Incurable Alzheimer's disease (AD) plagues many elderly people and families. It is important to accurately diagnose and predict it at an early stage. However, the existing methods have shortcomings, such as inability to learn local and global information and the inability to extract effective features. In this paper, we propose a lightweight classification network Local and Global Graph ConvNeXt. This model has a hybrid architecture of convolutional neural network and Transformers. We build the Global NeXt Block and the Local NeXt Block to extract the local and global features of the structural magnetic resonance imaging (sMRI). These two blocks are optimized by adding global multilayer perceptron and locally grouped attention, respectively. Then, the features are fed into the pixel graph neural network to aggregate the valid pixel features using mask attention. In addition, we decoupled the loss by category to optimize the calculation of the loss. This method was tested on slices of the processed sMRI datasets from ADNI and achieved excellent performance. Our model achieves 95.81% accuracy with fewer parameters and floating point operations per second (FLOPS) than other classical efficient models in the diagnosis of AD.