Foundations and seed rank

Proof of Theorem 2.3

Because \(Q_0\) is finite, \(1=\sum_{v\in Q_0}e_v\) in \({\Bbbk} Q\), the \(e_v\) are pairwise orthogonal, and the same identities hold in \(\Lambda_{\mathfrak G}\). Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of \(I\) acts as zero, so the action factors through \(\Lambda_{\mathfrak G}\). The vertex idempotents give

Conversely, if \(M\) is unital, then

Orthogonality makes the sum direct. For an arrow \(a:v\to w\), the identities \(a=e_wae_v\) imply

The relations in \(I\) act as zero because \(M\) is a module over \(\Lambda_{\mathfrak G}\). Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps. These identifications commute with morphisms, giving an equivalence of categories.

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Proof of Proposition 2.5

The action map \(\rho:\Lambda_{\mathfrak G}\twoheadrightarrow{\mathcal A}_X\) is surjective by definition and satisfies \(\lambda s=\rho(\lambda)s\). Hence every grammar orbit vector lies in \({\mathcal A}_XS\). The reverse inclusion holds because every element of \({\mathcal A}_X\) is \(\rho(\lambda)\) for some \(\lambda\).

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Proof of Proposition 2.9

For every \(a\in A\), one has \(a=a1_A\). Hence \(A1_A=A\). The number of operations needed to generate \(A\) as an algebra is a different invariant and does not enter the module calculation.

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Proof of Theorem 3.2

The image consists exactly of finite sums \(\sum_ja_js_j\), which is \({\operatorname{Hull}}_A(S)\). Because \(\varepsilon_S\) is linear over \(A\), its kernel is a left submodule. The first isomorphism theorem gives

The final equivalence is the equality \({\operatorname{im}}\varepsilon_S=M\).

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Proof of Corollary 3.3

A seed family of size \(r\) gives a surjection by Theorem 3.2. Conversely, the images of the standard basis vectors under any surjection \(A^{\oplus r}\twoheadrightarrow M\) form a generating family of size at most \(r\).

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Proof of Corollary 3.4

The evaluation map for one seed is \(a\mapsto as\). Its kernel is \(\operatorname{Ann}_A(s)\), which is therefore a left ideal, and Theorem 3.2 gives the quotient isomorphism.

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Proof of Proposition 3.5

The left ideal \(L\) is an ideal on both sides exactly when \(ab\in L\) for every \(a\in L\) and \(b\in A\). This is equivalent to

By additivity, the last identity is equivalent to \(L{\operatorname{Hull}}_A(s)=0\).

If \(s\) is cyclic, then \({\operatorname{Hull}}_A(s)=M\). Hence the preceding condition is \(L\subseteq\operatorname{Ann}_A(M)\). The reverse inclusion always holds because an operator annihilating all of \(M\) annihilates \(s\). Thus equality follows. Conversely, \(\operatorname{Ann}_A(M)\) is an ideal on both sides, so equality implies that \(L\) is an ideal on both sides.

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Proof of Theorem 3.6

By Theorem 3.2, \(j\) is an algebraic isomorphism of modules over \({\mathcal A}\) from \(E\) onto \({\operatorname{Hull}}_{{\mathcal A}}(S)\). The transported seminorms make it a topological isomorphism onto that subspace. For \(a\in{\mathcal A}\) and a continuous seminorm \(p\) on \(X\), continuity of \(a:X\to X\) supplies continuous seminorms controlling \(p(aj(e))\), so the induced action on \(E\) is continuous and extends uniquely to \(\widehat E\). The subspace is dense by hypothesis. A Cauchy net in \(E\) is mapped to a Cauchy net in \(X\), which converges because \(X\) is complete. This defines a continuous extension \(\widehat j:\widehat E\to X\). For any \(x\in X\), choose a net \((e_\lambda)\) in \(E\) with \(j(e_\lambda)\to x\). The transported topology makes \((e_\lambda)\) Cauchy, so it converges to some \(\xi\in\widehat E\), and continuity gives \(\widehat j(\xi)=x\). Thus \(\widehat j\) is surjective. If \(\widehat j(\xi)=0\), represent \(\xi\) by a Cauchy net \((e_\lambda)\). Then \(j(e_\lambda)\to0\), so every defining seminorm \(\widetilde p(e_\lambda)\) tends to zero. Thus \(e_\lambda\to0\) in the completion and \(\xi=0\). Hence \(\widehat j\) is bijective. Its inverse is continuous by the universal property of completion, or directly because the topology on \(E\) was transported from \(X\).

