variable (contravariant in A, covariant in B). For a commutative ring R and R-modules A and B, Ext(A, B) is an R-module (using that HomR(A, B) is an R-module in this case). For a non-commutative ring R, Ext(A, B) is only an abelian group, in general. If R is an algebra over a ring S (which means in particular that S is commutative), then Ext(A, B) is at least an S-module. Properties of Ext Here are some of the basic properties and computations of Ext groups. Ext(A, B) ≅ HomR(A, B) for any R-modules A and B. Ext(A, B) = 0 for all i > 0 if the R-module A is projective (for example, free) or if B is injective. The converses also hold: If Ext(A, B) = 0 for all B, then A is projective (and hence Ext(A, B) = 0 for all i > 0). If Ext(A, B) = 0 for all A, then B is injective (and hence Ext(A, B) = 0 for all i > 0). for all i ≥ 2 and all abelian groups A and B. If R is a commutative ring and u in R is not a zero divisor, then for any R-module B. Here B[u] denotes the u-torsion subgroup of B, {x ∈ B: ux = 0}. Taking R to be the ring of integers, this calculation can be used to compute for any finitely generated abelian group A. Generalizing the previous example, one can compute Ext groups when the first module is the quotient of a commutative ring by any regular sequence, using the Koszul complex. For example, if R is the polynomial ring k[x1,...,xn] over a field k, then Ext(k,k) is the exterior algebra S over k on n generators in Ext1. Moreover, Ext(k,k) is the polynomial ring R; this is an example of Koszul duality. By the general properties of derived functors, there are two basic exact sequences for Ext. First, a short exact sequence 0 → K → L → M → 0 of R-modules induces a long exact sequence of the form for any R-module A. Also, a short exact sequence 0 → K → L → M → 0 induces a long exact sequence of the form for any R-module B. Ext takes direct sums (possibly infinite) in the first variable and products in the second variable to products. That is: Let A be a