Engineering · Mathematics · Advanced Algebra Handbook
Graded, Tensor and Exterior Algebras
Advanced linear algebra turns linear maps into structural invariants. Bases and matrices are coordinates; the underlying map or module is the object. Canonical forms are useful because they expose invariants that do not depend on a particular basis. This page consolidates the relevant material from the supplied algebra source into a stand-alone handbook chapter.
Executive summary
This chapter develops graded, tensor and exterior algebras as part of a connected advanced-algebra learning sequence. The emphasis is on definitions, hypotheses, structural results and repeatable methods rather than historical narrative.
The source material is theorem-rich. Accordingly, the handbook presentation separates vocabulary from results and then adds a verification workflow so that each statement can be applied safely. Mathematical examples in the source are treated as examples, not as universal rules.
Problem-solving workflow
Identify the field or coefficient ring and the vector space or module.
Choose bases only after deciding what structure should be preserved.
Represent the map by a matrix and track how the matrix changes under a basis change.
Use invariant subspaces, cyclic decomposition or elementary divisors to reduce the problem.
Read structural information from the resulting normal or canonical form.
Translate the matrix conclusion back into a basis-independent statement.
Core definitions
Definition
An R-algebra A is a graded R-algebra if there are R-submodules Ap, for p ≥0, such that (i) A = p≥0 Ap; (ii) For all p, q ≥0, if x ∈Ap and y ∈Aq, then xy ∈Ap+q; that is, Ap Aq ⊆Ap+q. An element x ∈Ap is called homogeneous of degree p. Notice that 0 is homogeneous of any degree, but that most elements in a graded ring are not homogeneous and, hence, have no degree. Note also that any product of homogeneous elements is itself homogeneous.
Definition
Let R be a commutative ring and let M1, . . . , Mp be R-modules. An Rmultilinear function f : M1 × · · · × Mp →N, where N is an R-module, is a function that is additive in each of the p variables (when we fix the other p −1 variables) and if 1 ≤i ≤p, then f (m1, . . . ,rmi, . . . , m p) = r f (m1, . . . , mi, . . . , m p), where r ∈R and mℓ∈Mℓfor all ℓ.
Definition
Let R be a commutative ring, and let M be an R-module. Define T 0(M) = R, T 1(M) = M, T p(M) = M ⊗R · · · ⊗R M (p times) if p ≥2. Remark. Many authors denote T p(M) by :p M. In Proposition 9.97, T p(M) was originally denoted by U[M1, . . . , Mp] (here, all Mi = M), and we later replaced this notation by M1 ⊗· · · ⊗Mp, for this is easier to remember. We remind the reader that T p(M), however it is denoted, is generated by symbols m1 ⊗· · · ⊗m p in which no parentheses occur. ◀
Definition
If R is a commutative ring and V is a free R-module with basis X, then T (V ) is called the ring of polynomials over R in noncommuting variables X, and it is denoted by R⟨X⟩. Graded Algebras If V is the free R-module with basis X, then each element u in T (V ) has a unique expression u = p≥0 i1,...,i p ri1,...,i pxi1 ⊗· · · ⊗xi p, where ri1,...,i p ∈R and xi j ∈X. We obtain the usual notation for such a polynomial by erasing the tensor product symbols. For example, if X = {x, y}, then u = r0 + r1x + r2y + r3x2 + r4y2 + r5xy + r6yx + · · · .
Definition
A central polynomial identity on a k-algebra A is a polynomial identity f (X) ∈k⟨X⟩on A all of whose values f (a1, a2, . . .) (as the ai vary over all elements of A) lie in Z(A). There are theorems showing, in several respects, that PI-algebras behave like commutative algebras. For example, recall that a ring R is primitive if it has a faithful simple left R-module; if R is commutative, then R is a field. Another interesting area of current research involves noncommutative algebraic geometry. In essence, this involves the study of varieties now defined as zeros of ideals in k⟨x1, . . . , xn⟩instead of in k[x1, . . . , xn].
Definition
We call ;p(M) the pth exterior power of a k-module M. exterior used it in contrast to inner product. The first usage of the translation exterior can be found in work of E. Exterior Algebra
Definition
Let n be a positive integer and let 1 ≤p ≤n. An increasing p ≤n-list is a list H = i1, . . . , i p for which 1 ≤i1 < i2 < · · · < i p ≤n. If H = i1, . . . , i p is an increasing p ≤n-list, we write eH = ei1 ∧ei2 ∧· · · ∧ei p. Of course, the number of increasing p ≤n-lists is the same as the number of p-subsets of a set with n elements, namely, (n p ) .
Definition
The exterior derivative d p : $p(X) →$p+1(X) is defined as follows: (i) If f ∈$0(X) = A(X), then d0 f = n j=1 ∂f ∂x j dx j; (ii) If p ≥1 and ω ∈$p(X), then ω = i1...i p fi1...i pdxi1 ∧· · · ∧dxi p, and we define d pω = i1...i p d0( fi1...i p) ∧dxi1 ∧· · · ∧dxi p. If X is an open connected subset of Rn, the exterior derivatives give a sequence of A(X)-maps, called the de differential-form complex: 0 →$0(X) d0 →$1(X) d1 →· · · dn−1 →$n(X) →0.
