1 Tor in Homological Algebra
1.1 Motivation: Tensor Products and Exactness
In homological algebra, the tensor product is a basic operation for combining algebraic objects such as modules. While it is additive in each variable, it does not generally preserve exactness of sequences. This defect is the reason Tor enters the theory: Tor quantifies how much tensoring with a fixed module fails to keep a given exact sequence exact. Conceptually, it detects “obstructions” that are invisible at the level of ordinary tensor products.
1.2 Derived Functors Point of View
Tor is best understood as a derived functor of tensoring. Fix a module (or bimodule) and consider the functor that sends another module to its tensor product with the fixed one. Since tensoring is right exact but not necessarily left exact, its failure to be exact is measured by derived functors. The resulting sequence of functors, denoted \(\mathrm{Tor}_n\), extends tensoring to a homological setting where exactness phenomena are systematically organized.
1.3 Definition via Resolutions
A standard construction defines \(\mathrm{Tor}_n^R(M,N)\) using a projective (or more generally flat) resolution of one of the modules. One tensors the resolution with the other module and then takes homology of the resulting chain complex. This yields groups that depend only on the modules \(M\) and \(N\) and not on the particular chosen resolution, provided the resolution satisfies the required exactness and projectivity/flatness conditions.
1.4 Notation and Basic Properties
Common notation includes \(\mathrm{Tor}_n^R(M,N)\) for modules over a ring \(R\). The groups \(\mathrm{Tor}_n\) form a sequence indexed by \(n \ge 0\), where \(\mathrm{Tor}_0\) reproduces the ordinary tensor product: \[ \mathrm{Tor}_0^R(M,N) \cong M \otimes_R N. \] For \(n>0\), \(\mathrm{Tor}_n\) captures higher-order incompatibilities between \(M\) and \(N\) relative to exactness. These functors are additive and behave well under module maps, leading to a functorial framework.
2 Algebraic Setup
2.1 Modules over a Ring
Let \(R\) be a ring, and consider left \(R\)-modules and right \(R\)-modules (or suitable variants using bimodules). To form the tensor product \(M \otimes_R N\), one typically takes \(M\) as a right \(R\)-module and \(N\) as a left \(R\)-module, ensuring the balancing relation makes sense. Throughout the theory, the ring is fixed while homological constructions vary with the modules.
2.2 Bimodules and Module Structures
Bimodules provide a flexible way to organize actions from both sides. If \(B\) is an \(R\)-\(S\) bimodule and \(M\) is an appropriate module over one side, then one can tensor \(B\) with \(M\) over the relevant ring. Tor can then be defined in settings where the ring changes through bimodule structures, such as when working with change of rings or base change. In such contexts, careful tracking of left and right actions determines which tensor products and Tor groups are relevant.
2.3 Tensor Product Review
The tensor product \(M \otimes_R N\) is constructed as a quotient of the free abelian group on symbols \(m \otimes n\) by the bilinearity and balancing relations. It is characterized by a universal property: bilinear maps from \(M \times N\) that are balanced over \(R\) correspond uniquely to homomorphisms out of \(M \otimes_R N\). This universal viewpoint clarifies why tensoring is right exact: it respects colimits and cokernels, but may not preserve kernels.
2.4 Projective, Flat, and Free Resolutions
Resolutions are exact complexes used to replace a module by a more tractable one. A projective resolution uses projective modules, for which tensoring behaves particularly well with respect to exactness. A flat resolution uses flat modules, a weaker condition than projectivity but still sufficient to preserve exactness when tensoring. Free resolutions are special cases of projective resolutions in which modules have chosen bases. In computing Tor, one typically uses a projective or flat resolution, and the derived homology turns out independent of the chosen resolution type as long as the necessary conditions are met.
3 Computation of Tor
3.1 Using Projective Resolutions
Given a projective resolution \(P_\bullet \to M\), the chain complex \(P_\bullet \otimes_R N\) is formed by tensoring each term with \(N\). The \(n\)-th Tor group is then defined as the homology: \[ \mathrm{Tor}_n^R(M,N) = H_n(P_\bullet \otimes_R N). \] This method is conceptually direct because projective modules interact favorably with tensor products. The result depends only on \(M\) and \(N\), not on the resolution, because different projective resolutions are related by chain homotopies that induce the same homology groups.
