Module theory

Module theory extends linear algebra by allowing scalars from a ring rather than only a field.

An algebra representation is a unital algebra homomorphism into the endomorphisms of a vector space. It is equivalent to making that vector space a left module over the algebra, with $av=\rho (a)v$.

A finite-dimensional module $X$ degenerates to $Y$ when $Y$ has a representative in the closure of $X$'s change-of-basis orbit in the representation variety of an associative algebra. The full orbit of $Y$ then lies in that closure. The relation is transitive because orbit closures are closed and invariant under change of basis. A split extension as a degeneration shows that a module degenerates to the direct sum of its composition factors.

A ring is left hereditary if every submodule of a left projective module is projective. Equivalently, every left module has a projective resolution of length at most one. The standard projective resolution of a quiver representation, used also for arbitrary infinite-dimensional modules, proves that path algebras are left hereditary.

The standard projective resolution of a quiver representation has length one, even for quivers with oriented cycles. Thus every path algebra is a hereditary ring, and higher extension groups vanish. Applying a long exact sequence of Ext groups to $0\to I\to K\to C\to 0$ gives a surjection ${\mathrm{Ext}}^1(K,W)\twoheadrightarrow {\mathrm{Ext}}^1(I,W)$, since the next term ${\mathrm{Ext}}^2(C,W)$ is zero. This is the hereditary step in the Ringel lemma on bricks.

For central elements, each multiplication map on the preceding quotient must be injective; a usual convention also requires the final quotient to be nonzero. The augmented Koszul complex on central ring elements tensored with $M$ is then a quasi-isomorphism to the final quotient in degree zero.

A composition series of a module is a finite chain of submodules whose nonzero successive quotient modules are simple modules. Every finite-dimensional Lie algebra representation has such a chain, viewed as a module over the universal enveloping algebra.

A composition factor is an irreducible successive quotient in a composition series of a module. For a central Casimir operator, its scalar on a composition factor is an eigenvalue of the operator on the original Lie algebra representation.

A nonzero module is uniform if every pair of nonzero submodules has nonzero intersection. Equivalently, every nonzero submodule is an essential submodule. A right Noetherian domain is uniform as a right module: if $aA\cap bA=0$ for nonzero $a,b$, the right ideals $b^{i}aA$, $i\ge 0$, form an infinite direct sum, contradicting the ascending chain condition.

A nonzero module is indecomposable if it cannot be expressed as a direct sum of two nonzero submodules. The zero module is excluded. A module of finite composition length is indecomposable exactly when its endomorphism ring has no idempotents other than zero and one: an idempotent splits the module into its image and kernel.

For a nonzero finitely generated indecomposable module $P$ that is a projective module over a right Artinian ring $A$, its top $P/PJ(A)$ is a simple module. The Jacobson radical $J(A)$ is a nilpotent ideal, so the top is nonzero; the quotient is a semisimple module over the semisimple ring $A/J(A)$. If it decomposed, a nontrivial idempotent of its endomorphism ring would lift by projectivity to an endomorphism of $P$. The Hopkins-Levitzki theorem gives finite composition length, and the Fitting lemma makes the lift either invertible or nilpotent. Its induced map on the top would have the same property, impossible for that nontrivial idempotent.

For an endomorphism $f$ of a finite-length module $M$, sufficiently large $n$ givesIf $M$ is indecomposable, every endomorphism is therefore either invertible or nilpotent, so its endomorphism ring is local.

The Krull–Schmidt theorem says that a finite-length module decomposes as a finite direct sum of indecomposable modules, uniquely up to permutation and isomorphism of the summands.

A finite-dimensional module is ${⨁}_a{M}_a^{\oplus m_a}$ with pairwise nonisomorphic indecomposables and uniquely determined multiplicities. The Fitting lemma makes their endomorphism rings local. The multiplicity spaces provide the general linear factors in the Levi decomposition of a quiver automorphism group.

