1 Tor Functor Basics
1.1 Definition via derived functors
For a ring \(R\) and left \(R\)-modules \(M\) and right \(R\)-modules \(N\), the tensor product \(N\otimes_R M\) is right-exact in each variable separately. The Tor functor remedies the loss of exactness by passing to derived functors of the tensor product. Concretely, \(\operatorname{Tor}_i^R(N,M)\) is defined as the \(i\)-th homology group obtained from a derived tensor product that replaces either module by an appropriate resolution, ensuring that the tensor product is computed in a homotopically correct way.
1.2 Tor groups and grading
The index \(i\ge 0\) measures how far the tensor product deviates from preserving exact sequences. The collection \(\{\operatorname{Tor}_i^R(N,M)\}_{i\ge 0}\) forms a graded family of abelian groups. In many contexts (e.g., graded rings and graded modules), one also considers an internal grading coming from degrees of homogeneous elements; then Tor acquires both homological degree and internal degree.
1.3 Relationship to tensor product exactness
If \(N\) is flat as a right \(R\)-module, then functorial tensoring with \(N\) preserves injections, so \(\operatorname{Tor}_i^R(N,M)=0\) for all \(i\ge 1\) and all left \(R\)-modules \(M\). More generally, Tor detects obstructions: given a short exact sequence \(0\to A\to B\to C\to 0\), tensoring with \(N\) typically yields a sequence that is exact at the level of \(0\)-th homology but may fail to be exact elsewhere. The failure is recorded by the higher Tor groups.
1.4 Functoriality and naturality
Tor is functorial in both module arguments. A morphism \(M\to M'\) induces maps \(\operatorname{Tor}_i^R(N,M)\to \operatorname{Tor}_i^R(N,M')\), and similarly in the \(N\)-variable. These induced maps are natural with respect to compositions, and they are compatible with the homological structures used in defining Tor (resolutions, chain maps, and homotopies).
2 Computation of Tor
2.1 Using projective resolutions
A standard computational approach is to resolve one module by projective modules. For example, take a projective resolution \(P_\bullet \to M\). Tensor the complex \(N\otimes_R P_\bullet\) and take homology: \[ \operatorname{Tor}_i^R(N,M)\cong H_i(N\otimes_R P_\bullet). \] Projective resolutions exist over any ring and are often convenient because tensoring with a projective complex reflects the derived tensor product directly.
2.2 Using flat resolutions
Flat resolutions can also be used. If \(F_\bullet\to M\) is a flat resolution, then \[ \operatorname{Tor}_i^R(N,M)\cong H_i(N\otimes_R F_\bullet). \] Flat resolutions may be preferable when projectives are difficult to describe. Since flat modules preserve exactness under tensor product, a flat resolution controls the same derived information as a projective one.
2.3 Chain complexes and homology viewpoint
Tor is fundamentally a homology theory applied to a tensor product of chain complexes. The construction proceeds as follows: start with a resolution \(P_\bullet\) of a module \(M\), form the chain complex \(N\otimes_R P_\bullet\), and compute its homology. This viewpoint aligns Tor with broader homological methods, enabling tools like spectral sequences and comparison theorems.
2.4 Independence of resolution choice
Although computations involve a chosen resolution, the resulting Tor groups are well-defined up to canonical isomorphism. If two projective resolutions of \(M\) are used, there is a comparison chain map between them that is unique up to chain homotopy; tensoring with \(N\) preserves homotopy equivalences, yielding isomorphic homology groups. Thus, Tor depends only on \(N\) and \(M\), not on the particular model of their resolutions.
2.5 Common computation shortcuts
2.5.1 Vanishing criteria for Tor
Vanishing of Tor can simplify computations considerably. Typical criteria include:
- If one argument is flat, then all higher Tor groups vanish.
- If \(M\) has projective dimension at most \(d\), then \(\operatorname{Tor}_i^R(N,M)=0\) for \(i>d\).
- In settings such as modules over fields, nontrivial Tor often collapses to zero beyond degree \(0\), reflecting semisimplicity and exactness properties.
3 Structural Properties
3.1 Bilinearity in the module arguments
Tor is additive in each variable. For fixed \(N\), the functor \(M\mapsto \operatorname{Tor}_i^R(N,M)\) is additive on short exact sequences and commutes with finite direct sums; analogous statements hold for \(N\). This behavior comes from the fact that tensor products and homology are compatible with direct sums and that resolutions can be chosen compatibly with decompositions.
