LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Rational descent of a complex monodromy socle

Statement

Let $A \subseteq {End}_{Q} (V)$ be a finite-dimensional rational algebra. The socle (The socle as the sum of all simple submodules) of the $A_{C}$ -module $V_{C}$ is defined over $Q$ : it is $W \otimes C$ for a rational $A$ -submodule $W \subseteq V$ . If $V \neq 0$ , then $W \neq 0$ .

Facts & Assumptions

Given: A finite-dimensional unital rational algebra $A$ acting on a $Q$ -vector space $V$ , with $B = A_{C}$ and $J = J (B)$ its Jacobson radical; here $A_{C} = A \otimes_{Q} C$ acts on $V_{C}$ , and left multiplication $L_{b}$ of $B$ is the $C$ -linear endomorphism $x \mapsto b x$ of $B$ .

[F1]

For a finite-dimensional algebra over a field the Jacobson radical is a nilpotent two-sided ideal and the quotient by it is semisimple (For a finite-dimensional algebra, the Jacobson radical is nilpotent and the quotient by it is semisimple).

[F2]

Over $C$ the regular trace form $(b, c) = tr (L_{b c})$ is nondegenerate if and only if the finite-dimensional unital $C$ -algebra is semisimple; for a semisimple algebra $B ≅ \prod_{i} M_{d_{i}} (C)$ the simple left modules are the column modules and the trace form is the sum of the nondegenerate matrix trace pairings The trace form detects semisimplicity over the complex numbers. More generally a nonzero unital semisimple ring is a product of matrix rings over division rings (Wedderburn–Artin theorem for semisimple rings) whose simple left modules are the column modules, one isomorphism class per factor (Simple modules over a product of matrix rings over division rings).

[F3]

For a finite-length module the sum-of-simples, direct-sum and complement characterizations of semisimplicity are equivalent (Choice-free semisimple characterizations for finite-length modules); a module is semisimple when it is an internal direct sum of simple submodules, the empty sum included (Semisimple modules as direct sums of simple modules); the socle is the sum of all simple submodules (The socle as the sum of all simple submodules).

Proof

technique · direct

1.1F1F2algebra

Identify the radical of the regular trace form with $J$ . The ideal $J$ is nilpotent and $B / J$ is semisimple, by [F1]. Choose a composition series of the left regular $B$ -module; the simple factors are killed by $J$ , since $J S = S$ for a simple factor would give $J^{k} S = S$ for every $k$ , contradicting nilpotence. Each simple type of $B / J$ occurs among these factors: $B / J$ is a quotient of $B$ , so its left regular module is a quotient of the regular module and its composition factors, namely $d_{i} \geq 1$ copies of each column module, occur in the series. In a basis adapted to the series the diagonal blocks of $L_{b}$ are the factor actions, so $tr (L_{b}) = \sum_{i} m_{i} tr ({\overline{b}}_{i})$ with multiplicities $m_{i} \geq 1$ . Therefore the radical ${b : (b, c) = 0 for all c}$ of the trace form is exactly $J$ : it contains $J$ , and on $B / J$ the induced form is the sum over $i$ of $m_{i}$ times the nondegenerate matrix trace pairing of the factor $M_{d_{i}} (C)$ , which is nondegenerate because $m_{i} \neq 0$ and $C$ has characteristic zero, by [F2].

2.1step 1.1givenalgebra

Descend the kernel to a rational subspace. Evaluate the trace form in the rational basis $a_{1}, \dots, a_{d}$ of $A$ : the Gram matrix $T = (tr (L_{a_{i} a_{j}}))_{i, j}$ has rational entries, because each $L_{a_{i} a_{j}}$ is the extension to $C$ of the rational endomorphism of $A$ given by left multiplication by $a_{i} a_{j}$ , and the trace of such an extension is the trace of a rational matrix. By step 1.1 its kernel in $B$ is $J$ . Let $J_{0} = {a \in A : T a = 0} \subseteq A$ be the kernel computed in $Q$ -coordinates, a rational subspace; then $J_{0} \otimes C = J$ , since a rational linear system has the same solutions after extension of scalars and its solution space over $C$ is the complexification of its rational solution space. Since $J$ is a two-sided ideal of $B$ , the intersection $J_{0} = J \cap A$ is a two-sided ideal of $A$ , and $W = {v \in V : J_{0} v = 0}$ is a rational $A$ -submodule of $V$ .