3.2 Using Flat Resolutions
Flat resolutions work similarly. If \(F_\bullet \to M\) is a flat resolution, then \(F_\bullet \otimes_R N\) computes \(\mathrm{Tor}_n^R(M,N)\) as its homology. The key point is that flatness ensures tensoring preserves exactness in the directions needed to make the homological construction stable. This approach is often preferred when projective resolutions are cumbersome or unavailable.
3.3 Chain Complex Construction
In practical computations, one chooses a resolution and then tracks differentials after tensoring. The construction is systematic:
- Choose a resolution \( \cdots \to P_2 \to P_1 \to P_0 \to M \to 0\).
- Tensor to obtain \(\cdots \to P_2 \otimes_R N \to P_1 \otimes_R N \to P_0 \otimes_R N \to 0\).
- Compute homology \(H_n\) of this complex.
The degrees shift in the expected way: the homology in degree \(0\) recovers \(M \otimes_R N\), while higher degrees detect nontrivial cycles modulo boundaries formed by the tensoring process.
3.4 Computing Low-Degree Tor (0 and 1)
For low degrees, Tor admits concrete interpretations. As noted, \(\mathrm{Tor}_0^R(M,N)\) is the tensor product. For \(\mathrm{Tor}_1\), one can relate it to kernels and presentations. For example, if \(M\) has a presentation \(P_1 \to P_0 \to M \to 0\) with \(P_0,P_1\) projective, then tensoring yields a sequence \[ P_1 \otimes_R N \to P_0 \otimes_R N \to M \otimes_R N \to 0, \] and \(\mathrm{Tor}_1^R(M,N)\) measures the failure of this sequence to be exact at the left end, i.e., it is essentially the homology in degree \(1\). In practice, computing \(\mathrm{Tor}_1\) often reduces to identifying a kernel after tensoring a short part of a resolution.
4 Core Theorems and Relationships
4.1 Long Exact Sequences for Tor
Tor enjoys strong compatibility with short exact sequences of modules. If \[ 0 \to N' \to N \to N'' \to 0 \] is exact (with appropriate module structures), then applying Tor produces a long exact sequence relating \(\mathrm{Tor}_n^R(M,N')\), \(\mathrm{Tor}_n^R(M,N)\), and \(\mathrm{Tor}_n^R(M,N'')\) for all \(n\ge 0\). This theorem is central because it enables inductive computations: knowledge of Tor for two of the modules constrains the Tor for the third.
4.2 Change-of-Rings and Base-Change Ideas
When rings and bimodules enter, Tor can be compared across different base rings. Typical change-of-rings phenomena arise when \(R\to S\) is a ring homomorphism and one considers \(S\) as an \(R\)-module (or uses \(S\) as a base ring for tensoring). Under suitable hypotheses, Tor groups over \(R\) can be related to Tor groups over \(S\), often through spectral sequences or derived tensor product identities. The overarching theme is that Tor behaves naturally with respect to how module categories are related by functors induced from ring maps.
4.3 Tor Independence and Functoriality
The construction of \(\mathrm{Tor}_n\) is independent of the chosen resolution method (within the allowable class, such as projective or flat resolutions) and is functorial in both arguments. Functoriality means that module homomorphisms induce morphisms between Tor groups. Independence is ensured by homological comparison: different resolutions yield chain homotopy equivalent complexes after tensoring, leading to isomorphic homology.
4.4 Spectral Sequence Connections (Overview)
Spectral sequences provide a general mechanism for computing complicated derived invariants in stages. Tor can appear as an \(E\)-page term in spectral sequences associated with filtered complexes or derived functor compositions. While the specific form depends on the construction, the conceptual role is consistent: spectral sequences allow one to break down the computation of Tor into layers, often relating it to other homological data such as Ext groups or homology of intermediate constructions.
5 Structural Properties
5.1 Vanishing Criteria
Tor groups sometimes vanish entirely. For example, if one of the modules is flat over the ring \(R\), then \(\mathrm{Tor}_n^R(M,N)=0\) for all \(n\ge 1\). More generally, vanishing can indicate strong regularity in how the module sits inside the category of \(R\)-modules. Conversely, nonvanishing Tor often signals that the module cannot be approximated by flat behavior relative to the other module.
5.2 Symmetry and Bimodule Variants
When \(R\) is commutative and modules are taken in a compatible manner, Tor is symmetric in the sense that \(\mathrm{Tor}_n^R(M,N)\) is naturally isomorphic to \(\mathrm{Tor}_n^R(N,M)\). In noncommutative settings, symmetry typically fails without additional structure, but there are still corresponding bimodule formulations that preserve a form of duality or balanced behavior. Careful bookkeeping of left versus right actions replaces naive symmetry.
