🧮Commutative Algebra Unit 10 – Tensor Products and Flatness

Key Concepts and Definitions

• Tensor product $M \otimes_R N$ constructs a new module from two given modules $M$ and $N$ over a commutative ring $R$
• Universal property characterizes the tensor product as the "most general" bilinear map from $M \times N$
• Flatness measures how well the tensor product preserves exact sequences
  • Flat modules behave similarly to free modules under tensor products
• Torsion modules have elements that become zero when multiplied by some non-zero element of the ring
• Localization $S^{-1}R$ of a ring $R$ at a multiplicative set $S$ allows division by elements of $S$
• Noetherian rings satisfy the ascending chain condition on ideals (every ideal is finitely generated)
• Exact sequences generalize the notion of injectivity and surjectivity to sequences of maps between modules

Tensor Products: Basics and Construction

• Tensor product $M \otimes_R N$ is a quotient of the free module generated by symbols $m \otimes n$ for $m \in M$, $n \in N$
• Quotient is taken by the submodule generated by relations ensuring bilinearity:
  • $(m_1 + m_2) \otimes n = m_1 \otimes n + m_2 \otimes n$
  • $m \otimes (n_1 + n_2) = m \otimes n_1 + m \otimes n_2$
  • $(rm) \otimes n = m \otimes (rn) = r(m \otimes n)$ for $r \in R$
• Universal property states that any bilinear map $M \times N \to P$ factors uniquely through the tensor product
• Tensor product is unique up to unique isomorphism by the universal property
• Tensor products can be constructed over any commutative ring $R$ (integers, polynomial rings, etc.)
• Tensor products are associative: $(M \otimes_R N) \otimes_R P \cong M \otimes_R (N \otimes_R P)$
• Tensor products are symmetric: $M \otimes_R N \cong N \otimes_R M$

Properties of Tensor Products

• Right exact functor: tensoring preserves surjections and cokernels
  • If $M \to N \to P \to 0$ is exact, then so is $M \otimes_R Q \to N \otimes_R Q \to P \otimes_R Q \to 0$ for any $R$-module $Q$
• Commutes with direct sums: $(M \oplus N) \otimes_R P \cong (M \otimes_R P) \oplus (N \otimes_R P)$
• Base change: if $f: R \to S$ is a ring homomorphism, then $M \otimes_R S \cong S \otimes_R M$ as $S$-modules
• Tensor product with free modules: $M \otimes_R R^n \cong M^n$ (direct sum of $n$ copies of $M$)
• Tensor product with quotient modules: $(M/N) \otimes_R P \cong (M \otimes_R P) / (N \otimes_R P)$
• Tensor product localizes: $S^{-1}(M \otimes_R N) \cong (S^{-1}M) \otimes_{S^{-1}R} (S^{-1}N)$
• Hom-tensor adjunction: $\text{Hom}_R(M \otimes_R N, P) \cong \text{Hom}_R(M, \text{Hom}_R(N, P))$

Flatness: Introduction and Motivation

• Flat modules are a generalization of free modules that behave well under tensor products
• Motivation: understand when tensor products preserve exact sequences
  • Free modules always preserve exact sequences, but not all modules do
• Definition: an $R$-module $M$ is flat if the functor $- \otimes_R M$ is exact (preserves exact sequences)
  • Equivalently, tensoring with $M$ preserves injective maps
• Flat modules form a larger class than free modules, but still have good properties
• Flatness is a local property: $M$ is flat if and only if $M_{\mathfrak{p}}$ is flat over $R_{\mathfrak{p}}$ for all prime ideals $\mathfrak{p}$
• Flatness is preserved under base change, direct sums, and direct limits
• Projective modules (direct summands of free modules) are always flat, but the converse is not true

Criteria for Flatness

• Tensor criterion: $M$ is flat if and only if for every finitely generated ideal $I$, the natural map $I \otimes_R M \to M$ is injective
  • Intuition: tensoring with $M$ does not create new relations among elements of $I$
• Localization criterion: $M$ is flat if and only if for every finitely generated ideal $I$, the natural map $I \otimes_R M \to IM$ is an isomorphism
• Torsion-free criterion: if $R$ is an integral domain, then $M$ is flat if and only if $M$ is torsion-free
  • A module $M$ is torsion-free if $rm = 0$ for $r \in R$, $m \in M$ implies $r = 0$ or $m = 0$
• Local criterion: $M$ is flat if and only if $M_{\mathfrak{p}}$ is free over $R_{\mathfrak{p}}$ for all prime ideals $\mathfrak{p}$
• Finite presentation criterion: if $M$ is finitely presented, then $M$ is flat if and only if $M_{\mathfrak{p}}$ is free over $R_{\mathfrak{p}}$ for all maximal ideals $\mathfrak{p}$
• Over a local ring $(R, \mathfrak{m})$, an $R$-module $M$ is flat if and only if any of the following equivalent conditions hold:
  • $M$ is free
  • $M$ is projective
  • $\text{Tor}_1^R(R/\mathfrak{m}, M) = 0$