3.2 Long exact sequences in Tor
Given a short exact sequence of left \(R\)-modules \(0\to A\to B\to C\to 0\), tensoring with a fixed right module \(N\) and passing to derived homology yields a long exact sequence: \[ \cdots \to \operatorname{Tor}_{i}^R(N,A)\to \operatorname{Tor}_{i}^R(N,B)\to \operatorname{Tor}_{i}^R(N,C)\to \operatorname{Tor}_{i-1}^R(N,A)\to \cdots. \] This sequence provides an efficient way to compute Tor for complex modules by reducing to simpler ones.
3.3 Dimension shifting
Dimension shifting relates Tor groups in different degrees. If \(0\to K\to P\to M\to 0\) is exact with \(P\) projective, then for \(i\ge 1\) one obtains \[ \operatorname{Tor}_i^R(N,M)\cong \operatorname{Tor}_{i-1}^R(N,K). \] By repeatedly replacing a module with the kernel of a surjection from a projective module, one can reduce higher-degree Tor to lower-degree computations.
3.4 Change of rings and base change
Tor behaves predictably under ring maps. For a ring homomorphism \(R\to S\), one can compare \(\operatorname{Tor}^R\) and \(\operatorname{Tor}^S\) through base change, often using derived tensor products. In many algebraic-geometric applications, this allows computation after extending scalars or localizing, provided the relevant hypotheses (such as flatness of certain extensions) are satisfied.
3.5 Compatibility with module morphisms
Maps between modules induce maps between their Tor groups in a manner consistent with the long exact sequences and naturality of derived functors. Compatibility typically means that Tor commutes with boundary morphisms arising from short exact sequences, ensuring that diagrammatic constructions remain coherent.
4 Examples and Worked Computations
4.1 Tor over a field
If \(R=k\) is a field and \(M,N\) are vector spaces over \(k\), every module is flat and projective. Consequently, \[ \operatorname{Tor}_i^k(N,M)=0 \quad \text{for } i\ge 1, \] and \(\operatorname{Tor}_0^k(N,M)\cong N\otimes_k M\). This case illustrates how Tor measures non-flatness: when there is none, the higher groups disappear.
4.2 Tor with the ground ring as a module
For any ring \(R\), the module \(R\) is flat over itself. Hence for any right \(R\)-module \(N\) and left \(R\)-module \(M\), \[ \operatorname{Tor}_i^R(N,R)=0 \quad (i\ge 1), \] and \(\operatorname{Tor}_0^R(N,R)\cong N\otimes_R R\cong N\). Similarly, \(\operatorname{Tor}_i^R(R,M)=0\) for \(i\ge 1\), with \(\operatorname{Tor}_0^R(R,M)\cong M\).
4.3 Tor over principal ideal domains
Over a PID, finitely generated modules admit decompositions that make homological computations manageable. Cyclic modules such as \(R/(p^n)\) have short projective resolutions of length one, and Tor with another finitely generated module can be computed by decomposing the second module into primary components. The resulting Tor groups often reflect annihilators and \(p\)-power torsion lengths.
4.4 Tor for quotient modules R/I with R/J
A common family of examples involves quotient rings. Let \(M=R/I\) and consider \(\operatorname{Tor}_i^R(R/J, R/I)\). When \(I\) and \(J\) are such that one can use short resolutions (for instance, when \(I\) is generated by a nonzerodivisor or forms part of a regular sequence), Tor can be expressed in terms of intersections and sums of ideals and in related Ext computations. In general, one computes via resolutions of \(R/I\) and then tensors with \(R/J\).
4.5 Computing Tor for simple cyclic modules
Consider \(R\) a ring and modules \(M=R/I\) and \(N=R/J\). If \(I=(a)\) is principal and \(a\) is not a zero divisor in a suitable context, then \(R/(a)\) has a two-term resolution \[ 0\to R \xrightarrow{\cdot a} R \to R/(a)\to 0. \] Tensoring with \(R/J\) gives a complex whose homology yields
- \(\operatorname{Tor}_1^R(R/J, R/(a))\), which corresponds to elements killed by multiplication by \(a\) modulo \(J\),
- and \(\operatorname{Tor}_0^R(R/J, R/(a))\cong R/(J+a)\).
When \(a\) is a zero divisor or when multiple generators are involved, the resolution becomes longer and higher Tor groups may appear.
5 Tor and Homological Constructions
5.1 Tor versus Ext
Tor and Ext are dual in spirit but differ in variance and input. Ext is the right derived functor of \(\operatorname{Hom}\), whereas Tor is derived from \(\otimes\). While there is no universal isomorphism between \(\operatorname{Tor}\) and \(\operatorname{Ext}\), certain circumstances connect them, such as when modules are finite over a ring with appropriate duality or when one uses derived categories where both constructions arise from similar formalism.
