Positron Emission Tomography (PET) plays a vital role in the early diagnosis of Alzheimer’s Disease. However, its high radiation exposure and cost limit its clinical application. In contrast, structural Magnetic Resonance Imaging (sMRI) is widely accessible but lacks sensitivity in the early stage of AD. Given the simplicity of acquiring sMRI, a growing number of studies focus on synthesizing PET images from sMRI to reduce costs. However, most existing methods neglect the non-bijective mapping between different modalities, resulting in distorted metabolic information. To address this issue, we propose the Feature-Decoupled Cycle Generative Adversarial Network (FD-CycleGAN), which incorporates two pivotal innovations: (1) a dual-branch downsampling architecture featuring a Dual-Attention Decouple Module that disentangles modality-shared anatomical features from PET-specific metabolic patterns through channel-spatial attention operations; and (2) a Tri-Fusion Integrate Module that employs Global Fusion at bottleneck layers and Channel-Local Fusion in skip connections, progressively integrating features to maintain metabolic and structural consistency. Comprehensive experiments on the Alzheimer’s Disease Neuroimaging Initiative (ADNI) dataset demonstrate that our method outperforms state-of-the-art approaches, achieving significant improvements with a Mean Absolute Error (MAE) of 0.0213, a Peak Signal-to-Noise Ratio (PSNR) of 31.72 dB, and a Structural Similarity Index (SSIM) of 0.796. Furthermore, downstream classification tasks show that the synthesized PET images can effectively improve the classification accuracy of AD. This study provides a robust method for generating clinically reliable PET from sMRI, thereby advancing the field of medical image synthesis.