Tor functor

Definition

Properties

• Tor ≅ A ⊗_R B for any right R-module A and left R-module B.
• Tor = 0 for all i > 0 if either A or B is flat 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 resolution.
• There are converses to the previous statement:
• *If Tor = 0 for all B, then A is flat.
• *If Tor = 0 for all A, then B is flat.
• 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
• Symmetry: for a commutative ring R, there is a natural isomorphism Tor ≅ Tor.
• If R is a commutative ring and u in R is not a zero divisor, then for any R-module B,
• 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 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 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,
• For a commutative ring R and commutative R-algebras A and B, Tor has the structure of a graded-commutative algebra over R. Moreover, elements of odd degree in the Tor algebra have square zero, and there are divided power operations on the elements of positive even degree.

  Important special cases

• Group homology is defined by where G is a group, M is a representation of G over the integers, and is the group ring of G.
• For an algebra A over a field k and an A-bimodule M, Hochschild homology is defined by
• Lie algebra homology is defined by, where is a Lie algebra over a commutative ring R, M is a -module, and is the universal enveloping algebra.
• For a commutative ring R with a homomorphism onto a field k, is a graded-commutative Hopf algebra over k. As an algebra, is the free graded-commutative divided power algebra on a graded vector space π_*. When k has characteristic zero, π_* can be identified with the André-Quillen homology D_*.