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I am going to give a talk on the Koszul complex and its connection with local cohomology. We are using the book Residues and Duality for Projective Algebraic Varieties by Kunz and in the chapter that I shall talk about he introduces generalized fractions (a special sort of generalized fractions, he says these can be defined in greater generality) and then proves some rules which tell you how to calculate with these fractions.

I would like to be able to present a concrete example and demonstrate some of these calculation rules in action. However, that entales computing certain direct limits and I was not able to do that.

Let me give you a brief summary of generalized fractions. They are cooked up from the following ingredients:

  • A noetherian local Ring $(R,\mathfrak{m})$ of dimension $d > 0$
  • a system of parameters of $R$: $t_1,\dots,t_d$ (the order matters!)
  • a finitely generated $R$-module $M$ that is Cohen-Macaulay (I am not sure that I need the Cohen-Macaulay property, but since we are looking for an example it does not hurt to assume that $M$ is Cohen-Macaulay).

The recipy for cooking up generalized fractions is the following: For a positive integer $a$ let $t^a$ denote the system of parameters $t_1^a,\dots,t_d^a$ (again the order matters). Denote by $(t^a) \subset R$ the ideal generated by this sequence and let $M/(t^a)M \longrightarrow M/(t^{a + 1})M$ be the morphism of $R$-modules given by multiplication with the product $t_1\cdots t_d$. This gives us a direct system of $R$-modules. If we denote the morphism $M/(t^a)M \longrightarrow \lim_a M/(t^a)M$ from $M/(t^a)M$ to the direct limit $\lim_a M/(t^a)M$ by $\phi_{t^a}$, then the generalized fraction with numerator $x$ and denominator $t^a$ is defined to be $\phi_{t^a}(x)$. Note that with the assumptions above the module $\lim_a M/(t^a)M$ (which, by the way, is $H^d_{\mathfrak{m}}(M)$) is independent of the particular choice of the system of parameters, so we can compare generalized fractions with different denominators. Here are two rules for calculations with these:

  • For $x$, $x' \in M$ we have $\phi_t(x) = \phi_t(x') \Longleftrightarrow \exists a \geq 0$ such that $(t_1\cdots t_d)^a (x - x') \in (t_1\cdots t_d)^{a + 1} M$
  • Extension rule (comparing fractions with different denominators): If $t' = t'_1,\dots,t'_d$ is another system of parameters such that $(t') \subset (t)$, then we can find $c_{ij} \in R$ such that $t'_i = \sum_{j = 1}^d c_{ij} t_j$ for all $i = 1,\dots,d$. Let $\Delta = \det(c_{ij})$, then for all $x \in M$ we have $\phi_t(x) = \phi_{t'}(\Delta x)$ (note that this is independent of the choice of the $c_{ij}$ by the assumptions on the ring and module).

After all that general discussion let me remind you what I am looking for: I am looking for an explicit example to demonstrate these two rules. I tried to do it for $R = K[x,y]_{(x,y)}$, the localization of the polynomial ring in two variables over a field $K$ in the prime ideal $(x,y)$, $M = R$, $t = x, y$ and $t' = x, x^2 - y$, but I could not compute the direct limit in this case. If you do not want to use my example but your own one, go ahead. Of course you can also pick your favourite field $K$ to make my example more concrete if you wish.

Please note that this subject is new to me, so if I made any mistakes I would appreciate it if you pointed them out.

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