Today we’ll do a short post on some P.D. algebra properties and constructions. Let’s start with properties of P.D. ideals. Our first proposition is that given and as two P.D. ideals in , then is a sub P.D. ideal of both and . This is very straightforward to check using the criterion from last time, since is generated by the set of products where and . This proposition immediately gives us that powers of P.D. ideals are sub P.D. ideals and there is a natural choice for P.D. structure on them.
Another proposition is that given two P.D. ideals as above with the additional property that is a P.D. ideal of and and that and restrict to the same thing on the intersection, then there is a unique P.D. structure on such that and are sub P.D. ideals. Proving this would require developing some techniques that would lead us too far astray. We probably won’t use this one anyway. It just gives a sense of the types of constructions that are compatible with P.D. structures.
Another construction that requires no extra effort are direct limits. If is a directed system of P.D. algebras, then has a unique P.D. structure such that each natural map is a P.D. morphism.
Unfortunately, one common construction that doesn’t work automatically is the tensor product. It works in the following specific case. If and are -algebras, and and are augmentation ideals with P.D. structures and respectively, then form the ideal . We then get that has a P.D. structure such that and are P.D. Morphisms.
Next time we’ll start thinking about how to construct compatible P.D. structures over thickenings. Since we’ll be thinking a lot about I’ll just end this post by pointing out that actually has many choices of P.D. structure. But last time we said that actually has a unique one, so our convention is going to be to choice the “canonical” P.D. structure on induced from the unique one in .