Applications of Tensor Products and Flatness

• Change of base ring: tensor products allow changing the base ring of a module
  • If $f: R \to S$ is a ring homomorphism and $M$ is an $R$-module, then $S \otimes_R M$ is an $S$-module
• Extension of scalars: tensor products can be used to extend the scalars of a vector space
  • If $V$ is a vector space over a field $k$ and $K$ is an extension field of $k$, then $K \otimes_k V$ is a vector space over $K$
• Exterior algebra: tensor products are used to construct the exterior algebra of a module
  • The exterior algebra $\bigwedge M$ is the quotient of the tensor algebra $T(M)$ by the ideal generated by elements of the form $m \otimes m$
• Flatness and completions: if $R$ is a Noetherian ring and $I$ is an ideal, then the $I$-adic completion $\hat{R}$ is flat over $R$
  • Allows studying properties of $R$ by passing to its completion
• Flatness and fibers: if $f: X \to Y$ is a morphism of schemes, then $f$ is flat if and only if for every point $y \in Y$, the fiber $X_y$ is flat over the local ring $\mathcal{O}_{Y,y}$
  • Geometric interpretation of flatness as fibers varying continuously
• Flatness and deformations: flat families of schemes are used to study deformations and moduli spaces in algebraic geometry

Examples and Counterexamples

• $\mathbb{Z}/n\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/m\mathbb{Z} \cong \mathbb{Z}/\gcd(n,m)\mathbb{Z}$
  • Tensor product of cyclic groups computes their greatest common divisor
• $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z} \cong 0$ for any $n > 0$
  • Tensoring with $\mathbb{Q}$ kills torsion
• $k[x] \otimes_k k[y] \cong k[x,y]$ for a field $k$
  • Tensor product of polynomial rings gives the polynomial ring in two variables
• $\mathbb{Z}/2\mathbb{Z}$ is not flat over $\mathbb{Z}$
  • The sequence $0 \to 2\mathbb{Z} \to \mathbb{Z}$ is exact, but $0 \to 2\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/2\mathbb{Z}$ is not exact
• $\mathbb{Q}$ is flat over $\mathbb{Z}$
  • $\mathbb{Q}$ is torsion-free and $\mathbb{Z}$ is an integral domain
• $\mathbb{Z}_p$ (p-adic integers) is flat over $\mathbb{Z}$
  • Completion of $\mathbb{Z}$ at the prime ideal $(p)$ is flat

Advanced Topics and Connections

• Derived functors $\text{Tor}_i^R(M,N)$ measure the failure of the tensor product to be exact
  • $\text{Tor}_0^R(M,N) \cong M \otimes_R N$ and $\text{Tor}_i^R(M,N) = 0$ for all $i > 0$ if and only if $M$ or $N$ is flat
• Grothendieck spectral sequence relates the derived functors of a composite functor to the derived functors of its components
  • Generalizes the Künneth formula for the cohomology of a tensor product of chain complexes
• Flatness in algebraic geometry corresponds to the geometric notion of a flat morphism of schemes
  • Flat morphisms have well-behaved fibers and are used to study families of schemes
• Faithfully flat descent allows gluing objects and morphisms along a faithfully flat map
  • Generalizes the sheaf condition and Galois descent
• Cotangent complex $\mathbb{L}_{A/R}$ is a derived version of the module of Kähler differentials $\Omega_{A/R}$
  • Measures the infinitesimal deformation theory of a morphism of rings $R \to A$
• André-Quillen homology $D_*(R, A, M)$ is a derived version of the cotangent complex
  • Computes the homology of the cotangent complex and relates to deformation problems in algebraic geometry
• Hochschild homology $HH_*(A, M)$ is a derived version of the center of a ring
  • Computed using the derived tensor product $M \otimes^{\mathbb{L}}_A M$ and relates to deformation theory and cyclic homology