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Proof of Proposition 4.2

Both conditions say that there is \(m\in M\) with \(Am=M\).

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Proof of Proposition 4.3

For every \(S\subseteq X\), one has \(BS\subseteq AS\). Hence every generating family over \(B\) is also a generating family over \(A\). Taking closures gives the topological statement.

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Proof of Theorem 4.4

Let \(S\) generate \(M\). For every \(n\in N\), choose \(m\in M\) with \(\Phi(m)=n\) and write \(m=\sum_ia_is_i\). Then

so \(\Phi(S)\) generates \(N\).

For the topological statement, let \(S\) generate \(M\) densely. Given \(n=\Phi(m)\), choose a net \(m_\lambda\in AS\) converging to \(m\). Continuity gives

and every \(\Phi(m_\lambda)\) lies in \(B\Phi(S)\). Thus \(B\Phi(S)\) is dense in \(N\).

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Proof of Corollary 4.5

Apply Theorem 4.4 to the isomorphism and its inverse.

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Proof of Theorem 4.6

The lower inequality is quotient monotonicity. For the upper inequality, choose generating families \((\ell_i)_{i\in I}\) of \(L\) and \((n_j)_{j\in J}\) of \(N\) with the least possible cardinalities. Choose a lift \(m_j\in M\) of each \(n_j\). For any \(m\in M\), a finite subset \(F\subseteq J\) and coefficients \(a_j\in A\) give

The vector \(m-\sum_{j\in F}a_jm_j\) belongs to \(\iota(L)\), so it is a finite linear combination of the \(\iota(\ell_i)\). The union of the lifted family and the image of the generating family of \(L\) therefore generates \(M\). Its cardinality is at most \(|I|+|J|\), giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.

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Proof of Proposition 4.7

The coordinate projections are intertwining quotients and give the lower bound. For the upper bound, embed minimal generating families of \(M\) as \((m_i,0)\) and of \(N\) as \((0,n_j)\). Their union generates by finite sums and has cardinality at most the sum of the two seed ranks.

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Proof of Theorem 4.8

The coordinate projections give the lower bound. Let \(r\) be the maximum on the right, regarded as a cardinal, and choose an index set \(I\) of cardinality \(r\). Choose minimal generating families for \(M\) and \(N\), index them by subsets of \(I\), and fill any unused positions with zeros. Write the resulting families as \((m_i)_{i\in I}\) and \((n_i)_{i\in I}\). For any \(m\in M\) and \(n\in N\), their finite expressions involve a common finite subset \(F\subseteq I\) after zero coefficients have been added. Thus

and

The family of paired vectors generates and has cardinality at most \(r\). The empty family handles the case in which both modules are zero.

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Proof of Proposition 4.9

Choose minimal generating families \((m_i)_{i\in I}\) for \(M\) and \((n_j)_{j\in J}\) for \(N\). Each vector in either module is a finite linear combination of its generators. Bilinearity therefore expresses every simple tensor as a finite sum of terms

Simple tensors span the algebraic tensor product, so the family \((m_i\otimes n_j)_{(i,j)\in I\times J}\) generates it. This family has cardinality at most \(|I||J|\), which is the claimed bound. The proof also covers an empty generating family, when the corresponding module and tensor product are zero.

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Proof of Proposition 4.11

Let \(S=\{s_1,\ldots,s_r\}\) generate \(M\). Every pure tensor \(\lambda\otimes m\in M_K\) has an expression

so

Pure tensors span \(M_K\), hence \(1\otimes S\) generates \(M_K\) over \(A_K\). Taking the minimum over \(S\) proves the inequality.

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Proof of Proposition 4.13

For every \(n\in N\), the cyclic submodule \(An\) is contained in the scalar span \({\Bbbk} n\). Therefore \(q\) seed images span a subspace of dimension at most \(q\). Generating the quotient of dimension \(r\) requires at least \(r\) seeds. Quotient monotonicity transfers the lower bound to \(M\).