Principal results and structural facts
Key result
shows that det(P) = 1 for P ∈Sp(2, k) [it is true, for all m ≥1, that Sp(2m, k) ≤SL(2m, k)]. 9.60 If A is an m × m matrix with At A = I, prove that A A is a symplectic matrix. Conclude, if k is a finite field of odd characteristic, that O(m, k) ≤Sp(2m, k). 9.61 Let (V, f ) be an alternating space with f nondegenerate. Prove that T ∈GL(V ) is an isometry [i.e., T ∈Sp(V, f )] if and only if, whenever E = x1, y1, . . . , xm, ym is a symplectic basis of V , then T (E) = T x1, T y1, . . . , T xm, T ym is also a symplectic basis of V . 9.6 GRADED ALGEBRAS We are now going to use tensor products of many modules in order to construct some useful rings. This topic is often called multilinear algebra. Throughout this section, R will denote a commutative ring.
Key result
Let R be a commutative ring and let M1, . . . , Mp be R-modules. (i) There exists an R-module U[M1, . . . , Mp] that is a solution to the universal mapping problem posed by multilinearity: M1 × · · · × Mp h f *' ' ' ' ' ' ' ' ' ' ' U[M1, . . . , Mp] Δf + N There is a R-multilinear h such that, if f is R-multilinear, then there exists a unique R-homomorphism Δf making the diagram commute. (ii) If fi : Mi →M′ i are R-maps, then there is a unique R-map u[ f1, · · · , f p]: U[M1, . . . , Mp] →U[M′ 1, . . . , M′ p] taking h(m1, . . . , m p) ↦h′( f1(m1), . . . , f p(m p)), where h′ : M′ 1 × · · · × M′ p → U[M′ 1, . . . , M′ p]. Graded Algebras
Key result
However, we did not prove equality, A ⊗R (B ⊗R C) = (A ⊗R B)⊗R C; we only constructed an isomorphism. ◀
Key result
If R is a commutative ring and A and B are commutative R-algebras, then A ⊗R B is the coproduct in the category of commutative R-algebras.
Key result
If k is a commutative ring and A is a k-algebra, then A is a left Aemodule whose submodules are the two-sided ideals. If A is a simple k-algebra, then A is a simple Ae-module.
Key result
If A and B are R-modules, then for all p ≥0, T p(A ⊕B) ∼= p j=0 W(A, B) j ⊗R W ′(A, B)p−j, where W(A, B) j, W ′(A, B)p−j range over all words of length j and p −j, respectively.
Key result
If V is a free R-module with basis X, where R is a commutative ring, then T (V ) is a free R-algebra with basis X.
Key result
Let k be a commutative ring, and let M be a k-module. (i) If m, m′ ∈M, then in ;2(M), we have m ∧m′ = −m′ ∧m. (ii) If p ≥2 and mi = m j for some i ̸= j, then m1 ∧· · · ∧m p = 0 in ;p(M).
Key result
If M is a k-module, x ∈;p(M), and y ∈ ;q(M), then x ∧y = (−1)pq y ∧x. Remark. This identity holds only for products of homogeneous elements. ◀
Key result
Let V be a free k-module with basis e1, . . . , en, where n ≥1. (i) There exists a exterior algebras G(V ) with an algebra automorphism u ↦u, called conjugation, such that u = u; e0 = e0; v = −v for all v ∈V. (ii) The exterior algebras G(V ) is a graded k-algebra G(V ) = p G p(V ), where G p(V ) = ⟨eH : where H is an increasing p-list⟩ [we have extended the notation eH = ei1 ∧· · · ∧ei p in ;p(V ) to eH = ei1 · · · ei p in G p(V )]. Moreover, G p(V ) is a free k-module with rank(G p(V )) = n p .
Key result
If V is a free k-module with basis e1, . . . , en, then <n(V ) = ⟨e1 ∧· · · ∧en⟩∼= k.
Key result
For all p ≥0 and all k-modules A and B, where k is a commutative ring, <p(A ⊕B) ∼= p i=0 <i(A) ⊗k <p−i(B) .
Key result
Let k be a field, let V be a vector space over k, and let v1,. . .,vp be vectors in V . Then v1 ∧· · · ∧vp = 0 in ;(V ) if and only if v1, . . . , vp is a linearly dependent list.
