  We construct GLSM invariants for a general choice of stability in both the
narrow and broad sector cases and prove they form a Cohomological Field Theory.
This is obtained by forming the analogue of a virtual fundamental class which
lives in the local cohomology of the twisted Hodge complex. This general
construction comes from the use of two new ingredients. First, the use of the
Thom-Sullivan and Godement resolutions applied to matrix factorizations are
introduced to handle poorly behaved (non-separated) moduli spaces. Second, a
localized Chern character map built from the Atiyah class of a matrix
factorization is utilized to forgo the use of Hochschild homology.