For an $R$-module $M$, its dual module is $M^{\ast }={\mathrm{Hom}}_R(M,R)$, with pointwise addition and scalar multiplication.

The double dual module is $M^{\ast \ast }={\mathrm{Hom}}_R(M^{\ast },R)$. The evaluation homomorphism sends $m\in M$ to the functional $f\mapsto f(m)$.

An $R$-module is finitely generated when there are finitely many elements $m_1,\dots ,m_r$ such that every element is an $R$-linear combination of them.

Let a finitely generated module over an integral domain have a finite generating set $S$ and a maximal independent subset $T$. For each $s\notin T$, dependence of $T\cup \{s\}$ gives a nonzero coefficient $a_s$ with $a_{s}s\in ⟨T⟩$. The product of those coefficients is nonzero and carries the whole module into $⟨T⟩$.

An R-module is finitely generated exactly when it is the image of a surjection from a finite free module $R^n$. Given generators $m_i$, send $(r_i)$ to ${\sum }_i{r}_i{m}_i$; conversely, the images of the standard free generators generate every such image. This criterion does not assert that the module is free: a quotient can have nontrivial relations.

An $R$-module is Artinian when every descending chain of submodules stabilizes. Equivalently, every nonempty family of submodules has a minimal member.

A filtered colimit of modules is a direct limit indexed by a filtered category. Filtered colimits preserve exact sequences, and tensor products commute with them.

A direct system indexed by a directed set $A$ consists of abelian groups $G_a$ and homomorphisms ${\varphi }_{ab}:G_a\to G_b$ for $a\le b$, with ${\varphi }_{aa}=1$ and ${\varphi }_{bc}{\varphi }_{ab}={\varphi }_{ac}$.

The direct limit of a direct system of abelian groups is the quotient of ${⨁}_a{G}_a$ by the relations identifying $g\in G_a$ with ${\varphi }_{ab}(g)\in G_b$ whenever $a\le b$.

For nonzero integers $a_i$ and $P_n=a_0\cdots a_{n-1}$, the compatible embeddings of stage $n$ into $\mathbb{Q}$ send $m$ to $m/P_n$. They identify the direct limit with

The additive group ${\mathbb{Q}}_{\mathrm{sq}}$ consists of the rational numbers whose reduced denominators are square-free integers. If $p_0,p_1,\dots$ lists the primes without repetition, then

An $(R,S)$-bimodule is simultaneously a left $R$-module and a right $S$-module, with commuting scalar actions $(rm)s=r(ms)$.

For a group $G$ acting on a module $M$, the invariant submodule is $M^G=\{m\in M:g\cdot m=m\text{ for all }g\in G\}$.

For a group $G$ acting on a module $M$, the coinvariant module is

Suppose $1\to X\to \mathcal{G}\to \Gamma \to 1$ is an exact sequence of pro-p groups, with $X$ abelian and $\Gamma \cong {\mathbb{Z}}_p$. Conjugation gives $X$ a compact Galois module structure. The closed commutator subgroup is $(\gamma -1)X$: its image is closed by compactness, and after quotienting by it, a lift of $\gamma$ centralizes $X$ and topologically generates the remaining quotient. Hence $0\to X_{\Gamma }\to {\mathcal{G}}^{\mathrm{ab}}\to \Gamma \to 0$ is exact.

The support of an $R$-module $M$ isFor a nonzero finitely generated module, the support is nonempty and contains a maximal ideal.

The tensor product of an $R$-module $M$ and an $R$-module $N$ is an $R$-module $M{\otimes }_{R}N$ generated by pure tensors $m\otimes n$, subject to additivity in both variables and the balancing relation

For a homomorphism of commutative rings $R\to A$ and two $A$-modules, the tensor product of modules over $A$ is the quotient of the tensor product of modules over $R$ imposing all relations $am\otimes n=m\otimes an$. The quotient map is $A$-linear when $A$ acts through the first factor on the source. This expresses the extra balancing imposed by change of rings.

