Oakland filmmaker Peter Nicks understands the ins and outs of his city’s public institutions, and he wants audiences to gain as much insight and clarity about its inner workings as he did. From observing the overwhelmed ER at Highland Hospital during a chaotic few hours in 2012’s “The Waiting Room” to shadowing the Oakland Police Department during a precarious time of national unrest and an internal sex scandal in 2017’s “The Force,” Nicks’ fly-on-the-wall technique paints a multihued portrait of a vibrant but challenged city. With “Homeroom,” the finale in his decade-in-the-making Oakland trilogy, Nicks burrows into the public education system and discovers something that’s eluded him on previous journeys: hope.