On projections of the tails of a power.

Let 휅 be an inaccessible cardinal, 픘 a universal algebra, and ∼ the equivalence relation on U κ of eventual equality. From mild assumptions on 휅, we give general constructions of E ∈ End (U κ / ∼) satisfying E ∘ E = E which do not descend from Δ ∈ End ⁡ (U κ) having small strong supports. As an application, there exists an E ∈ End (Z κ / ∼) which does not come from a Δ ∈ End ⁡ (Z κ) . [ABSTRACT FROM AUTHOR]