For commutative $R$-algebras, their module tensor product becomes a commutative algebra by $(a\otimes b)(a^{\prime }\otimes b^{\prime })=a{a}^{\prime }\otimes b{b}^{\prime }$. Its universal property is that pairs of compatible $R$-algebra maps from $A$ and $B$ into a commutative algebra $C$ correspond to maps $A{\otimes }_{R}B\to C$. This coproduct of algebras gives, contravariantly, the fibre product of schemes on affine charts.

If $L,L^{\prime }$ are field extensions of $K$, choose a $K$-basis of $L$ containing $1$. After tensoring with $L^{\prime }$, that basis expresses $L{\otimes }_K{L}^{\prime }$ as a nonzero direct sum of copies of $L^{\prime }$, and $1\otimes 1$ is nonzero. A prime ideal of this nonzero ring supplies a point used in the point-lifting property of a scheme fibre product.

A pure tensor is an element of a tensor product that can be written as $m\otimes n$. General tensors are finite sums of pure tensors and need not themselves be pure.

If $M{\otimes }_{R}N\cong R^n$ for a positive integer $n$, then both $M$ and $N$ are projective modules. Choose a finite expression for the inverse image of one basis vector. It produces a split surjection $N^r\to R$; tensoring the splitting with $M$ exhibits $M$ as a direct summand of the finite free module $(M{\otimes }_{R}N{)}^r$. Symmetry gives the result for $N$.

An $R$-module $M$ is tensor-nilpotent when $M^{\otimes k}=0$ for some positive integer $k$. A nonzero finitely generated module cannot be tensor-nilpotent: localize at a maximal ideal in its support and reduce modulo that maximal ideal, obtaining a nonzero tensor power of a nonzero vector space.

A map $b:M\times N\to P$ is balanced when it is additive in each variable and satisfies $b(rm,n)=b(m,rn)$.

Every balanced map $b:M\times N\to P$ factors uniquely through an $R$-module homomorphism $\tilde{b}:M{\otimes }_{R}N\to P$ satisfying $\tilde{b}(m\otimes n)=b(m,n)$.

For modules over a commutative ring, there is a natural isomorphismgiven by $f\mapsto (m\mapsto (n\mapsto f(m\otimes n)))$.

Tensor product need not commute with an infinite direct product. For example,The element $(1\mathrm{mod}n{)}_n$ has infinite order in the product, so it survives rational localization.

An $R$-module $M$ is flat when tensoring with it preserves injections, equivalently when $-{\otimes }_{R}M$ is an exact functor.

A flat module $M$ is faithfully flat when $N{\otimes }_{R}M=0$ implies $N=0$. Every field extension is faithfully flat over its base field.

An $R$-module $M$ is flat if and only if $M_{\mathfrak{p}}$ is flat over $R_{\mathfrak{p}}$ for every prime ideal $\mathfrak{p}$, equivalently for every maximal ideal.

Over a principal ideal domain, a module is flat exactly when it is torsion-free. Hence the tensor product of two torsion-free modules over a principal ideal domain is torsion-free: both tensor functors are exact, so their composite is exact.

If an $R$-algebra $A$ is flat as an $R$-module, then for ideals $I,J\subseteq R$,Tensor the exact sequence $0\to I\cap J\to I\oplus J\to I+J\to 0$ with $A$.

A nonzero commutative ring $R$ has the property that $M{\otimes }_{R}N=0$ forces $M=0$ or $N=0$ exactly when $R$ is local with maximal ideal $\mathfrak{m}$ and $M=\mathfrak{m}M$ forces $M=0$ for every $R$-module. Reduction modulo $\mathfrak{m}$ turns a tensor product into a tensor product of vector spaces.

If a finitely generated faithful module $M$ over a commutative ring $R$ satisfies $xM\subseteq IM$, the determinant of a matrix presenting these relations gives a monic equationIn particular, an element of an extension field that preserves a nonzero finitely generated ideal is integral over $R$.

