Today we’ll look at the P.D. envelope of an ideal. To do this properly would take many pages of gory calculations, so we’ll be a little sketchy in order to get the idea out there. Before we do that we need to look at a construction I’ve been avoiding on purpose. Suppose is an -module. Then there is a P.D. algebra and an -linear map satisfying the universal property that given any an -P.D. algeba and an -linear map there is a unique P.D. map with the property .

Let’s be a little more explicit what this is now. First, is a graded algebra with , and . Let’s denote for and for . In fact, by abusing notation we often just write in place of for the P.D. structure. This is because is generated as an -module by . This should just be thought of as a “generalized P.D. polynomial algebra”. We’ll soon see its importance. Now back to the regularly planned post.

Let be a P.D. algebra and an ideal of which is an -algebra. There exists a P.D. envelope of which is a -algebra denoted (or sometimes just when the rest is understood) with a P.D. ideal such that compatible with and satisfying a universal property:

For any -algebra, , containing an ideal containing with a compatible (with ) P.D. structure , there is a unique P.D. morphism which makes the diagram commute:

Where that vertical right arrow should be dotted. Like I said, the existence of this thing takes a lot of tedious calculations. You basically show by brute force that it exists in a few special cases and then reduce the general case to these. These calculations actually bear out a few interesting things that we’ll just list:

First, . Next, if our structure map actually factors and we have an extension to on , then , which essentially tells us that it really is an “envelope” or some sort of minimal construction.

More importantly, we have a more concrete way to think about the construction. Namely, is generated as a -algebra by and any set of generators of gives a set of P.D. generators for .

Let’s do two examples to get a feel for what this thing is. Let and , then we’ll consider the trivial P.D. structure on given by the ideal. This gives us the P.D. polynomial algebra. Along the same lines but slightly more generally, suppose is an -module and . Let be the ideal of generated by the postive graded part and do everything with respect to the trivial P.D. structure on again. We get .

The other example is that when we have extending to with a section , then the compatibility condition is irrelevant. This just means that . This is just because the section gives us that and an application of the universal property.

I’m not sure what to do next. I’m sure I’ve alienated all my readers with all this. At this point I could shift over and at least define crystalline cohomology. We have enough to cover some of the basic definitions and actually show that all this has a purpose. On the other hand, we definitely have a ton more properties we should do if we want to get anywhere with crystalline cohomology.