5.3 Behavior Under Direct Sums and Products
Tor is additive with respect to direct sums: tensoring distributes over direct sums, and homology respects this additivity. As a result, Tor of a direct sum in either variable decomposes as a direct sum of Tor groups. For direct products, behavior is subtler because tensor products do not generally commute with infinite products. In many algebraic contexts, one instead uses finiteness assumptions or replaces products with derived limits when necessary.
5.4 Depth and Regularity Connections (Algebraic View)
In commutative algebra, Tor is tied to depth and regularity phenomena. Modules whose resolutions have certain growth patterns or controlled homological dimensions often correspond to vanishing patterns in Tor. While the precise relationship depends on the framework (such as local rings and the language of depth, Cohen–Macaulay conditions, or Castelnuovo–Mumford regularity), the recurring idea is that Tor acts as an indicator of how far a module is from being “homologically simple.”
6 Examples
6.1 Tor over a Field
If \(R\) is a field \(k\), then every \(k\)-module is a vector space and is automatically flat (indeed free). Consequently, \(\mathrm{Tor}_n^k(M,N)=0\) for all \(n\ge 1\), and \(\mathrm{Tor}_0^k(M,N)\cong M\otimes_k N\). This example highlights that Tor measures the failure of flatness and exactness, which disappears over fields.
6.2 Tor for Modules with Known Resolutions
When modules admit explicit projective or free resolutions, Tor can be computed by direct homology calculations. Common settings include quotient modules defined by ideals in polynomial rings, or modules presented by generators and relations. Given a short free resolution, tensoring with the second module yields a computable chain complex, often turning \(\mathrm{Tor}_n\) into kernels and cokernels derived from matrices representing module maps.
6.3 Example: Tor and Short Exact Sequences
Suppose \(N'\to N\to N''\) fits into a short exact sequence. The long exact Tor sequence then relates \(\mathrm{Tor}_1^R(M,N'')\) to the failure of exactness of tensoring the original sequence. If, for instance, \(\mathrm{Tor}_1^R(M,N')\) vanishes, the long exact sequence simplifies and yields information about \(\mathrm{Tor}_1^R(M,N'')\) from the induced maps on tensors. This illustrates how Tor turns local extension data into homological invariants.
6.4 Example: Comparing Different Resolutions
Consider two different projective resolutions of the same module \(M\). After tensoring with \(N\), one obtains two chain complexes whose homology groups are both \(\mathrm{Tor}_n^R(M,N)\). The comparison is established via chain maps between resolutions, which are homotopy equivalences up to the relevant notions of derived equivalence. In computation, this independence allows one to choose whichever resolution is easiest, knowing the resulting Tor groups coincide.
7 Applications and Interpretations
7.1 Interpreting Tor as “Derived Tensor” Data
Tor is frequently interpreted as the homology of the derived tensor product. In this viewpoint, one replaces an ordinary tensor product—insufficiently exact—with a construction that accounts for higher homological information. The groups \(\mathrm{Tor}_n\) then represent the layers of this derived object, measuring how non-flat behavior propagates through the tensoring process.
7.2 Module-Theoretic Consequences
Tor provides criteria and consequences about module extensions, presentations, and flatness. Vanishing of higher Tor often signals that one module is flat relative to another context, while nonvanishing Tor can be used to detect hidden torsion in tensor products. Through long exact sequences, Tor also controls how module properties change when passing to submodules or quotients.
7.3 Connections to Homological Invariants
Although Tor is distinct from Ext, both belong to the family of derived functor invariants and interact through standard homological constructions. Tor frequently pairs with Ext in computations involving resolutions and in duality frameworks. Even when the focus is purely on tensoring, the surrounding machinery of homological algebra often brings Tor into dialogue with other invariants that measure different failure modes of exactness.
7.4 Computational Approaches and Tools
Computing Tor ranges from hand calculations in low degrees to algorithmic methods in commutative algebra. Practical approaches include using minimal resolutions, exploiting symmetry or grading structures (especially in graded rings), and applying long exact sequences to reduce complexity. Modern computational algebra systems can also assist by automating resolution generation and homology calculations in settings where input data is finite and structured. In all cases, the guiding strategy is to choose resolutions that make the resulting tensor complex tractable.