A short exact sequence is an exact sequenceThus $j$ is injective, $q$ is surjective, and $\mathrm{im}j=\ker q$.

A short exact sequence with specified kernel module $A$ and quotient module $C$. Its equivalence class is determined by the degree-one Ext functor; equivalence must preserve both end modules.

With trivial action on both end modules, an extension has underlying abelian group ${\mathbb{Z}}^r\oplus {\mathbb{Z}}^s$ and generator action $\left(\begin{array}{cc}I& T\\ 0& I\end{array}\right)$. The integer matrix $T$ is invariant under equivalences fixing the ends. The one-element projective resolution with differential $z-1$ computes the same classification over the group ring.

Pushing an inclusion $i:A\to B$ along $f:A\to A\prime$ gives a new module extension with kernel $A\prime$ and the same quotient. This pushout realizes the covariance of the degree-one Ext functor in its second argument.

A commuting isomorphism between two module extensions that is the identity on both end modules. Classifying middle modules up to arbitrary isomorphism is a different and generally coarser problem.

A short exact sequence splits when its injection has a retraction, equivalently when its surjection has a section. In that case the middle module is isomorphic to the direct sum of the two outer modules.

The Five lemma has a short form: in a commutative diagram of short exact sequences, if the maps on both outer terms are isomorphisms, then the map on the middle terms is an isomorphism.

An inverse system is a family $(N_i)$ with transition maps $g_{ij}:N_j\to N_i$ for $i\le j$, satisfying $g_{ii}=1$ and $g_{ik}=g_{ij}{g}_{jk}$.

The inverse limit is the module of compatible families

Maps from a module $X$ to ${\underleftarrow{\lim }}_i{N}_i$ correspond naturally to compatible families of maps $X\to N_i$.

An inverse system of nonempty finite sets over a directed index set has nonempty inverse limit. Give each set the discrete topology: the compatibility conditions are closed subsets of the compact product, and directedness gives them the finite intersection property.

A decreasing filtration of a module is a chain $M=M_0\supseteq M_1\supseteq \cdots$ of submodules.

For a filtered algebra $A$ with finite-dimensional filtration pieces, a good filtration of a module on a finitely generated module is compatible with multiplication and has finitely generated associated graded module over the associated graded ring. Taking bounded-degree translates of a finite set of module generators supplies such a filtration of a module when the associated graded ring is Noetherian.

Two decreasing filtrations $(M_n)$ and $(N_n)$ are equivalent when each contains a fixed shift of the other: $M_{n+a}\subseteq N_n$ and $N_{n+b}\subseteq M_n$ for fixed $a,b$ and every $n$.

For an ideal $I$, the $I$-adic filtration of a module $M$ is $M\supseteq IM\supseteq I^{2}M\supseteq \cdots$.

If $R$ is a Noetherian local ring with maximal ideal $\mathfrak{m}$ and $I\subseteq \mathfrak{m}$, thenMore generally, the intersection of the powers acts trivially on every finitely generated module.

The $x$-adic filtration is the filtration by powers of the principal ideal $(x)$.

An $I$-filtration $(M_n)$ is stable when $I{M}_n=M_{n+1}$ for all sufficiently large $n$. Every stable $I$-filtration is equivalent to the $I$-adic filtration.

For an ideal $I\subseteq R$, the Rees ring is the graded subringIt packages all powers of $I$ into one algebra and turns stable filtrations into finitely generated graded modules.

The Rees module of an $R$-module $M$ with respect to an ideal $I$ is ${\mathcal{R}}_I(M)={⨁}_{j\ge 0}{I}^{j}M{t}^j$, a graded module over the Rees ring $\mathcal{R}(I)$. Generators of $M$ in degree zero generate this module. For $N\subseteq M$, the graded submodule ${⨁}_j(I^{j}M\cap N)t^j$ is finitely generated when $R$ is a Noetherian ring; a bound on its generators' degrees yields the Artin-Rees lemma.

