BRST theory emerged in the framework of the quantum theory of gauge fields, where the timing of a degeneration of the Yang - Mills Lagrangian led to its replacement in a generating functional with some modified Lagrangian, depending on ghost fields and invariant under BRST transformations. These BRST transformations resulted from the replacement of parameter functions in gauge transformations with odd ghost fields, and their extension to action on these ghost fields. BRST theory was mainly developed in the framework of Hamiltonian formalism, but its Lagrangian variant also was under consideration. The main works in this direction were the articles of J.Gomis, J.Paris, S.Samuel in 1995 and G.Barnish, F.Brandt, M.Henneaux in 2000 in *Physics Reports*, as well as preceding works of these authors in the *Communication in Mathematical Physics*. These works, however, involved the so-called regularity condition which came from BRST theory of Hamiltonian systems with constraints, and which was not appropriate for Noether identities. The latter, in contrast to the algebraic constraint conditions, are the differential identities. Moreover, this BFST theory was developed for fields on .

I was interested in BRST theory, as a kind of prequantum field theory which is a necessary step in the procedure of BV-quantization of fields. Because a BRST operator is nilpotent, I spent the calculation of its relative and iterated cohomology on an arbitrary manifold *X* [95] in 2000. In 2005, I returned to BRST theory in connection with a consideration of a general type of gauge transformations, depending on the derivatives of fields of arbitrary order [118]. Later, when studying Noether identities, I gave up on the above-mentioned conditions of regularity and introduced a new cohomology condition. In 2008, after constructing a complete description of reducible degenerate Lagrangian systems, I began exploring their BRST extension. Such an extension was proved to be possible if the gauge operator continues to a nilpotent BRST operator acting on ghost fields. In this case, the above-mentioned cochain sequence becomes a complex, called the BRST complex, and an original Lagrangian admits the BRST extension, depending on original fields, antifields, indexing the zero and higher order Noether identities, and ghost fields, parameterizing zero and higher order gauge symmetries [129]. The BRST extension of some basic field models was built.

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