 This lecture is part of an online commutative algebra course and will be about Gorenstein rings, or more precisely it will be about Gorenstein local rings because all rings are going to be local and notarian. So Gorenstein rings are rings with a sort of duality property. So in order to define them let's first do the zero-dimensional case, which is a little bit easier than the than the general case. So if r is zero-dimensional and notarian and local then r is Gorenstein. It turns out to be equivalent to saying that Hom over r from k to r is one-dimensional over the field k as a vector space. So here this dimension is the dimension over the vector space k and here the dimension is the dimension, the ring theoretic definition. k is going to be the field r over m where m is the maximal ideal. And more generally we can define the dual of a module m to be Hom over r from m to r. And if r is Gorenstein this implies that the dual of m behaves well. For instance the dual of m, the dual of the dual of m should be the should be isomorphic to m if m is a finitely generated module over r and so on. For general rings this definition of duality doesn't behave at all well and in fact for general zero-dimensional rings there's an alternative definition of duality where you define that the dual of m to be Hom over r from m to omega. So this is going to be the dual of m where omega is some special module called the dualizing module. So for Gorenstein rings that they're the ones where the dualizing module happens to be isomorphic to the ring you first thought of at least in the nought dimensional case. High dimensions things get get more complicated. So let's see a few examples of zero-dimensional rings that are or aren't Gorenstein. So first of all we could take say a ring of formal power series modulo x to the five and here Hom over r from k to r just as dimension equals one because it's spanned by say it's spanned by x to the four and it's sort of useful to kind of draw a picture of these of zero-dimensional rings. There's a lot of blocks. So for instance this ring can be pictured as a series of five blocks where where this bottom block is generated by x to the four. The two bottom blocks are generated by x cubed by x squared x and one. So each of these blocks sort of represents a sub quotient isomorphic to k. So this this ring has length five as a module over itself and you can picture pictures looking like this and then the reason it's Gorenstein is that there's only one block on the bottom. So Hom from k to r is sort of the blocks on the bottom of this tower and hopefully the pictures of these towers will become a little bit clearer in the next example. So now let's look at r equals k x y and I'm going to quotient out by x squared x y y squared. Now let's draw a picture of this ring as a module over itself and this time it is length three so we get three blocks and we can sort of picture the blocks a bit like this. So this is going to be generated that the whole tower is generated by one and this block might be generated by x and this might be generated by y say. And now you notice that there are two I've drawn two blocks on the bottom. This corresponds to the fact that Hom over r from k to r this time has dimension equal to two because it's a two dimensional space spanned by x and y. So this one is not Gorenstein. We can sort of make it Gorenstein by quotienting out a bit less so now if I take r equals k x y modulo x squared and y squared now this is length four and I can draw a picture of it a bit like this. So on top I have the one and then I have the submodules generated by x and y which are all two dimensional because they both contain x y. And now that you can see that there's only one block on the bottom so this corresponds to the fact that and the dimension from Hom over r from k to r is now equal to one again. So this one is Gorenstein. And another informal way of thinking of a Gorenstein ring that's zero dimensional is it looks the same if you turn it upside down. So if I take five blocks here and flip them upside down I get the same thing and that corresponds to the fact that r is kind of self-dual in some sense. If I take this collection and flip it upside down then it looks like that which is definitely different. Now there's only one thing on the bottom that sort of corresponds to the fact that r is not self-dual. If you dualize r you get a different module and finally this one is the same upside down again so it is indeed self-dual. So for zero dimensions the duality property of r is reasonably easy to understand. In higher dimensions the definition is a little bit more complicated. Growth and Dick defined Gorenstein rings so as usual these are going to be local notarian rings so I suppose r has dimension r. It's called Gorenstein if an x to the i of kr is zero for i not equal to d and has dimension 1 for i equals d. So for i equals zero so for d equals zero which is just the zero dimensional case x zero is just HOM so this is sort of saying HOM from k to r is one dimensional which is what we had before. Now the first thing you notice about this definition is that it's really a bit of a heavyweight definition. It's not immediately obvious what these higher order x's are. You