Generator-offset property: Difference between revisions

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For (6) and (7), we know SGA scales are MV3 and that MV3s project to mosses when one removes all instances of one step size, in particular s. Thus it suffices to prove that the result of equating any two step sizes is a mos. Equating Y and Z equates the swung alternants, resulting in the mos aX 2bW (this is a known fact).

It remains to prove that equating X and Y equates the generator with the difference between the start of the 2nd chain with the end of the 1st one or vice versa; similarly for equating X and Z. Consider the swung-generator-alternant chain g1 g2 .... g1 g2 g3. Assume that the offset g1 = δ has one more Y (and one fewer Z) than g2 = g - δ. g3 (which becomes the imperfect generator of aX 2bW) either has one more X or one fewer X than both g1 and g2. If g3 has one more X, then g3 + g1 is equated to g1 + g2 = g, as desired. If g3 has one fewer X, then simply use the inverted ~~the~~ generator class. <math>\square</math>

It remains to prove that equating X and Y equates the generator with the difference between the start of the 2nd chain with the end of the 1st one or vice versa; similarly for equating X and Z. Consider the swung-generator-alternant chain g1 g2 .... g1 g2 g3. Assume that the offset g1 = δ has one more Y (and one fewer Z) than g2 = g - δ. g3 (which becomes the imperfect generator of aX 2bW) either has one more X or one fewer X than both g1 and g2. If g3 has one more X, then g3 + g1 is equated to g1 + g2 = g, as desired. If g3 has one fewer X, then simply use the inverted generator class. <math>\square</math>

=== Proposition 2 ===

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 For (6) and (7), we know SGA scales are MV3 and that MV3s project to mosses when one removes all instances of one step size, in particular s. Thus it suffices to prove that the result of equating any two step sizes is a mos. Equating Y and Z equates the swung alternants, resulting in the mos aX 2bW (this is a known fact).
-It remains to prove that equating X and Y equates the generator with the difference between the start of the 2nd chain with the end of the 1st one or vice versa; similarly for equating X and Z. Consider the swung-generator-alternant chain g1 g2 .... g1 g2 g3. Assume that the offset g1 = δ has one more Y (and one fewer Z) than g2 = g - δ. g3 (which becomes the imperfect generator of aX 2bW) either has one more X or one fewer X than both g1 and g2. If g3 has one more X, then g3 + g1 is equated to g1 + g2 = g, as desired. If g3 has one fewer X, then simply use the inverted the generator class. <math>\square</math>
+It remains to prove that equating X and Y equates the generator with the difference between the start of the 2nd chain with the end of the 1st one or vice versa; similarly for equating X and Z. Consider the swung-generator-alternant chain g1 g2 .... g1 g2 g3. Assume that the offset g1 = δ has one more Y (and one fewer Z) than g2 = g - δ. g3 (which becomes the imperfect generator of aX 2bW) either has one more X or one fewer X than both g1 and g2. If g3 has one more X, then g3 + g1 is equated to g1 + g2 = g, as desired. If g3 has one fewer X, then simply use the inverted generator class. <math>\square</math>
 === Proposition 2 ===