 This lecture is part of an online course on commutative algebra and will be about blow-up algebras. Actually, it won't really be about blow-up algebras. What it will be is a survey of algebras constructed from a filtration, but the term blow-up algebras sounds kind of cooler. So for a filtration, we have various subsets of a ring, closed under addition, such that r i times r j is contained in r i plus j. And there are two sorts of filtration. We can have a decreasing filtration where the subsets of the filtration decrease. And a typical example of this is where you take r n to be some power of an ideal r i. On the other hand, we can have an increasing filtration where r naught is contained in r one, which is contained in r two and so on. And a typical example of this is where we fix some subring r zero and we take r i to be spanned by products of at most n, sorry, at most i elements taken from some set, then a one up to a k. So you might fix some set of generators for the ring and then r i is going to be the set of monomials of degree at most i. And there are two or three ways of constructing algebras from an increasing or decreasing filtration. We can construct a blow up algebra or we can construct a graded algebra or we can construct a sort of inverse limit. So what I'm going to do is I'm going to go through all these possibilities and give a few examples of them. So first of all, let's do the blow up algebras. And here we have a decreasing filtration, r naught contained in r one and so on. And usually we take our n to be i to the n for some ideal, although you don't really have to do this. And then the blow up algebra is just the algebra where you take the direct sum of these and so on. So this is graded because we can just set the degree of these pieces to be zero, one, two and so on. So this algebra was used in the proof of the art in re-slammer. So you remember we discussed the notion of a stable filtration of a module and showed that a module had a stable filtration if and only if a corresponding blow knot module was a finer e-generation module over this ring. And these correspond to blow-ups in algebraic geometry. So I'll say a little bit about what a blow-up in algebraic geometry is. So if we take a sub-variety of a variety v, so let's take w contained in v, let's suppose these are varieties, then we have a concept of a blow-up of v along w. And the idea is that roughly speaking, suppose w, v are both smooth and w is nicely embedded in v, then the blow-up replaces each point of w by the projective space of its normal bundle. So for instance, if v is a plane and w is a point, say the origin, then the blow-up of the plane at the point w is as a construction that ends up replacing this point by an entire copy of projective, one-dimensional projective space because that's the projective space of the normal bundle of w. I'm not going to describe this construction in detail because it's really algebraic geometry rather than commutative algebra in that it involves non-affine varieties. Anyway, this is closely related to the blow-up algebra because we take r to be the coordinate ring of v and we take i to be so that r over i is the coordinate ring of w. And then we look at the blow-up algebra, r plus i plus i squared and so on. And this is a graded ring. Let's call it r twiddle. And then the blow-up of w, sorry, the blow-up of v along w is a construction where you, there's a construction algebra where you sort of take a sort of projective, it's not exactly a projective space, but it's called prodge because it's related to the construction of a projective space and you apply it to a graded ring. And if you apply this construction to the blow-up algebra constructed like this, you get the blow-up in algebraic geometry. So that's where the name blow-up for this ring comes from. It's very closely related to blow-ups in algebraic geometry. But next we can look at the graded algebra. So here again, I'm going to take a decreasing filtration and we can take a graded algebra corresponding to this filtration to be the following ring. We just take r0 modulo r1 plus r1 modulo r2 plus r2 modulo r3 and so on. So you can see this has turned the ring r into a graded ring. And you can see that in some sense, this graded ring is about the same size as r because both of them have filtration such the quotients of these pieces here. So if we take r to be a local ring and i to be the maximal ideal, then this turns the local ring into k plus m over, let's call the maximal ideal m because that's what I'm used to calling it, m over m squared plus m squared over m cubed and so on, where this is the quotient field of r, r over its maximal ideal. So for example, if you take the ring k xy and suppose we localize it at the origin, which means we localize with respect to the ideal generated by x and y, then if we take the corresponding graded ring, we get back the polynomial ring we first started with. So this is not quite an inverse to localization but something a little bit like it. Another example might be supposed to take the localization of z at the prime two. So this is things of the form a over b with b odd. And let's take i to be the ideal of all multiples of two. Then the graded ring turns out to be z modulo two z plus z modulo two z modulo z modulo four z and so on, which is just the ring of polynomials over the field with two elements. So we've somehow turned this local ring into a polynomial ring over a field with two elements. There's a sort of relation between the graded algebra of a ring and the Reese algebra. There's something actually called the extended Reese algebra which you occasionally come across, which is given by r plus i plus i squared plus i cubed and so much you may think is just the ordinary Reese algebra, but then you extend it to the left and just add lots of copies of r. So the degree is given by let me let this have nought minus one, minus two, one, two, three and so on. By the way, the extended Reese algebra is sometimes called the Reese algebra but by authors just to confuse everybody. And what you find is the extended Reese algebra is an algebra if r is an algebra over some field k, for example, then you find that the extended Reese algebra is an algebra over the ring of polynomials where you let t have degree minus one. Let's call this the extended Reese algebra. And geometrically, you find that this is a sort of affine line. The fiber over the affine line is more or less the the fiber over the origin of the affine line is the graded algebra of r with respect to the ideal and the fiber over any other point of the affine line is just the ring r. So what the extended Reese algebra does is it shows the graded algebra of r is a deformation of r. In fact, it's a flat deformation but we haven't really covered flatness yet. So I won't worry about that. So this is a sort of precise way of saying the graded algebra of r is about the same size as r. It's not just the same size as