Модульдердің тензорлық көбейтіндісі және абелдік топтар

Operation that pairs a left and a right 𝑅‐module into an abelian group
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e. g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right module and a left module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.

In practice, it is sometimes more difficult to show that a tensor product of R modules is nonzero than it is to show that it is 0. The universal property gives a convenient way for checking this. To check that a tensor product is nonzero, one can construct an R bilinear map to an abelian group such that 1= f(m,n) \neq 0 This works because if 1= m \otimes n = 0 , then 1= f(m,n) = \bar{f}(m \otimes n) = \bar{(f)}(0) = 0
For example, to see that 1= \Z/p\Z \otimes Z \Z/p\Z , is nonzero, take to be and 1= (m,n) \mapsto mn This says that the pure tensors as long as is nonzero in 1= \Z / p\Z .

Let R be an integral domain with fraction field K.
For any R module M, as R modules, where is the torsion submodule of M.
If M is a torsion R module then and if M is not a torsion module then 1= K \otimes R M \ne 0 If N is a submodule of M such that is a torsion module then as R modules by 1= x \otimes n \mapsto x \otimes n In 1= K \otimes R M , if and only if or 1= m \in M {\operatorname{tor} } In particular, where 1= m \mapsto 1 \otimes m where is the localization of the module at the prime ideal (i. e., the localization with respect to the nonzero elements).

1= R^n \otimes R S = S^n , for any R algebra S (i. e., a free module remains free after extending scalars.) For a commutative ring and a commutative R algebra S, we have: in fact, more generally, where is an ideal. Using 1= \Complex = \R [x]/(x^2 + 1) , the previous example and the Chinese remainder theorem, we have as rings This gives an example when a tensor product is a direct product. 1= \R \otimes {\Z} \Z[i] = \R[i] = \Complex .

In the general case, not all the properties of a tensor product of vector spaces extend to modules. Yet, some useful properties of the tensor product, considered as module homomorphisms, remain.

The dual module of a right R module E, is defined as HomR(E, R) with the canonical left R module structure, and is denoted E∗. The canonical structure is the pointwise operations of addition and scalar multiplication. Thus, E∗ is the set of all R linear maps E → R (also called linear forms), with operations

The dual of a left R module is defined analogously, with the same notation. There is always a canonical homomorphism E → E^(∗∗) from E to its second dual. It is an isomorphism if E is a free module of finite rank. In general, E is called a reflexive module if the canonical homomorphism is an isomorphism.

In the general case, each element of the tensor product of modules gives rise to a left R linear map, to a right R linear map, and to an R bilinear form. Unlike the commutative case, in the general case the tensor product is not an R module, and thus does not support scalar multiplication. Given right R module E and right R module F, there is a canonical homomorphism θ : F ⊗R E^(∗) → HomR(E, F) such that θ(f ⊗ e′) is the map Given left R module E and right R module F, there is a canonical homomorphism θ : F ⊗R E → HomR(E^(∗), F) such that θ(f ⊗ e) is the map
Both cases hold for general modules, and become isomorphisms if the modules E and F are restricted to being finitely generated projective modules (in particular free modules of finite ranks). Thus, an element of a tensor product of modules over a ring R maps canonically onto an R linear map, though as with vector spaces, constraints apply to the modules for this to be equivalent to the full space of such linear maps. Given right R module E and left R module F, there is a canonical homomorphism θ : F^(∗) ⊗R E^(∗) → LR(F × E, R) such that θ(f′ ⊗ e′) is the map (f, e) ↦ ⟨f, f′⟩ ⋅ ⟨e′, e⟩. Thus, an element of a tensor product ξ ∈ F∗ ⊗R E∗ may be thought of giving rise to or acting as an R bilinear map F × E → R.

The most prominent example of a tensor product of modules in differential geometry is the tensor product of the spaces of vector fields and differential forms. More precisely, if R is the (commutative) ring of smooth functions on a smooth manifold M, then one puts

where Γ means the space of sections and the superscript means tensoring p times over R. By definition, an element of is a tensor field of type (p, q). As R modules, is the dual module of 1= \mathfrak{T}^p q
To lighten the notation, put and so 1= E^* = \Gamma(M, T^* M) When p, q ≥ 1, for each (k, l) with 1 ≤ k ≤ p, 1 ≤ l ≤ q, there is an R multilinear map:

where means and the hat means a term is omitted. By the universal property, it corresponds to a unique R linear map:

It is called the contraction of tensors in the index (k, l). Unwinding what the universal property says one sees:

Remark: The preceding discussion is standard in textbooks on differential geometry (e. g., Helgason). In a way, the sheaf theoretic construction (i. e., the language of sheaf of modules) is more natural and increasingly more common; for that, see the section .