# A cohomology theory of grading-restricted vertex algebras - Mathematics > Quantum Algebra

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Abstract: We introduce a cohomology theory of grading-restricted vertex algebras. Toconstruct the {\it correct} cohomologies, we consider linear maps from tensorpowers of a grading-restricted vertex algebra to -rational functions valued inthe algebraic completion of a module for the algebra,- instead of linear mapsfrom tensor powers of the algebra to a module for the algebra. One subtlecomplication arising from such functions is that we have to carefully addressthe issue of convergence when we compose these linear maps with vertexoperators. 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 anadditional $n$-th cohomology $H {\infty}^{n}V, W$ of a grading-restrictedvertex 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 anadditional second cohomology denoted by $H^{2} {\frac{1}{2}}V, W$ which willbe shown in a sequel to the present paper to correspond to what we callsquare-zero extensions of $V$ and to first order deformations of $V$ when$W=V$.

Author: ** Yi-Zhi Huang**

Source: https://arxiv.org/