10.1 Construction and properties of tensor products

Tensor products are a powerful tool in commutative algebra, combining modules to create new structures. They're defined by a universal property and have key properties like associativity and distributivity. Understanding tensor products is crucial for grasping advanced algebraic concepts.

Computations with tensor products involve free modules, quotients, and cyclic groups. These calculations reveal important relationships between algebraic structures and help solve complex problems in module theory and beyond. Mastering tensor products opens doors to deeper algebraic insights.

Tensor Product Fundamentals

Tensor product definition and properties

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• Tensor product $M \otimes_R N$ combines modules $M$ and $N$ over commutative ring $R$, creating new algebraic structure
• Construction starts with free module on $M \times N$, quotients by submodule generated by relations preserving bilinearity
• Relations ensure $(m + m', n) \sim (m, n) + (m', n)$, $(m, n + n') \sim (m, n) + (m, n')$, and $(rm, n) \sim (m, rn)$ for $r \in R$
• Universal property establishes existence of bilinear map $\phi: M \times N \to M \otimes_R N$
• For any bilinear map $f: M \times N \to P$, unique linear map $\tilde{f}: M \otimes_R N \to P$ exists, represented by commutative diagram

Computation of tensor products

• Free modules $F_1 \otimes_R F_2$ with bases $\{e_i\}$ and $\{f_j\}$ yield free module with basis $\{e_i \otimes f_j\}$
• $M \otimes_R R \cong M$ for any $R$-module $M$, simplifying computations
• Quotient modules follow $(M/N) \otimes_R P \cong (M \otimes_R P) / (N \otimes_R P)$
• $\mathbb{Z}/m\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/\gcd(m,n)\mathbb{Z}$ demonstrates tensor product of finite cyclic groups
• $\mathbb{Q} \otimes_\mathbb{Z} \mathbb{Q} \cong \mathbb{Q}$ showcases tensor product of infinite-dimensional vector spaces

Properties and Applications

Proofs for tensor product properties

• Associativity $(L \otimes_R M) \otimes_R N \cong L \otimes_R (M \otimes_R N)$ proved using universal property
• Commutativity $M \otimes_R N \cong N \otimes_R M$ established by isomorphism $m \otimes n \mapsto n \otimes m$
• Distributivity $L \otimes_R (M \oplus N) \cong (L \otimes_R M) \oplus (L \otimes_R N)$ shown using universal property and direct sum properties
• Zero module property $M \otimes_R 0 \cong 0$ holds for any $R$-module $M$
• Scalar multiplication compatibility ensures $r(m \otimes n) = (rm) \otimes n = m \otimes (rn)$ for $r \in R$

Tensor products vs bilinear maps

• Bilinear maps $f: M \times N \to P$ linear in each argument separately
• Tensor product serves as universal bilinear map, factoring all bilinear maps
• Multilinear maps represented as linear maps on tensor products
• Dual spaces relationship $(M^* \otimes_R N^*) \cong (M \otimes_R N)^*$ holds for finite-dimensional vector spaces
• Change of rings allows $M \otimes_R S$ to become $S$-module for $R$-algebra $S$
• Flatness of modules determined by exactness of $M \otimes_R -$ functor
• Problem-solving employs universal property, reduction to simpler tensor products, and identification of bilinear maps