Sardanashvily theorems - CiteSeerX

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Theorem GS3. Let Z be a smooth manifold. A graded commutative C∞(Z)- algebra A is isomorphic to the structure ring of a graded manifold with a body Z if.
G. Sardanashvily

My main mathematical theorems As a mathematical and theoretical physicist, I have proved a lot of theorems and assertions. Here, there is a list of my twenty three most relevant original theorems.

1

Modification of the abstract De Rham theorem

Theorem GS1. Let h

h0

h1

hp−1

hp

0 → S −→ S0 −→ S1 −→ · · · −→ Sp −→ Sp+1 ,

p > 1,

be an exact sequence of sheaves on a paracompact topological space Z, where the sheaves Sp and Sp+1 are not necessarily acyclic, and let h

h0

h1

hp−1

hp

∗ ∗ ∗ ∗ ∗ Γ(Z, Sp+1 ) (1.1) Γ(Z, Sp ) −→ 0 → Γ(Z, S) −→ Γ(Z, S0 ) −→ Γ(Z, S1 ) −→ · · · −→

be the corresponding cochain complex of structure groups of these sheaves. The qcohomology groups of the cochain complex (1.1) for 0 ≤ q ≤ p are isomorphic to the cohomology groups H q (Z, S) of Z with coefficients in the sheaf S.  References [1] G.Giachetta, L. Mangiarotti, G. Sardanashvily, Cohomology of the variational complex, arXiv: math-ph/0005010v4; Theorem 3.

2

Generalization of the Serra–Swan theorem for non-compact manifolds

Theorem GS2. Let X be a smooth manifold. A C ∞ (X)-module P is isomorphic to the structure module of a smooth vector bundle over X if and only if it is a projective module of finite rank.  References [1] G. Giachetta, L. Mangiarotti, G. Sardanashvily, Advanced Classical Field Theory (World Scientific, Singapore, 2009); Theorem 10.9.3.

1

3

Serra–Swan-like theorem for graded manifolds

Combining Theorem GS2 and the well-known Batchelor theorem leads to the following one. Theorem GS3. Let Z be a smooth manifold. A graded commutative C ∞ (Z)algebra A is isomorphic to the structure ring of a graded manifold with a body Z if and only if it is the exterior algebra of some projective C ∞ (Z)-module of finite rank.  References [1] D.Bashkirov, G.Giachetta, L.Mangiarotti and G.Sardanashvily, The antifield Koszul– Tate complex of reducible Noether identities, J. Math. Phys. 46 (2005) 103513; Theorem 1. [2] G. Giachetta, L. Mangiarotti, G. Sardanashvily, Advanced Classical Field Theory (World Scientific, Singapore, 2009); Theorem 3.3.2.

4

Cohomology of differential forms on an infinite order jet manifold

Theorem GS4. Let Y → X be a smooth fibre bundle and J ∞ Y its infinite order jet manifold. Cohomology of differential forms on J ∞ Y in the class of finite jet order equals that of differential forms in the class of locally finite jet order.  References [1] G. Sardanashvily, Cohomology of the variational complex in the class of exterior forms of finite jet order, Int. J. Math. and Math. Sci. 30 (2002) 39-48; Theorem 5.1 [2] G. Giachetta, L. Mangiarotti, G. Sardanashvily, Advanced Classical Field Theory (World Scientific, Singapore, 2009); Theorem 2.5.2.

5

Cohomology of the variational bicomplex on fibre bundles

The following is based on Theorems GS1 and GS4.

2

Theorem GS5. Given a fibre bundle Y → X, the variational bicomplex .. . dV

dV

6 1,0 O∞

0→ dV

0→R→

.. . dH

dV dH

0 O∞



6

0→R→

dV

6 dH

1,1 O∞



6

.. .

dV

6 dH

%



→···

6

E1 → 0 −δ

6 0,n O∞

6 d

0

−δ

6 1,n O∞

→···

0,1 O∞

.. .



6 0,n O∞

(5.1)

6 d

1

d

O (X) →

O (X) → · · ·

On (X) → 0

6

6

6

0

0

0

∗ of a graded differential algebra O∞ on an infinite order jet manifold J ∞ Y possesses the following cohomology. (i) The second row from the bottom and the last column of the variational bicomplex (5.1) make up the variational complex d

d

δ

δ

H H 0,1 0,n 0 0 → R → O∞ −→ O∞ · · · −→ O∞ −→ E1 −→ E2 −→ · · · .

Its cohomology is isomorphic to the de Rham cohomology of a fibre bundle Y , namely, k≥n ∗ H k≥n (δ; O∞ ) = HDR (Y ).

∗ k