I would say both antigen and antibody tests are covered per footnote 10 in FAQ Set 51.
FAQ Set 51 - PDF page 3
…since June 2020, when FAQs Part 43 were issued, the FDA has authorized additional diagnostic tests for COVID-19, including tests that can be self-administered and self-read at home or elsewhere without the involvement of a health care provider, sometimes referred to as self-tests or at-home tests.10
10 The FDA provides information on which at-home tests are authorized for use at https://www.fda.gov/medical-devices/coronavirus-disease-2019-covid-19-emergency-use-authorizations-medical-devices/in-vitro-diagnostics-euas .
At that webpage, the introduction lists both types of tests .
There are several types of SARS-CoV-2 and COVID-19 related IVDs:
Diagnostic Tests: Tests that detect parts of the SARS-CoV-2 virus and can be used to diagnose infection with the SARS-CoV-2 virus. These include molecular tests and antigen tests.
Serology/Antibody and Other Adaptive Immune Response Tests: Tests that detect antibodies (for example, IgM, IgG) to the SARS-CoV-2 virus or that measure a different adaptive immune response (such as, T cell immune response) to the SARS-CoV-2 virus. These types of tests cannot be used to diagnose a current infection.