A cohomology theory of grading-restricted vertex algebras

We introduce a cohomology theory of grading-restricted vertex algebras. To construct the {\it correct} cohomologies, we consider linear maps from tensor powers of a grading-restricted vertex algebra to "rational functions valued in the algebraic completion of a module for the algebra," instead of linear maps from tensor powers of the algebra to a module for the algebra. One subtle complication arising from such functions is that we have to carefully address the issue of convergence when we compose these linear maps with vertex operators. In particular, for each $n\in \mathbb{N}$, we have an inverse system $\{H^{n}_{m}(V, W)\}_{m\in \mathbb{Z}_{+}}$ of $n$-th cohomologies and an additional $n$-th cohomology $H_{\infty}^{n}(V, W)$ of a grading-restricted vertex algebra $V$ with coefficients in a $V$-module $W$ such that $H_{\infty}^{n}(V, W)$ is isomorphic to the inverse limit of the inverse system $\{H^{n}_{m}(V, W)\}_{m\in \mathbb{Z}_{+}}$. In the case of $n=2$, there is an additional second cohomology denoted by $H^{2}_{\frac{1}{2}}(V, W)$ which will be shown in a sequel to the present paper to correspond to what we call square-zero extensions of $V$ and to first order deformations of $V$ when $W=V$.