when the ring R is a group ring over a field k, or an augmented k-algebra, we can impose a ring structure on Ext*
R(k, k). The multiplication has quite a few equivalent interpretations, corresponding to different interpretations of the elements of Ext*
R(k, k).

One interpretation is in terms of these homotopy classes of chain maps. Then the product of two elements is represented by the composition of the corresponding representatives. We can choose a single resolution of k, and do all the calculations inside Hom_R(P_*,P_*), which is a differential graded algebra, with cohomology precisely Ext_R(k,k).

The Ext groups can also be interpreted in terms of exact sequences; this has the advantage that it does not rely on the existence of projective or injective modules. Then we take the viewpoint above that an element of Extn
R(A, B) is a class, under a certain equivalence relation, of exact sequences of length n + 2 starting with B and ending with A. This can then be spliced with an element in Extm
R(C, A), by replacing ... → X₁ → A → 0 and 0 → A → Y_n → ... with:

where the middle arrow is the composition of the functions X₁ → A and A → Y_n. This product is called the Yoneda splice.

These viewpoints turn out to be equivalent whenever both make sense.

Using similar interpretations, we find that Ext*
R(k, M) is a module over Ext*
R(k, k), again for sufficiently nice situations.