S is commutative), then Tor(A, B) is at least an S-module. Properties Here are some of the basic properties and computations of Tor groups. Tor(A, B) ≅ A ⊗R B for any right R-module A and left R-module B. Tor(A, B) = 0 for all i > 0 if either A or B is flat (for example, free) as an R-module. In fact, one can compute Tor using a flat resolution of either A or B; this is more general than a projective (or free) resolution. There are converses to the previous statement: If Tor(A, B) = 0 for all B, then A is flat (and hence Tor(A, B) = 0 for all i > 0). If Tor(A, B) = 0 for all A, then B is flat (and hence Tor(A, B) = 0 for all i > 0). By the general properties of derived functors, every short exact sequence 0 → K → L → M → 0 of right R-modules induces a long exact sequence of the form for any left R-module B. The analogous exact sequence also holds for Tor with respect to the second variable. Symmetry: for a commutative ring R, there is a natural isomorphism Tor(A, B) ≅ Tor(B, A). (For R commutative, there is no need to distinguish between left and right R-modules.) If R is a commutative ring and u in R is not a zero divisor, then for any R-module B, where is the u-torsion subgroup of B. This is the explanation for the name Tor. 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 Tor groups that involve 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 is the exterior algebra over k on n generators in Tor1. for all i ≥ 2. The reason: every abelian group A has a free resolution of length 1, since every subgroup of a free abelian group is free abelian. For any ring R, Tor preserves direct sums (possibly infinite) and filtered colimits in each variable. For example, in the first variable, this says that Flat base change: for a commutative flat R-algebra T, R-modules A and B, and an integer i, It