The question ¡°Does Q have finite projective dimension?¡± is intimately connected to the question ¡°Does have the Weak Lefschetz Property?¡±. The second question is connected to the enumeration of plane partitions.
When the field k has positive characteristic, we investigate three questions about the Frobenius powers of Q. When does there exist a pair so that Q has infinite projective dimension and has finite projective dimension? Is the tail of the resolution of the Frobenius power eventually a periodic function of t (up to shift)? In particular, we exhibit a situation where the tail of the resolution of , after shifting, is periodic as a function of t, with an arbitrarily large period. Can one use socle degrees to predict that the tail of the resolution of is a shift of the tail of the resolution of Q?
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