can also wonder why are they called Gorenstein. I can't spell Gorenstein. Why is it called a Gorenstein ring if the definition was invented by growth and dick? Well Gorenstein was best known as the group theorist who was sort of directed the classification of finite simple groups but before he started doing group theory he actually worked an algebraic geometry and he proved a theorem about plane curve singularities and he found a certain property of them which turns out to be equivalent to the Gorenstein property so growth and dick named them after Gorenstein because Gorenstein essentially showed that plane curve singularities are Gorenstein rings. Incidentally mathematical folklore says that Gorenstein himself went around claiming that he didn't actually understand the definition of a Gorenstein ring. I'm not buying this I mean Gorenstein was a smart guy and was perfectly capable of understanding this definition and for heaven's sake if someone names something after you of course you go and read up what the definition is so it's a nice story but you should take it with a grain of salt. I think that the name Gorenstein ring was really popularized by Bass who wrote this famous paper on ubiquity of Gorenstein rings shortly after Gorenstein invented them. Okay so obviously Bass had a word day calendar or something and he came up with this word ubiquity which means everywhere present and it's used a lot by theologians and I discovered this when I was trying to search on Google for the paper ubiquity of Gorenstein rings and when you get as far as ubiquity of G.O. Google suddenly throws up a lot of search suggestions about papers on the ubiquity of God so anyway unfortunately Bass's paper is behind a paywall if you try and search for it but if you have a university account you may be able to get hold of a copy. So anyway let's get back to this problem that this definition is rather rather heavy going fortunately in order to test whether a ring is Gorenstein we can use a much simpler criterion. If R is greater if R has dimension greater than 0 then R is Gorenstein if and only if R has a zero divisor so a non-zero divisor X in M so that R over X is Gorenstein. So you remember exactly the same thing held for Cohn-McCauley rings a ring is Cohn-McCauley if and only if when you question it out by a non-zero divisor the result is is also Cohn-McCauley. So you can sort of reduce your ring to a zero dimensional ring like this and then test the zero dimensional ring to see if it's Gorenstein and if you can't get down to zero dimensions then your ring isn't even Cohn-McCauley and it's not Gorenstein and you may sort of worry a bit that you have to choose this non-zero divisor carefully but in fact it doesn't matter so any non-zero divisor will do provided it's in M. I'm not going to prove this but it's quite easy to prove what you do is if X is a non-zero divisor we look at the exact sequence nought goes to R goes to R goes to R over X and then we've got a short exact sequence so you remember from homological algebra you can get lots and lots of long exact sequences involving the X groups from that and by playing around with those you can show that this condition here is equivalent to this condition about X groups. You may wonder why we don't use this simpler definition. Well the answer is with the simpler definition it's not at all clear that the result is independent of the choice of X which is a bit of a nuisance. This more high-powered definition using X groups it's theoretically a bit easier to use because you don't have to choose an element to check it's independent of the choice of element. So what we're going to do now is just give some examples of higher dimensional rings that are or aren't Goronstein. It turns out that being Goronstein is actually a really subtle property in making a apparently harmless change to the ring changes whether or not it's Goronstein. So the first example is we're going to take the ring of power series in two variables and we're going to let this be acted on by a group of order three a cyclic group of order three and we're going to look at the fixed subring and see whether this is Goronstein. Well there's more than one way this group of order three can act on this it can change for instance it can change X to omega X where omega is a cube root of unity so omega cube equals one we're going to work over a field of characteristic not three so let's just say the characteristic K is not equal to three and we could map Y to omega Y and that would give us one ring. Alternatively we can map X to omega X and we can map Y to omega squared Y and this looks like a completely trivial change there doesn't really seem to be much difference between these two actions but we'll see that one of them is Goronstein and one of them isn't. So in order to see whether they're Goronstein or not we should start by drawing a picture of the ring and we're going to sort of start by drawing well it's not quite a basis but let's pretend it's a basis so we draw one X, X squared, X cubed, Y, Y squared, Y cubed and so on so so we draw a point for each monomial and this isn't quite a basis because it's formal power series so whatever but we won't worry about that too much. Now what we