3.1F2F3step 2.1algebra

Identify $W_{C}$ with the socle. Because $J_{0} \otimes C = J$ , one has $W_{C} = {v \in V_{C} : J v = 0}$ : choose a finite rational basis $j_{1}, \dots, j_{a}$ of $J_{0}$ , so $W$ is the kernel of $V \to V^{a}$ , $v \mapsto (j_{1} v, \dots, j_{a} v)$ . Write an element of $V_{C}$ as $\sum_{i = 1}^{k} v_{i} \otimes λ_{i}$ with the finitely many $λ_{i}$ rationally independent, obtained by reducing the span of its original coefficients. The equations $j_{a} \sum_{i} v_{i} \otimes λ_{i} = 0$ imply $j_{a} v_{i} = 0$ for each $a, i$ , by comparing coordinates in a finite rational basis of the span of the vectors $j_{a} v_{i}$ . Thus it lies in $W \otimes C$ ; the reverse inclusion is immediate. No basis of an infinite-dimensional space is selected. Every simple $B$ -submodule $S \subseteq V_{C}$ is killed by $J$ : otherwise $J S = S$ , and then $J^{k} S = S$ for all $k$ , contradicting nilpotence of $J$ . Hence the socle of $V_{C}$ is contained in ${v : J v = 0}$ . Conversely ${v : J v = 0}$ is a module over the semisimple algebra $B / J$ , and it is a sum of simple modules: for each of its vectors $v$ , the cyclic module $(B / J) v$ is a quotient of the regular $B / J$ -module, and that regular module is a finite direct sum of the column modules of [F2], so its image is a sum of simple modules. Each cyclic module is finite-dimensional, so [F3] makes its sum of simple images a finite direct sum. Every vector of ${v : J v = 0}$ therefore belongs to a sum of simple $B$ -submodules, and this annihilator, containing every simple submodule, is exactly the socle. This argument does not require $V$ to be finite-dimensional. Thus the socle of $V_{C}$ equals $W_{C} = W \otimes C$ , a rational $A$ -submodule complexified.

4.1F1F3step 3.1algebra∎

Nonvanishing for nonzero $V$ . If $V \neq 0$ then $V_{C} \neq 0$ . Let $n$ be an exponent with $J^{n} = 0$ , which exists by nilpotence, and let $r \geq 1$ be least with $J^{r} V_{C} = 0$ ; such an $r$ exists since $J^{n} = 0$ . Then $J^{r - 1} V_{C} \neq 0$ by minimality of $r$ , and $J \cdot J^{r - 1} V_{C} = J^{r} V_{C} = 0$ , so the nonzero subspace $J^{r - 1} V_{C}$ is contained in the socle, which is therefore nonzero. By step 3.1 the socle is $W \otimes C$ , so $W \neq 0$ .

Source notes

This explicit regular-trace argument descends the full complex socle, used here in place of Deligne II 4.2.7’s sum of minimal-dimensional complex subsystems. It does not identify these two sums. It is separate from the characteristic-zero scalar-extension lemma and does not assume that $V_{C}$ is semisimple; no choice principle is used.

Depends on

Used by

Semisimplicity of geometric Gauss-Manin local systems Theorem

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pierre Deligne, Theorie de Hodge II, Publ. Math. IHES 40 (1971) 5-57; SS4.2.6-4.2.7, printed pp. 45-46 (the rational-descent step abbreviated there and supplied here) (standard reference, not scraped)