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By Agostino Prastaro
This publication provides, for the 1st time, a scientific formula of the geometric conception of noncommutative PDE's that is compatible sufficient for use for a mathematical description of quantum dynamics and quantum box idea. a geometrical idea of supersymmetric quantum PDE's is additionally thought of, in an effort to describe quantum supergravity. Covariant and canonical quantizations of (super) PDE's are proven to be based at the geometric concept of PDE's and to provide quantum (super) PDE's via functors from the class of commutative (super) PDE's to the class of quantum (super) PDE's. worldwide homes of options to (super) (commutative) PDE's are got via their crucial bordism teams.
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U u Let U be an abelian category. A dz%ferentialobject in U is a pair (A,d) where AEOb(U) and dEHomu(A;A) such t h a t d 2 = 0 . let D ( U ) be the category of differential objects in U. We call homology t h e additive functor H:D(U)+U, given by H(A,d)=ker(d)/im (d)=Z(A)/B(A), where Z ( A ) is the set of cycles of A and B is the set of boundaries of A. H(A) is the homology of (A,d). Then, a spectral sequence in the category U is a sequence of differential objects of U : { E n , & } , N=1,2,. , such t h a t ff(E,,d,)=E,+~, n=1,2,....
In fact we have the following. 17. (Pontrjagin-Thom-Wall). , V = d M , for some smooth (n+l)-dimensional manifold M , i f f the Stiefel-Whitney numbers < wi,' . wi,,pv > axe all zero, where i l + . . i, = n is any partition o f n and pv is the fundamental class o f V . 17 Two smooth closed n-dimensional manifolds belong t o the same bordism class i f fall their corresponding Stiefel-Whitney numbers are equal. Furthermore, the bordism group +On o f closed n-dimensional oriented smooth manifolds is a finitely generated abelian group o f the form +on= Z @ .
A). Let us denote yet by and 5. the boundaries on the reduced complexes. Then one has: 8. h and similarly for In this way we get the chain complexes {fi*(A), & } and {@(A), &}. Then we can use such complexes to compute the Hochschild homologies. , where h B = A @K Aopp, B = A @Z A'pp, that are algebras with products ( a l @ and P. is a projective resolution of the left hmodule, (resp. 5-module), E . (A two-sided A-module E can be considered a left-B-module, (resp. B-module), by means of the formula: (a1 €3 a2)s = alsaz).