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We study the category of (complex-valued) finitely-generated smooth representations of a p-adic group G and its K theory. We show that every representation has a resolution by representations induced from finitely-generated representations of open compact subgroups. We do this by studying another category, the compactified representation category recently defined by Bezrukavnikov and Kazhdan, and using techniques from toric mirror symmetry to send it functorially into a geometric category of equivariant constructible sheaves on the Bruhat-Tits building.