7.1 Definition and properties of Tor functor

The Tor functor measures how tensor products deviate from exactness. It takes two modules and produces a new module for each non-negative integer, with Tor₀ being the tensor product itself. Higher Tor functors capture the non-exactness.

Tor is derived from the tensor product functor using projective resolutions. It's additive, preserves direct sums, and vanishes for flat modules. Tor helps detect flatness and appears in long exact sequences, making it a powerful tool in homological algebra.

Definition and Properties

Tor as a Derived Functor

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• Tor functor $\operatorname{Tor}_n^R(M,N)$ measures the degree to which the tensor product $M \otimes_R N$ fails to be exact
  • Takes two R-modules $M$ and $N$ as input and produces a new R-module $\operatorname{Tor}_n^R(M,N)$ for each non-negative integer $n$
  • $\operatorname{Tor}_0^R(M,N)$ is isomorphic to the tensor product $M \otimes_R N$
  • Higher Tor functors $\operatorname{Tor}_n^R(M,N)$ for $n > 0$ measure the deviation from exactness
• Tor is a derived functor obtained by taking a left derived functor of the tensor product functor $- \otimes_R N$
  • Derived functors are a way to extend a functor that is not exact to a sequence of functors that preserve exactness
  • Left derived functors are computed using projective resolutions of the first argument ($M$ in this case)
• Tor is an additive functor covariant in both arguments and preserves direct sums
  • $\operatorname{Tor}_n^R(M_1 \oplus M_2, N) \cong \operatorname{Tor}_n^R(M_1, N) \oplus \operatorname{Tor}_n^R(M_2, N)$
  • $\operatorname{Tor}_n^R(M, N_1 \oplus N_2) \cong \operatorname{Tor}_n^R(M, N_1) \oplus \operatorname{Tor}_n^R(M, N_2)$

Flatness and Vanishing of Tor

• An R-module $M$ is flat if the functor $M \otimes_R -$ is exact
  • Equivalently, $M$ is flat if $\operatorname{Tor}_n^R(M,N) = 0$ for all $n > 0$ and all R-modules $N$
  • Examples of flat modules include free modules, projective modules, and localizations of flat modules
• If either $M$ or $N$ is a flat R-module, then $\operatorname{Tor}_n^R(M,N) = 0$ for all $n > 0$
  • In this case, the tensor product $M \otimes_R N$ is exact, and there is no need for higher Tor functors
• Tor can be used to detect flatness: an R-module $M$ is flat if and only if $\operatorname{Tor}_1^R(M,N) = 0$ for all R-modules $N$
  • This provides a practical way to check flatness using a single Tor functor instead of all higher Tor functors

Computation and Applications

Computing Tor using Resolutions

• Tor functors can be computed using projective resolutions or flat resolutions
  • To compute $\operatorname{Tor}_n^R(M,N)$, take a projective resolution $P_\bullet \to M$ of $M$ and tensor it with $N$ to get a chain complex $P_\bullet \otimes_R N$
  • The homology of this chain complex at the $n$-th position is $\operatorname{Tor}_n^R(M,N)$
• Alternatively, Tor can be computed using flat resolutions of the second argument
  • Take a flat resolution $F_\bullet \to N$ of $N$ and tensor it with $M$ to get a chain complex $M \otimes_R F_\bullet$
  • The homology of this chain complex at the $n$-th position is $\operatorname{Tor}_n^R(M,N)$
• The choice of resolution (projective or flat) depends on the properties of the modules involved and the convenience of computation

Applications and Properties of Tor

• Tor is related to the concept of torsion in module theory
  • An element $x \in M$ is a torsion element if $rx = 0$ for some nonzero $r \in R$
  • The set of torsion elements in $M$ forms a submodule called the torsion submodule $\operatorname{Tor}(M)$
  • $\operatorname{Tor}_1^R(R/I, M)$ is isomorphic to the $I$-torsion submodule of $M$
• Tor appears in the long exact sequence of Tor associated with a short exact sequence of modules
  • Given a short exact sequence $0 \to A \to B \to C \to 0$ and an R-module $M$, there is a long exact sequence of Tor functors: $\cdots \to \operatorname{Tor}_{n+1}^R(C,M) \to \operatorname{Tor}_n^R(A,M) \to \operatorname{Tor}_n^R(B,M) \to \operatorname{Tor}_n^R(C,M) \to \cdots$
  • This long exact sequence can be used to compute Tor functors and study the relationships between modules
• Tor satisfies the dimension shifting property: $\operatorname{Tor}_n^R(M,N) \cong \operatorname{Tor}_{n-1}^R(M,\Omega N)$, where $\Omega N$ is the first syzygy of $N$
  • This property allows for the computation of higher Tor functors using lower ones and syzygies
• Tor is balanced, meaning that $\operatorname{Tor}_n^R(M,N) \cong \operatorname{Tor}_n^R(N,M)$ for all $n \geq 0$
  • This symmetry property simplifies computations and proofs involving Tor functors