have to do is identify the fixed elements so the elements that are fixed under X goes to omega X and Y goes to omega Y are going to be those such that the total degree is divisible by three so our ring will really be sort of all these orange things here and now this is a two-dimensional ring so what we have to do is quotient out by non-zero divisor so let's pick a non-zero divisor so X cubed is a good non-zero divisor and let's quotient out by X cubed well that means we throw out everything in this region here and this reduces us to a one-dimensional ring and now we can ask is this one-dimensional ring Goronstein well to do that we again have to quotient out by non-zero divisor and a non-zero divisor in this quotient is Y cubed so here's a non-zero divisor and let's quotient out by this non-zero divisor well that means we throw away everything here and what are we left with well now we're left with a zero dimensional ring looking like this and now you see if we draw it's in block form it kind of looks like this it here this element would be XY squared and this element here would be X squared Y so the we see that HOM over R from K to R where R is now this bit here has has it is now two-dimensional so this is not Goronstein now let's do the same with this ring here and I'll do this a bit much more quickly because it's very similar so again we draw out the sort of not quite a basis for it looking like this and this time we mark an orange the fixed point elements and this time it looks a little bit different now looks like this so fixed point elements are going to be lines looking like that and now again we kill off a non-zero divisor like this so we kill off all this stuff and then we take a second non-zero divisor here and kill off all the stuff that's a multiple of it and now we see we're left with something that subtly different if we draw this in block form it now looks like this so this is one XY X squared Y squared so now the dimension of HOM over R from K to R is just one in fact it's spanned by this bit here whereas in the previous example it was spanned by these two bits here so although these two rings look almost identical one of them is Goronstein and one of them isn't this is also illustrates the fact it's actually quite hard to tell whether a ring is Goronstein just by sort of looking at it and you know if a ring is regular or Cohen Macaulay it's usually pretty obvious whether or not it's regular or Cohen Macaulay I mean regular sort of means non singular like a manifold and Cohen Macaulay has something to do with bits of different dimension not meeting but but Goronstein property is much subtler so now I'm going to give another example where it's really quite hard to guess whether a ring is Goronstein or not this time I'm going to look at singularities of curves in four-dimensional affine space and I'm writing down curves by writing down equations and four-dimensional affine space is a bit tedious because you've got to write down lots of equations so what I'm going to do is I'm just going to define the curve by the image under an embedding so what I can do I can map the curve a1 to a4 by mapping the point t to say t to the 4, t to the 5, t to the 6, t to the 7 and then the image of this will have some sort of funny cusp but at the origin or another thing I can do is I can map it to t to the 5, t to the 6, t to the 7, t to the 8 or I could map it to t to the 6, t to the 7, t to the 8, t to the 9 and I think you will have to agree that these three curves all look very similar I mean if I asked you to guess which of them were Goronstein and which were not Goronstein you would probably have rather a hard job trying to see any difference at all between them and if you draw pictures there in four-dimensional space it's very hard to distinguish them and what we're going to do is we're going to show the middle one is Goronstein but the two outer ones are not Goronstein so it's again illustrates that the Goronstein property is very subtle so let's first look at the first one and for this again I'm going to draw a basis of the ring so we're going to draw one t, t squared, t cubed, t to the 4, t to the 5 and so on I forgot to say we're looking at the local ring at the origin of course at all points other than the origin that this is just a regular curve so it's not very interesting and now the local ring at the origin or let's say the completion of the local ring at the origin doesn't actually contain t or t squared or t cubed it just contains 1, t to the 4, t to the 5, t to the 6, t to the 7 then it contains t to the 8 because that's t to the 4 squared and it contains everything beyond that so the local ring well let's take its completion for simplicity then it's going to be sort of spanned by the in some formal sense by by these elements here so now let's work out whether it's Goronstein so we pick a non-zero divisor in the maximal ideal which will be this element here and we then have to kill off all elements in the ideal generated by this we have to multiply this by 1 and then we have to multiply it by t to the 4 and t to the 5 and t to the 6 and so on so if we look at r over t to the 4 we need to check whether this is Goronstein and it just consists of this element