Key result
now gives the familiar identities from advanced calculus: curl · grad = 0 and div · curl = 0. Exterior Algebra We call a 1-form ω closed if dω = 0, and we call it exact if ω = grad f for some C∞-function f . More generally, call a p-form ω closed if d pω = 0, and call it exact if ω = d p−1ω′ for some (p −1)-form ω′. Thus, ω ∈$p(X) is closed if and only if ω ∈ker d p and ω is exact if and only if ω ∈im d p−1. Therefore, the de differential-form complex is an exact sequence of A(X)-modules if and only if every closed form is exact; this is the etymology of the adjective exact in “exact sequence.” It can be proved that the de differential-form complex is an exact sequence whenever X is a simply connected open subset of Rn. For any (not necessarily simply connected) space X, we have im grad ⊆ker curl and im curl ⊆ker div, and the R-vector spaces ker curl/ im grad and ker div/ im curl are called the cohomology groups of X. ◀
Source-grounded examples
Worked source example
(i) The polynomial ring A = R[x] is a graded R-algebra if we define Ap = {rx p : r ∈R}. The homogeneous elements are the monomials and, in contrast to ordinary usage, only monomials (including 0) have degrees. On the other hand, x p has degree p in both usages of the term degree. (ii) The polynomial ring A = R[x1, x2, . . . , xn] is a graded R-algebra if we define Ap = { rxe1 1 xe2 2 · · · xen n : r ∈R and ei = p } ; that is, Ap consists of all monomials of total degree p. Graded Algebras (iii) In algebraic topology, we assign a sequence of (abelian) cohomology groups H p(X, R) to a space X, where R is a commutative ring and p ≥0, and we define a multiplication on p≥0 H p(X, R), called cup product, making it a graded R-algebra. ◀ Just as the degree of a polynomial is often useful, so, too, is the degree of a homogeneous element in a graded algebra.
Worked source example
Consider the special case of the de differential-form complex for n = 3. 0 →$0(X) d0 −→$1(X) d1 −→$2(X) d2 −→$3(X) →0 If ω ∈$0(X), then ω = f (x, y, z) ∈A(X), and d0 f = ∂f ∂x dx + ∂f ∂y dy + ∂f ∂z dz, a 1-form resembling grad( f ). If ω ∈$1(X), then ω = f dx + gdy + hdz, and a simple calculation gives d1ω = ∂g ∂x −∂f ∂y dx ∧dy + ∂h ∂y −∂g ∂z dy ∧dz + ∂f ∂z −∂h ∂x dz ∧dx, a 2-form resembling curl(ω). If ω ∈$2(X), then ω = Fdy ∧dz + Gdz ∧dx + Hdx ∧dy. Now d2ω = ∂F ∂x + ∂G ∂y + ∂H ∂z , a 3-form resembling div(ω). These are not mere resemblances. Now $2(X) is a free A(X)- module, but we now choose a basis dx ∧dy, dy ∧dz, dz ∧dx instead of the usual basis dx ∧dy, dx ∧dz, dy ∧dz; it follows that d1ω is curl(ω) in this case. Finally, $3(X) has a basis dx ∧dy ∧dz, and so d3ω is div(ω) when ω is a 2-form. We have shown that the de differential-form complex is 0 →$0(X) grad −→$1(X) curl −→$2(X) div −→$3(X) →0.
How to reason with these results
Most advanced-algebra problems become manageable when the representation is separated from the invariant structure. Begin with the definition, then decide whether the problem is asking for an elementwise calculation, a statement about a morphism, or a classification up to isomorphism. That choice determines the correct proof language.
When a theorem gives a structural conclusion, do not jump directly to the conclusion. Write the hypotheses next to the object you are studying and check them one by one. If a hypothesis fails, either strengthen the object, pass to a quotient or localisation where the theorem applies, or use a more elementary argument.
For computational work, record each transformation together with the equivalence relation it preserves. In algebra, row operations, similarity, quotienting, localisation and isomorphism preserve different kinds of information. A calculation is useful only when the preserved structure matches the question.
Common failure modes
| Failure mode | Control |
|---|---|
| Confusing a linear map with one particular matrix representing it. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Changing basis on only the domain or codomain when similarity requires a coordinated change. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Assuming diagonalisation when the polynomial or field conditions do not permit it. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Ignoring characteristic-dependent behaviour. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Treating canonical-form calculations as mere row reduction without tracking the allowed equivalence relation. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
Verification checklist
- The ambient set, ring, field, group, module or category has been stated.
- Every operation and map used is well-defined in that setting.
- The hypotheses of each structural result have been checked before use.
- Representatives, coordinates or generators have not been confused with the underlying object.
- Existence and uniqueness have been separated where both matter.
- The final result has been checked against the original defining relation or universal property.
Quick questions
What should I identify first in a problem about graded, tensor and exterior algebras?
Start with the ambient algebraic structure, its operation or maps, and the exact hypotheses. Most incorrect solutions begin by using a familiar rule that is not valid in the stated structure.
How should definitions be used in proofs?
Expand the definition at the point where it becomes useful. Definitions are not background prose; they are the conditions that determine what must be proved and which implications are available.
When is a structural theorem safer than direct calculation?
Use a structural theorem when its hypotheses are satisfied and the calculation would otherwise depend on arbitrary coordinates, representatives or generators. The theorem usually identifies an invariant that survives those choices.
How can a final answer be checked?
Substitute the result back into the defining relation, verify any required closure or map property, and check edge cases such as zero, the identity, the empty object or degenerate quotients where relevant.