Let the vector bundle \(\mathcal{E}\) be a deformation of the tangent bundle over the Grassmannian \(G(k,n)\). We compute the ring structure of sheaf cohomology valued in exterior powers of \(\mathcal{E}\), also known as the polymology. This is the first part of a project studying the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle, a generalization of ordinary quantum cohomology rings of Grassmannians. A companion physics paper [arXiv:1512.08586] describes physical aspects of the theory, including a conjecture for the quantum sheaf cohomology ring, and numerous examples.