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Differential Graded Algebras and Homological Foundations

1 · Prerequisites

2 · Summary

This page fixes the cohomological differential graded algebra and module interfaces used by the adjacent higher-algebra pages. It adds only these signed definitions; chain-complex, derived-category, and Hochschild results remain in their canonical pages.

Companion: dg-and-homological-algebra-foundations-examples.

Reading prerequisites

Read the declared prerequisites in order: algebraic-representation-and-tensor-symmetry-map, derived-categories, and hochschild-homology-and-diagonal-koszul-resolutions. The first is a separate draft page defining unital associative-algebra representations over a field and their equivalence with bilinear unital module actions. Its ordinary module convention agrees with the underlying action here; the cohomological grading and differential signs are specified locally in the DG module definition over a commutative ring. This reading route adds no logical dependency on the draft representation theorem. The latter two pages are published; their exact hypotheses remain local to those pages.

Existing library route

chain-complexes-and-homology: the established chain and cochain convention, with cochain differential of degree $+ 1$ .
derived-categories: localization of complexes at quasi-isomorphisms, under the page's exact-category and size hypotheses.
hochschild-homology-and-diagonal-koszul-resolutions: Hochschild homology and the polynomial diagonal Koszul calculation over a field; it supplies neither a Gerstenhaber bracket nor formality.

Scope and local proof obligations

Local definitions

Cohomological differential graded algebra fixes a cohomological differential of degree $+ 1$ , $d^{2} = 0$ , an associative unital graded product, and the signed Leibniz rule.
Cohomological left and right differential graded modules gives separate left and right module conventions; it does not identify the two actions for a noncommutative algebra.

The polynomial DGA and the failure of a general degree-one derivation to square to zero are verified on the companion page. No new chain-complex, derived- category, or Hochschild theorem is claimed. No A-infinity identities, BV identities, deformation classification, or formality statement is supplied.

Canonical suppliers and reuse boundaries

def-cochain-complex-in-an-abelian-category: cohomological indexing $d^{n} : C^{n} \to C^{n + 1}$ and $d^{n + 1} d^{n} = 0$ .
def-chain-complex-in-an-abelian-category: existing homological convention, with differential of degree $- 1$ .
thm-a-chain-homotopy-equivalence-is-a-quasi-isomorphism: a chain homotopy equivalence induces a homology isomorphism; no converse is asserted.
def-algebra-over-a-commutative-ring: unital associative multiplication with central structure map from the commutative base ring.
def-left-and-right-modules: unital left/right module axioms.
def-bimodule: commuting left/right actions when both sides are present.

Conventions, diagrams, and sources

Use cohomological grading throughout: $d : A^{n} \to A^{n + 1}$ and $d (a b) = d (a) b + (- 1)^{∣ a ∣} a d (b)$ . The DGA is associative and unital but is not assumed graded commutative. The two source treatments and their inspected locators are recorded in batch 41 source coverage.

The original audit's reuse disposition and its no-new-item summary were superseded for this run by the owner direction to author the exact 81-pair frontier. The original claim scope remains binding; its source boundaries are kept in the batch notes.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Cohomological differential graded algebra

Definition

Let $R$ be a commutative ring with unit. A cohomological differential graded algebra over $R$ (cohomological DGA) consists of a $Z$ -graded $R$ -module $A = ⨁_{n \in Z} A^{n}$ , an $R$ -bilinear associative multiplication $A^{p} \times A^{q} \to A^{p + q}$ with a unit $1 \in A^{0}$ and central $R$ -action, and $R$ -linear maps $d^{n} : A^{n} \to A^{n + 1}$ such that $d^{n + 1} d^{n} = 0$ . The degree convention and square-zero differential are those of a cochain complex (Cochain complex in an abelian category); the ungraded multiplication and unit satisfy the $R$ -algebra convention in Algebras over a commutative ring, central structure maps, and algebra homomorphisms.

For homogeneous $a \in A^{p}$ and $b \in A^{q}$ , require the signed Leibniz rule $d^{p + q} (a b) = d^{p} (a) b + (- 1)^{p} a d^{q} (b) .$ No symmetry condition $a b = (- 1)^{p q} b a$ is imposed on homogeneous elements of degrees $p$ and $q$ . In particular, this definition fixes the cohomological sign convention and does not assert any formality, derived-category, or Hochschild-cohomology result.

The unit is automatically closed: applying the Leibniz rule to $1 \cdot 1$ gives $d (1) = d (1) + d (1)$ , so cancellation in the additive group gives $d (1) = 0$ .

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Cohomological left and right differential graded modules

Definition

Let $A$ be a cohomological DGA over a commutative ring $R$ . A cohomological left differential graded $A$ -module is a graded unital left $A$ -module $M = ⨁_{q \in Z} M^{q}$ (with module action as in Unital left and right modules over a ring; unqualified module means left module) whose action has degree zero, $A^{p} M^{q} \subseteq M^{p + q}$ , together with $R$ -linear maps $d_{M}^{q} : M^{q} \to M^{q + 1}$ satisfying $d_{M}^{q + 1} d_{M}^{q} = 0$ and, for homogeneous $a \in A^{p}$ and $m \in M^{q}$ , $d_{M} (a m) = d_{A} (a) m + (- 1)^{p} a d_{M} (m) .$

A cohomological right differential graded $A$ -module is a graded unital right $A$ -module $M$ with $M^{q} A^{p} \subseteq M^{q + p}$ and a degree-one square-zero $R$ -linear differential satisfying, for homogeneous $m \in M^{q}$ and $a \in A^{p}$ , $d_{M} (m a) = d_{M} (m) a + (- 1)^{q} m d_{A} (a) .$

The left and right conventions are separate. A differential graded bimodule uses both actions with the commuting-action condition of $(S, R)$ -bimodules and commuting left and right scalar actions; both differential compatibility identities then apply. A left module is not silently identified with a right module when $A$ is noncommutative.

5 · Examples, counterexamples and false statements

None yet.

Differential Graded Algebras and Homological Foundations

1 · Prerequisites

2 · Summary

Reading prerequisites

Existing library route

Scope and local proof obligations

Local definitions

Canonical suppliers and reuse boundaries

Conventions, diagrams, and sources

3 · Logical flowchart

4 · Definitions, theorems and proofs

Cohomological differential graded algebra

Definition

Cohomological left and right differential graded modules

Definition

5 · Examples, counterexamples and false statements

Sources