5.2 Tor and projective/flat dimension
The vanishing range of Tor encodes homological dimensions. If \(\operatorname{Tor}_i^R(N,M)=0\) for all sufficiently large \(i\) and for all \(N\), then \(M\) has finite projective dimension bounded by that range. Dually, one can use Tor-vanishing with varying \(N\) to test flatness of modules and to estimate the flat dimension in relative settings.
5.3 Flatness criteria via Tor
A module \(M\) is flat if and only if \(\operatorname{Tor}_1^R(N,M)=0\) for all right \(R\)-modules \(N\). Often it is enough to test this vanishing on a restricted class of modules (for example, finitely presented modules under mild hypotheses). This characterization provides an effective diagnostic: Tor transforms a qualitative notion (flatness) into computable conditions.
5.4 Relation to Tor spectral sequences
In complex situations where Tor groups arise from iterated derived tensors, spectral sequences often appear. A Tor spectral sequence can be constructed from a double complex built out of resolutions of both variables or from filtered derived tensor products. Spectral sequences provide a systematic way to compute Tor by successive approximations, yielding information about higher Tor groups from the data on simpler pages.
6 Applications and Uses
6.1 Commutative algebra applications (depth and regularity context)
Tor is widely used to study invariants of modules over commutative rings. In particular, Tor can detect whether sequences behave like regular sequences and can relate to depth and Castelnuovo–Mumford regularity through homological algebra of graded modules. In practice, Betti numbers in a minimal free resolution are encoded by Tor groups, making Tor central for understanding growth of syzygies.
6.2 Behavior under localization
Localization is compatible with Tor under suitable hypotheses. If \(R\to R_\mathfrak{p}\) is localization at a prime \(\mathfrak{p}\), then Tor groups localize: \[ \operatorname{Tor}_i^{R_\mathfrak{p}}(N_\mathfrak{p}, M_\mathfrak{p}) \cong \left(\operatorname{Tor}_i^R(N,M)\right)_\mathfrak{p} \] in standard settings. This allows one to reduce global questions to local computations and to compare module behavior at different primes.
6.3 Connections to Künneth-type reasoning
Künneth formulas relate homology of tensor products to Tor and tensor of homologies. While Künneth statements often appear in topological settings, algebraic versions for chain complexes similarly involve Tor terms that account for extension phenomena. In commutative algebra, Tor frequently plays the role of the “correction term” that completes naive tensor computations.
6.4 Detecting module interactions in algebraic invariants
Tor measures how two modules “overlap” through relations that cannot be resolved by tensoring alone. This detection is reflected in invariants such as Betti numbers, syzygies, and degrees of minimal generators. Consequently, Tor is used as a diagnostic tool for understanding whether modules share hidden dependencies induced by the ring’s structure.
7 Conceptual Extensions
7.1 Tor in graded settings
When \(R\) is a graded ring and \(M,N\) are graded modules, resolutions can be chosen to respect grading. Tor then becomes a bigraded object, with one grading given by the homological index \(i\) and another by internal degree. This refinement is crucial in computations related to Hilbert series, regularity, and minimal graded free resolutions.
7.2 Tor over noncommutative rings (module sidedness)
In the noncommutative setting, sidedness matters: the tensor product \( -\otimes_R-\) requires a right module and a left module. Accordingly, Tor is defined for a right \(R\)-module \(N\) and a left \(R\)-module \(M\), producing \(\operatorname{Tor}_i^R(N,M)\). The definitions and formal properties remain similar, but careful attention to module structures is required for correctness.
7.3 Higher Tor and iterative derived tensor products
Higher Tor groups can be interpreted as the homology of derived tensor products. Iterating derived tensor constructions yields information about more intricate interactions between modules, often controlled by the structure of resolutions. From a conceptual standpoint, the higher-degree groups capture successively deeper failure of exactness and more subtle extension patterns.
7.4 Derived categories perspective (conceptual overview)
In derived categories, Tor arises from the derived tensor product \(N \otimes_R^{\mathbb{L}} M\). The homology of this derived object recovers Tor: \[ \operatorname{Tor}_i^R(N,M)\cong H_i\!\left(N \otimes_R^{\mathbb{L}} M\right). \] This perspective clarifies functoriality, base change, and the compatibility of Tor with other derived constructions. It also explains why spectral sequences naturally emerge: they reflect the filtration of derived tensor products or the passage from one resolution level to another.