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Rank bookkeeping for a long exact sequence of finite-dimensional vector spaces

Statement

Let $F$ be a field (Field) and let $\dots \to A_{k} \overset{α_{k}}{\to} B_{k} \overset{β_{k}}{\to} C_{k} \overset{γ_{k}}{\to} A_{k - 1} \to \dots$ be a long exact sequence of $F$ -vector spaces (Exact sequence and short exact sequence in an abelian category), indexed by the integers. Assume that $A_{k}$ and $C_{k}$ are finite-dimensional over $F$ for every $k$ (Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis) and vanish for $k < 0$ and for all $k > N$ , where $N$ is a fixed integer. Then the spaces $B_{k}$ are also finite-dimensional and vanish for $k < 0$ and $k > N$ , and there is a unique polynomial $Q (t) = \sum_{k = 0}^{N} q_{k} t^{k} \in Z [t]$ with $q_{k} \geq 0$ for every $k$ such that $P_{A} (t) + P_{C} (t) = P_{B} (t) + (1 + t) Q (t),$ where $P_{A} (t) = \sum_{k} (\dim_{F} A_{k}) t^{k}$ and similarly for $B, C$ . Explicitly $q_{k} = \dim_{F} \ker α_{k}$ , so that for every $k$ $\sum_{i = 0}^{k} (- 1)^{k - i} (\dim_{F} A_{i} + \dim_{F} C_{i} - \dim_{F} B_{i}) = q_{k} \geq 0,$ and in particular $\dim_{F} B_{k} \leq \dim_{F} A_{k} + \dim_{F} C_{k}$ for every $k$ .

Facts & Assumptions

Given: A field $F$ , an integer $N$ , a long exact sequence as displayed with $A_{k}, C_{k}$ finite-dimensional for all $k$ and zero for $k < 0$ and $k > N$ , and the notation $a_{k} : = \dim_{F} \ker α_{k} \geq 0$ .

[F1]

A sequence of morphisms is exact when at every interior node the image of the incoming map equals the kernel of the outgoing map (Exact sequence and short exact sequence in an abelian category).

[L1]

For a linear map $T : V \to W$ with $V$ finite-dimensional, $\dim_{F} V = nullity T + rank T = \dim_{F} \ker T + \dim_{F} im T$ (Rank-nullity: $\dim_{F} V = nullity T + rank T$ , Rank and nullity of a linear map with finite-dimensional domain, Kernel and image of a linear map).

[L2]

If $U$ is a linear subspace of a finite-dimensional space $V$ , then $U$ is finite-dimensional with $\dim_{F} U \leq \dim_{F} V$ (If $\dim_{F} V = n$ and $U$ is a linear subspace of $V$ , then $U$ is finite-dimensional, $\dim_{F} U \leq n$ , and $\dim_{F} U = n$ if and only if $U = V$ ).

[L3]

If $U, W$ are finite-dimensional linear subspaces of a vector space, then $\dim_{F} (U + W) + \dim_{F} (U \cap W) = \dim_{F} U + \dim_{F} W$ (The dimension formula: for finite-dimensional linear subspaces $U$ and $W$ of $V$ , the subspaces $U + W$ and $U \cap W$ are finite-dimensional and $\dim_{F} (U + W) + \dim_{F} (U \cap W) = \dim_{F} U + \dim_{F} W$ ).

[F2]

$Z [x]$ is the set of finitely supported functions $N \to Z$ , with pointwise addition and convolution product; two polynomials are equal exactly when all coefficients agree, and $x$ is the sequence with coefficient $1$ at index $1$ (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution).

[F3]

$(Z, +, \cdot, 0, 1)$ is a commutative ring with multiplicative identity (The integers form a commutative ring).

Proof

technique · rank bookkeeping

1.1F1L1L2given

If $N < 0$ , the vanishing hypotheses force every $A_{k}, C_{k}$ to be zero, and exactness forces every $B_{k}$ to be zero; all formulas then hold with $Q = 0$ . Henceforth assume $N \geq 0$ . Exactness gives $im α_{k} = \ker β_{k}$ , $im β_{k} = \ker γ_{k}$ , and $im γ_{k} = \ker α_{k - 1}$ . Put $r_{k} : = \dim_{F} im α_{k}$ and $s_{k} : = \dim_{F} \ker γ_{k}$ ; these are finite by [L1] and [L2], since $A_{k}, C_{k}$ are finite-dimensional. The maps retain the names $α_{k}, β_{k}, γ_{k}$ .