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Proof of Theorem 5.2

If \(S\) generates \(M\), its image generates every quotient. Conversely, suppose the image of \(S\) generates \(M/JM\) and put \(N=AS\). Then

Nakayama's lemma gives \(M=N\). Thus the same finite families generate before and after passage to the top, and the minimal cardinalities agree.

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Proof of Corollary 5.3

The module \(M/JM\) over \(A/J\) is a left vector space over \(D\). Its minimal number of generators is its dimension over \(D\).

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Proof of Lemma 5.5

For one factor \(M_d(D)\), let \(e_{jj}\) be the diagonal matrix units. The left regular module decomposes as

Each minimal left ideal \(M_d(D)e_{jj}\) is isomorphic to the natural column module \(D^d\). Taking the direct product of factors gives the stated decomposition.

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Proof of Theorem 5.6

By Corollary 3.3, \(M\) is generated by \(r\) elements exactly when it is a quotient of \(A^{\oplus r}\). By Lemma 5.5,

Every short exact sequence of modules over a semisimple ring splits. Therefore a quotient isomorphic to \(M\) exists if and only if

The smallest such \(r\) is the displayed maximum. For a nonsemisimple algebra of finite dimension, Theorem 5.2 reduces the calculation to the semisimple module \(M/{\operatorname{Jac}}(A)M\) over \(A/{\operatorname{Jac}}(A)\).

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Proof of Corollary 5.7

By Theorem 5.2, \({\operatorname{srank}}_A(M)={\operatorname{srank}}_{A/J}(M/JM)\). Apply Theorem 5.6 to the displayed decompositions of \(A/J\) and \(M/JM\), with every \(D_i={\Bbbk}\). This gives the stated maximum. If \(q=0\), the unital \(A/J\)-module \(M/JM\) is zero, and Theorem 5.2 gives seed rank zero.

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Proof of Corollary 5.8

The formula gives \(\lceil1/d\rceil=1\). Directly, let \(v\neq0\) and choose a coordinate \(v_j\neq0\). For any target \(w=(w_i)\), the matrix whose \(j\)th column has entries \(w_iv_j^{-1}\) and whose other columns vanish sends \(v\) to \(w\).

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Proof of Corollary 5.9

Apply Theorem 5.6 to the single simple type with multiplicity \(m\).

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Proof of Theorem 6.2

Let the minimal polynomial be

The acting algebra is

by the Chinese remainder theorem. The factor \(R_p\) acts on \(X_p\) and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives

Each \(R_p\) is local with maximal ideal generated by the class of \(p\) and residue division field \(K_p={\Bbbk}[t]/(p)\). By Corollary 5.3,

Finally, every summand \({\Bbbk}[t]/(p^e)\) contributes one copy of \(K_p\) after quotienting by \(p\), so the dimension equals the number of summands in the elementary divisor decomposition.

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Proof of Corollary 6.3

For each irreducible \(p\), the number of invariant factors divisible by \(p\) equals the number of elementary divisors for \(p\). Because \(f_1\mid\cdots\mid f_r\) and \(f_1\neq1\) after zero summands are removed, some irreducible divisor of \(f_1\) divides all \(r\) invariant factors. Hence \(\max_p\mu_p(T)=r\).

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Proof of Corollary 6.4

The irreducibles are \(p=t-\lambda\). On the primary component for \(\lambda\), the quotient by \((T-\lambda I)X_\lambda\) has dimension equal to the number of Jordan blocks, which is also \(\dim{\operatorname{Ker}}(T-\lambda I)\).

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Proof of Corollary 6.5

The equivalence of (i) and (ii) is Proposition 4.2. By Theorem 6.2, (ii) holds exactly when every \(\mu_p(T)\) belonging to a nonzero primary component equals one, which is (iii). Relatively prime primary cyclic modules combine into one cyclic module by the Chinese remainder theorem, so (iii) is equivalent to (iv). In rational canonical form, one invariant factor means that the unique invariant factor is both the minimal and characteristic polynomial, proving (iv)\(\Leftrightarrow\)(v). Scalar extension preserves the equality of minimal and characteristic polynomials. Over an algebraic closure, that equality is equivalent to one Jordan block for each eigenvalue, which is (vi).

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