r but it's a nice deformation of r. And you can use this to go backwards and forwards between r and the corresponding graded algebra. Another thing we can do with decreasing filtration is take the completion. And I'm not gonna say very much about this because we're going to discuss this in much more detail later on. So what happens is if you've got a ring r with an ideal i, what we can do is we can take r modulo i and r modulo i squared and r modulo i cubed and there are maps between these and the completion r hat is a set of elements in the product. So you're taking element a, b, c and so on, such that the image of b is a and the image of c is b and so on. So the standard example of this is if we take r to be the ring of polynomials over x and i to be the ideal x, then r over i is just r and this is rx over x squared and this is rx modulo x cubed and so on. So here we're taking a constant and here we're taking a polynomial of degree one with that constant and here we're taking a polynomial of degree two with that linear term and so on. And if we put these all together, we see that what we get is a power series in r. So the completion of a polynomial ring with respect to this ideal is the ring of formal power series. And you can do the same instruction with any ring and any filtration in particular, you can take the decreasing filtration of an ideal. So that's given several examples of decreasing filtrations. Now let's look at some examples of rings constructed from increasing filtrations are not contained in r one, contained in r two and so on. So first of all, we can do an analog of the blow up algebra where we just take r zero plus r one plus r two and so on. For example, let's take r to be k x y and let's take our n to be spanned by monomials of degree less than or equal to n. Then you can see that this algebra here is not difficult to see. This turns out to be isomorphic to a polynomial ring k x y t where t is now an extra element. It's t is kind of corresponds to the element one in r one. And this is also graded. If you let x, y and z have degree one and you can see the grading turns out to be essentially the same as the grading here. So in this particular case, this is just a rather funny way of defining a polynomial ring. We can also form the graded algebra of a decreasing filtration. So now we have to take r. So we're now doing increasing filtrations. So we now have to take r zero plus r one over r zero plus r two over r one and so on. And this is quite a useful way of turning non commutative rings into commutative rings. So I'm mostly talking about commutative algebra this course, but sometimes we can prove things about non commutative rings using techniques of commutative algebra by turning them into commutative rings. And let's see an example where you can do this. Let's take r to be the ring of differential operators over k and I'm going to take k to be a characteristic zero in order to avoid, there are some weird complications you get with differential operators and fields of characteristic p which I don't want to think about quite right now. And this is generated as a k algebra by, well, let's take a ring of differential operators and n variables. So I'm going to take x one up to x n and I'm also want d by dx one up to d by dx n. And I'm going to write this as d one up to d n because that's easier to write. And then you see these are not commutative. We have d i x j equals x j d i if i is not equal to j, but d i x i is equal to x i d i plus one. If you think about this, this is just the Leibnitz rule. You have to be a bit careful here because when you write d by d x i of x i, you might mean the derivative of x i with respect to x i, which is just one, or you might mean the differential operator, which is the composition of multiplication by x i by this differential operator here. And the notation is a bit ambiguous. And when I write it like this, I don't mean d i applied to x i, I mean the composition of these differential operators. Anyway, you see that d by d x i of x i times g is equal to, if I applied the operator multiplication by x i, then this operator, this is equal to x i d by d x i of g plus g, which is y, which, if you think about it, is just this rule here. So differential operators are not quite commutative, but they're pretty close to being commutative because the obstruction to these two commuting is something simpler than them. So what we can do is we can do the following. Let's put r i to be the space spanned by products of at most i of these generators, x one up to x n, d one up to d n. And what you notice is that if a is an r i and b is an r j, then a b is an r i plus j and b a is an r i plus j, but a b minus b a is an r i plus j minus one. So this can be easily proved by induction, starting with the case we did here. So you see that here, if these are in r one, and you see the commutator of these two is a constant, which is an r zero. And from that, it's easy to show by induction, i and j, this is true for all i and j. And now this means that r zero over r one plus r one over r two plus r two over r three and so on, is, sorry, that's getting the increased and decreasing filtrations muddled up. So that's r two over r one and so on. So this graded algebra is commutative because the commutator always lies in this bit here, which we're just quotienting out by. So what we've done is we've managed to turn a non-commutative algebra into a commutative algebra by replacing it with the corresponding graded algebra. So for example, the graded algebra of a ring of differential operators now becomes polynomial algebra on x one bar up to x n bar where I'm writing a bar over for the image of this element in the corresponding graded algebra. As an application of this, let's just show the ring of differential operators has no zero divisors. Well, this sort of follows almost immediately because if we've got a, b in the ring of differential operators, then they've got a highest degree part which you can think of as being the image in the graded ring of differential operators. And now if a and b are not equal to zero then a bar and b bar obviously not equal to zero. So a bar, b bar is not equal to zero because the ring of all polynomials in two n variables has no zero divisors. But we notice the top degree part of a b is equal to the top degree part of a times the top degree part of b. So the fact that this product here is non-zero because it's non-zero in a polynomial ring implies that this must also be non-zero. So rings of differential operators tend not to have zero divisors. If you know about enveloping algebras of Lie algebras you can apply a similar argument to show that if you take the universal enveloping algebra of a Lie algebra then it's got no zero divisors again because the corresponding graded ring also has no zero divisors. Okay, so next lecture will be about the different types of module over a ring you can have that are similar to free modules.