If $R$ is Noetherian, $I$ is an ideal, and $N\subseteq M$ are finitely generated modules, then some $k$ satisfiesfor every $n\ge k$.

A filtered algebra has a nested family of vector subspaces whose index is compatible with multiplication.

A filtered ring is a ring with subgroups $(R_n)$ satisfying $R_m{R}_n\subseteq R_{m+n}$.

The length of a module is the number of simple factors in a composition series. It is additive across short exact sequences.

A finitely generated module over a principal ideal domain decomposes aswith finitely many irreducibles $p$. The free rank and primary cyclic factors are unique up to associates and ordering.

An $R$-module is an abelian group equipped with scalar multiplication by a ring $R$, satisfying the usual distributive and associative laws.

A module is finitely presented when it admits the displayed exact sequence for finite integers $r,s$. Thus it has finitely many generators and finitely many relations. On a Noetherian ring, every finitely generated module has a finite presentation.

Changing the coefficient ring of a module along a ring homomorphism can mean restriction of scalars, extension of scalars, or coextension of scalars. The underlying module may be retained with a smaller ring action, or replaced by a tensor or Hom construction with the new ring.

For a ring homomorphism $R\to A$ and a left $R$-module $M$, the left $A$-action on ${\mathrm{Hom}}_R(A,M)$ is $(a\cdot h)(b)=h(ba)$. This construction is right adjoint to restriction of scalars.

For a ring homomorphism $R\to A$, tensoring an $R$-module with the $(A,R)$-bimodule $A$ gives an $A$-module. This functor is left adjoint to restriction of scalars.

For a ring homomorphism $\theta :R\to A$, an $A$-module becomes an $R$-module by the displayed action. Its underlying abelian group stays the same.

A module isomorphism is a bijection of modules over the same ring preserving addition and scalar multiplication. Its inverse preserves these operations too. When a linear operator $T$ gives a vector space an $F[t]$-module structure by $t\cdot v=T(v)$, a module isomorphism is exactly an invertible linear map intertwining the two operators. This translates classification of modules into matrix similarity and rational canonical form.

A torsion module over an integral domain is a module in which every element is annihilated by some nonzero scalar.

For a module $M$ over an integral domain $R$, its torsion submodule isThe domain condition makes this a submodule: products of nonzero annihilators remain nonzero and annihilate sums.

An $R$-module homomorphism is an additive map $f:M\to N$ satisfying $f(rm)=rf(m)$.

A module retraction onto a submodule $U\stackrel{i}{\to }X$ is a module homomorphism $r:X\to U$ with $ri=I_U$. It gives $X=i(U)\oplus \ker r$: write $x=i(r(x))+(x-i(r(x)))$. Thus a nonzero proper submodule admitting a retraction contradicts indecomposability. Extending a projection from a larger submodule can produce such a retraction through the long exact sequence of Ext groups.

The endomorphism ring ${\mathrm{End}}_R(M)$ consists of the $R$-module homomorphisms $M\to M$, with pointwise addition and composition as multiplication.

An endomorphism ring is local when its nonunits form a proper two-sided ideal; commutativity is not required. The Fitting lemma gives this property for an indecomposable finite-length module. If two nonunits had an invertible sum, multiplication by its inverse would give $I=f+g$ with noninvertible $f,g$. Fitting makes $f$ nilpotent, so $I-f=g$ would be invertible, a contradiction. Products with a nonunit are nonunits by finite-length injectivity/surjectivity. The quotient by the nonunit ideal is a division ring.

For a finite-dimensional module over an algebraically closed field, write a Krull-Schmidt decomposition $X={⨁}_a{M}_a^{\oplus m_a}$ with distinct indecomposable types. Each endomorphism ring of $M_a$ is a local endomorphism ring with residue division algebra $k$, by the finite-dimensional division algebra over an algebraically closed field result. Modulo the Jacobson radical of $\mathrm{End}(X)$, the blocks of one type become $M_{m_a}(k)$ and all maps between different types vanish. A composite through a different indecomposable type cannot be invertible, since that would make one type a direct summand of the other. This yields the displayed product and the Levi decomposition of a quiver automorphism group.