and this and this and this and you see the product of any of these two is zero so we see that HOM over whatever it is r over t to the 4 I guess from k to r over t to the 4 has dimension 3 so it's not Goronstein well now let's do this case where we take t to the 5 t to the 6 t to the 7 t to the 8 and see this apparently trivial change from 4 to 5 does actually make a difference so again we draw a picture of a basis for the local ring so we can take 1 2 3 4 5 6 7 8 so it's going to be 5 9 10 12 13 14 and so on so our ring the local ring contains 1 it contains t to the 5 6 7 and 8 it doesn't contain t to the 9 contains t to the 10 t to the 11 t to 12 and so on so it contains everything beyond that I can't quite remember how far I need to go so so we want to test whether this ring this orange ring is Goronstein and to do that we pick a non-zero divisor so here's a non-zero divisor t to the 5 and now we need to work out the ideal generation by t to the 5 so we need t to the 5 and then it contains t to the 10 t to the 11 t to the 12 t to the 13 it doesn't contain t to the 14 though but it does contain t to the 15 and everything beyond that so let's summarise what we've got so this is t to the 14 so so let's look at what r over t to the 5 looks like well it contains t to the 1 it contains t to the 6 t to the 7 t to the 8 and it also contains t to the 14 and now we want to work out what is home over r over t to the 5 from k to r over t to the 5 well we notice that none of these elements are can be in the image of k because t to the 6 times t to the 8 equals t to the 14 which is not zero and t to the 7 times t to the 7 equals t to the 14 which is not zero so this space here is one dimensional spanned by t to the 14 so the dimension of this is equal to 1 so it is Gorenstein and you notice we are getting a sort of duality on this three dimensional space we get a sort of bilinear form mapping this three dimensional space times itself to this one dimensional space and this is this is part of the duality that you always expect when working with Gorenstein rings so the third example now let's look at what happens if you take six seven eight and nine so here we've got one so our ring contains the elements one and then it contains one two three so t to one two three four five and t to six seven eight and nine write these down I'm using tracks six seven eight nine and then it doesn't contain 10 11 but it contains t to the 12 13 14 15 16 17 18 and so I'm actually have to go a bit further and we write 9 10 11 12 13 14 15 16 17 and so on and now as usual we have to question up by a zero device which we choose to be t to the 6 and this kills off t to the 6 and then we kill off that and then we have to kill off t to the 12 t to the 13 14 15 but we keep 16 17 and then we kill off t to the 18 we kill off everything beyond that so if we look at r over t to the 6 we end up with one t to the 7 t to the 8 t to the 9 t to the 16 t to the 17 and the home from K to this is now two dimensional spanned by these two elements here so again this example is not Gorenstein so you see being Gorenstein is a sort of very subtle arithmetic property of exactly which elements we have which of these elements appear in a basis for the ring so I'll just finish by sketching the fact that regular rings in our Gorenstein and for this we use the causal complex so you remember if we've got a if we've got a regular local ring R this time we're going to use growth index rather high powered definition of regular rings of course you could do it just by using the just by quotienting out by a regular sequence and getting down to a one dimensional to a zero dimensional ring that's one dimensional over case that would prove it but I want to do it using the causal complex so we want to show that X to the D over R of K and R has length one and we want to show that all the ones other than this vanish so what we do is we pick X1 up to XD to be a basis for M over M squared which we can do as R is regular and then we look at the causal complex so we get naught goes to R goes to R to the D goes to R to the D choose to and so on so it goes all the way down to R to the D goes to R goes to R over X1 up to X to the D which is in fact isomorphic to K and now we can use this to compute the X groups so we want to compute X to the D over R of K and R and you remember for this we can we've got we see that this X group is given by the homology of the sequence naught goes to R goes to R to the D and so on so we want the homology at this point here and if we remember what the causal complex is this dual map will take a basis a1 up to a D to X1 a1 plus X2 a2 and so on so the homology of this bit here will be R over X1 X2 up to XD which is equal to K so so we can show you some the causal complex that this piece here has dimension one and in fact if this group here has dimension one then it in fact implies all the other X groups have dimension zero so in fact you don't really need to calculate them okay next lecture next two lectures we're going to be studying the other sort of ring which is complete intersection rings and what we're first going to be doing is studying fitting ideals which are a useful technical tool for studying complete intersection rings