What's a fragment? Some languages allow the speaker to omit pronouns clearly implied by context, a feature called "null subject" or "pro-drop". WALS lists this as feature 101: Expression of pronominal subjects. Spanish and Italian allow a null subject, but French and English do not. This means that all complete sentences in English must have an explicit subject; otherwise, you end up with not a sentence but a fragment. Fragments occur often in spoken English due to conversational deletion, but formal writing avoids conversational deletion outside dialogue.
Is this a fragment? No. This sentence has a subject: "an inflated perception of knowledge". Microsoft Word just can't find it because the writer inverted the constituent order from English's more common order, which puts the subject before the predicate. The sentence starts with "identified", the passive participle form of "identify". This form is identical to the past tense form "identified", and Word sees an initial finite verb and assumes that what follows it isn't a subject. So it guesses that the writer may have accidentally performed conversational deletion in an inappropriate context.
Undoing the the inversion would make the sentence easier for Word to parse, and probably for human readers as well, as jlawler pointed out in a comment:
An inflated perception of knowledge is identified as the central problem.
Is the inversion the only thing tripping up Word? This sentence would likely have a green squiggly for another reason: the passive voice. In addition to a subject and object, a verb also has an agent (someone performing an action) and patient (the thing on which the action is performed). English is an "accusative" language, meaning that the sentence's subject is usually the agent. It does allow promoting the patient to a clause's subject using be followed by a participle, and writers often do this when the agent is unknown or unimportant. However, overuse of passive voice tends to sound unclear or evasive, as in "mistakes were made". For this reason, many style guides recommend avoiding excessive use of the passive voice. Thus Word will probably draw a squiggly for that too. In fact, one style guide called "E-Prime", promoted by the General Semantics movement, eschews all use of be as a blunt instrument to avoid passing off judgments as facts.