A nonzero module is a brick when every endomorphism is either zero or an isomorphism. Equivalently, its endomorphism ring is a division ring.

An indecomposable finite-dimensional quiver representation that is not a brick module contains a brick with nonzero self-extensions. The Ringel form then gives $q_Q(\dim B)=1-\dim {\mathrm{Ext}}_Q^1(B,B)\le 0$, impossible for a positive definite Tits form of a quiver.

The Fitting lemma supplies a nonzero nilpotent endomorphism $f$ of a non-brick indecomposable $X$. Minimize its nonzero rank; nilpotence and minimality give $f^2=0$. Set $I=\mathrm{im}f\subset K=\ker f={⨁}_j{K}_j$. Choose a nonzero component $u:I\to K_j$. The square-zero endomorphism $X\stackrel{f}{\to }I\stackrel{u}{\to }{K}_j\hookrightarrow X$ has rank at least $\dim I$, so $u$ is injective.

If ${\mathrm{Ext}}^1(I,K_j)=0$, the projection $K\to K_j$ extends to $X\to K_j$ by the long exact sequence of Ext groups, giving a module retraction and contradicting indecomposability. Thus this extension group is nonzero. Since path algebras are hereditary, $u$ induces a surjection ${\mathrm{Ext}}^1(K_j,K_j)\twoheadrightarrow {\mathrm{Ext}}^1(I,K_j)$. Hence $K_j$ is a proper indecomposable submodule with self-extensions. Iterate until a brick module is reached; dimensions strictly decrease.

This minimal-rank argument is given in section 2 of William Crawley-Boevey's quiver lectures.

A submodule $N$ of an $R$-module $M$ is a subset closed under addition, additive inverses, and scalar multiplication by every element of $R$.

A submodule $N\subseteq M$ is an essential submodule if it intersects every nonzero submodule of $M$ nontrivially. A nonzero module is a uniform module precisely when every nonzero submodule is an essential submodule.

A primary decomposition expresses an ideal or submodule as an intersection of primary components.

Every proper ideal in a Noetherian ring is a finite intersection of primary ideals. This existence theorem does not give uniqueness of the individual components: embedded primary components may vary.

A proper submodule $N⊊M$ is primary when $rm\in N$ and $m\notin N$ imply $r^{k}M\subseteq N$ for some positive integer $k$.

For a submodule $N\le M$, the quotient module $M/N$ consists of additive cosets with scalar multiplication $r(m+N)=rm+N$.

If an R-module homomorphism $f:M\to P$ vanishes on a submodule $N$, there is a unique homomorphism $\bar{f}:M/N\to P$ satisfying $f=\bar{f}\circ q$, where $q:M\to M/N$ is the quotient map. It is given by $\bar{f}(m+N)=f(m)$.

A free module has a basis: every element has a unique finite linear combination in that basis with coefficients in the scalar ring.

Order the linearly independent subsets of a vector space by inclusion. A chain's union is independent, since every finite relation is contained in one chain member. Zorn's lemma gives a maximal independent subset. A vector outside its span could be adjoined while preserving independence, so it spans. Sending finite-support coefficient families to their linear combinations gives an isomorphism from the free module on that subset to $V$. This supplies an arbitrary, possibly infinite-dimensional basis using the axiom of choice; no pre-existing basis theorem is invoked.

Elements of a module are linearly independent if every finite linear relation between them has all coefficients zero. An independent finite set freely generates its span, a finite free module. Over an integral domain, maximal independence within a finite generating set enables clearing denominators relative to an independent module subset.

A finite free module is a free module with a finite basis, equivalently a module isomorphic to $R^n$ for some nonnegative integer $n$.