Noether identities of a generic differential operator. The Koszul-Tate complex

Remarks on two approaches to the horizontal cohomology: compatibility complex and the Koszul-Tate resolution

The Koszul-Tate resolution is described in the context of the geometry of jet spaces and differential equations. The application due to Barnich, Brandt, and Henneaux of this resolution to computing the horizontal cohomology is analyzed. Relations with the Vinogradov spectral sequence are discussed.

Noether's inverse second theorem in homology terms

A generic degenerate Lagrangian system of even and odd variables on an arbitrary smooth manifold is examined in terms of the Grassmann-graded variational bicomplex. Its Euler-Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. However, non-trivial higher-stage Noether identities are ill defined, unless a certain homology condition holds. We show that, under this condition, there exists the exact Koszul-Tate chain complex whose boundary operator produces all non-trivial Noether and higher-stage Noether identities of an…

关于微分 $\mathbb Z$-graded 变种的构造

On construction of differential $\mathbb Z$-graded varieties

本文提出了一种构造微分 $\mathbb Z$-graded 变种的方法，通过树状Koszul-Tate解析和同伦 retract 数据减少同调计算，同时关联了李-里纳尔特代数与坐标环的解析。