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\def \Ass{\mathop{\rm Ass}}
\def \Fun{\mathop{\rm Fun}}
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\def\inverselimit{{\lim\limits_{\textstyle \longleftarrow}}}
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{\bf Math 8212\hfill Commutative and Homological Algebra 2\hfill Spring 2022}
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{\bf Homework Assignment 4} Due {\bf Saturday 5/7/2022}, uploaded to Gradescope.
Each question part is worth 1 point. There are 8 question parts. You are on target for an A if you make a genuine attempt on at least half of them. This homework has fewer parts than previous homeworks. If you can find a way to do the calculation of your overall score so that it comes to be more than 50\% (e.g. by weighting each of the four homeworks so that they count equally, or by something else), I will accept that.
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1. Although we did not define explicitly the higher differentials in a spectral sequence, it is possible to deduce what they must be in this example. Consider the double complex
$$
\diagram{
\ZZ&\umapleft{p}&\ZZ&\umapleft{}&0\cr
\rmapdown{}&&\rmapdown{1}&&\rmapdown{}\cr
0&\umapleft{}&\ZZ&\umapleft{p}&\ZZ\cr
}
$$
with terms that are 0 except as shown, the nonzero terms being in bidegrees
$$(0,0), (1,0), (-1,1)\hbox{ and }(0,1).$$
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a. Compute the homology of the total complex of this double complex. [You may want to use standard facts having to do with structure of finitely generated modules over a Euclidean domain, Smith Normal Form etc, including the fact that the order of a quotient of a free abelian group by the subgroup spanned by columns of a square matrix is the determinant of that matrix.]
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b. Filtering the double complex by rows (so that the modules in each row describe the quotient of two consecutive terms in the filtration), we get a spectral sequence. Find all the numbers $n$ for which $E^n = E^{n+1}$. Determine whether the naturally given grading on the $E^\infty$ term is the same as the naturally given grading on the homology of the total complex from a.
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c. Filtering the double complex by columns (so that the modules in each column describe the quotient of two consecutive terms in the filtration), we get a spectral sequence. Find all the numbers $n$ for which $E^n = E^{n+1}$. Determine whether the naturally given grading on the $E^\infty$ term is the same as the naturally given grading on the homology of the total complex from a.
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d. Describe how to construct a spectral sequence for which $E^5\ne E^6 = E^\infty$.
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2. (Similar to Exercise 10.12 of Eisenbud) Let $S=k[x_1,\ldots ,x_r]$ be a polynomial ring over a field $k$ in $r$ variables, where the indeterminate $x_i$ has degree $d_i$. Let $M$ be a finitely generated graded $S$-module and put $H_M(n) = \dim_k M_n$ and $h_M(t) = \sum_{n\ge 0} H_M(n) t^n$.
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a. Show that $h_M(t)$ is a rational function of $t$, and that in fact $h_M(t)$ may be written as a polynomial divided by $\prod_{i=1} (1-t^{d_i})$.
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b. Show that there is a number $d$ (which may be taken to be the least common multiple of the degrees $d_i$) such that for each $s$, $H_M(dn+s)$ agrees with a polynomial in $n$ for all $n >> 0$; that is, $H_M(n)$ is an `almost PORC function' of $n$.
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3. Let $k$ be a field. The ring $A=k[x,y]/(x^2-y^3)$ that we studied in class has maximal ideal ${\frak m} = (\bar x, \bar y)$, where bars denote the images of elements in $A$. We saw in class that the ideal $(\bar y)$ is $\frak m$-primary. For any $\frak m$-primary ideal $J$ we may form the (Hilbert-)Samuel function $\chi_J(n) = \dim_k (A/J^n)$. Note that if we were to localize at $\frak m$ we would have $A_{\frak m}/(J_{\frak m})^n \cong A/J^n$, so we don't have to localize before computing the dimension.
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a. Compute the values of $\chi_{\frak m}(n)$. Find the polynomial $f(t)$ such that $\chi_{\frak m}(n)=f(n)$ for large $n$.
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b. Compute the values of $\chi_{(\bar y )}(n)$. Find the polynomial $f(t)$ such that $\chi_{(\bar y )}(n)=f(n)$ for large $n$.
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