lemma A.651 Acyclicity of $\delta$

open in the book · appendices/A-long-proofs.tex:31679 · p. 3108

Rests on

depends_on definition A.647 Koszul–Tate differential ¶
- depends_on equation 26.16 eq:cd-first-class-algebra ¶
- depends_on notation A.646 Gradings and brackets ¶
  - depends_on definition 26.50 The BRST charge ¶
    - depends_on equation 26.16 eq:cd-first-class-algebra ¶ ↺
    - depends_on postulate 26.47 Dirac quantization ¶
      - depends_on postulate 25.27 Dirac's correspondence rule ¶
      - depends_on theorem 26.21 Properties of the Dirac bracket ¶
      - depends_on theorem 26.14 First-class constraints generate gauge transformations ¶
  - depends_on equation 26.97 eq:cd-brst-charge ¶
  - depends_on equation 26.16 eq:cd-first-class-algebra ¶ ↺
depends_on lemma A.650 Flattening the constraints ¶
- depends_on remark 26.4 The constant-rank theorem is owed by Part II ¶
- depends_on remark 26.7 The regularity assumption, stated once ¶
- depends_on theorem A.286 Implicit function theorem ¶
  - depends_on definition 7.99 Differentiability at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on equation 6.12 eq:top-euclidean-metric ¶
  - depends_on proposition 7.104 Chain rule in several variables ¶
    - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on definition 7.99 Differentiability at a point ¶ ↺
    - depends_on definition 7.102 Differential of a function ¶
      - depends_on definition 7.99 Differentiability at a point ¶ ↺
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - proves proof ch:05-real-analysis@proof-63 ¶
  - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶ ↺
    - depends_on definition 7.99 Differentiability at a point ¶ ↺
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-62 ¶
  - proves proof app:A-long-proofs@proof-185 ¶
- proves proof app:A-long-proofs@proof-390 ¶
depends_on remark A.649 The regularity hypothesis, in the form the proof needs ¶
proves proof app:A-long-proofs@proof-391 ¶

Supports

depends_on proposition A.655 Uniqueness up to a canonical transformation ¶
depends_on theorem A.653 Existence of a nilpotent BRST charge ¶
- depends_on proposition A.655 Uniqueness up to a canonical transformation ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Koszul–Tate differential	declared	appendices/A-long-proofs.tex:31686
`depends_on`	→	Flattening the constraints	declared	appendices/A-long-proofs.tex:31686
`depends_on`	→	The regularity hypothesis, in the form the proof needs	declared	appendices/A-long-proofs.tex:31686
`depends_on`	←	Uniqueness up to a canonical transformation	declared	appendices/A-long-proofs.tex:31981
`depends_on`	←	Existence of a nilpotent BRST charge	declared	appendices/A-long-proofs.tex:31867
`proves`	←	app:A-long-proofs@proof-391	declared	appendices/A-long-proofs.tex:31690