2.1L1step 1.1

Rank-nullity for $α_{k} : A_{k} \to B_{k}$ and $γ_{k} : C_{k} \to A_{k - 1}$ gives $\dim_{F} A_{k} = a_{k} + r_{k}$ and $\dim_{F} C_{k} = s_{k} + a_{k - 1}$ .

3.1L1L2L3step 1.1step 2.1choose

Choose a finite basis $y_{1}, \dots, y_{m}$ of $im β_{k} = \ker γ_{k}$ and lifts $x_{i} \in B_{k}$ with $β_{k} (x_{i}) = y_{i}$ . Then $B_{k} = \ker β_{k} + span (x_{1}, \dots, x_{m})$ : subtract the corresponding linear combination of the lifts from any element. The kernel has dimension $r_{k}$ by exactness and step 2.1, so this sum is finite-dimensional by [L3]. Rank-nullity now applies to $β_{k}$ and gives $\dim_{F} B_{k} = r_{k} + s_{k}$ . Only finitely many lifts are chosen.

4.1F1step 2.1step 3.1algebra

The three dimension identities give $\dim_{F} A_{k} + \dim_{F} C_{k} - \dim_{F} B_{k} = a_{k} + a_{k - 1} \geq 0$ . In particular the weak inequality holds. Outside $0 \leq k \leq N$ , exactness and $A_{k} = C_{k} = 0$ force $B_{k} = 0$ . Also $a_{N} = 0$ , since exactness identifies $\ker α_{N}$ with the image of the zero space $C_{N + 1}$ .

5.1F2F3step 4.1algebra

Set $Q (t) = \sum_{k = 0}^{N} a_{k} t^{k}$ . With $a_{k} = 0$ outside this range, the coefficient of $(1 + t) Q$ in degree $k$ is $a_{k} + a_{k - 1}$ ; at degree $N + 1$ it vanishes because $a_{N} = 0$ . Thus step 4.1 and coefficient comparison give $P_{A} + P_{C} = P_{B} + (1 + t) Q$ . Its coefficients are nonnegative integers.

6.1step 4.1step 5.1algebra

Telescoping gives $\sum_{i = 0}^{k} (- 1)^{k - i} (a_{i} + a_{i - 1}) = a_{k}$ , since $a_{- 1} = 0$ . This proves the partial-sum formula and identifies $q_{k} = a_{k}$ . For negative $k$ read the sum as empty and set $q_{k} = 0$ .

7.1F2F3algebra∎

Finally, uniqueness of $Q$ : if $(1 + t) Q = 0$ with $Q = \sum_{k} q_{k} t^{k} \in Z [t]$ , then by [F2] the coefficients satisfy $q_{0} = 0$ and $q_{k} = - q_{k - 1}$ for every $k \geq 1$ , so all $q_{k} = 0$ and $Q = 0$ ; hence two polynomials with $(1 + t) Q = (1 + t) Q^{'}$ agree.

Remarks

The vanishing convention. The hypothesis that the sequence vanishes in degrees $k < 0$ is the one used by the Morse applications, where the graded pieces are homology groups in nonnegative degrees; it is exactly what makes the alternating partial sums land on the coefficient $q_{k}$ of $Q$ without a leftover boundary term from below. The finite-range hypothesis gives the finite sums and the degree bound $N$ .
Field versus ring coefficients. The proof uses rank-nullity, which needs a field; over a general ring the numerical inequality can fail. This is the algebraic source of the coefficient-field dependence recorded on the examples page.

Depends on

Used by

Strong Morse inequalities Corollary
Perfectness, vanishing correction, and vanishing handle boundaries Lemma
Finiteness and additivity of the Euler characteristic Proposition
Relative Morse inequalities for a cobordism Proposition
Morse polynomial identity Theorem

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Sources

Liviu Nicolaescu, An Invitation to Morse Theory (2nd ed.), Chapter 2 Section 2.3, printed pp. 46-53 (PDF pp. 56-63) (standard reference, not scraped)
Alexander Ritter, Morse Homology (Cambridge Part III lecture notes), Lecture 21, PDF pp. 96-101 (standard reference, not scraped)