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# Tardily projective? You’re free then

Suppose $A$ is a $\mathbb Z$-graded connected algebra over a field $k$ concentrated in positive degrees, and suppose $P$ is a projective graded $A$-module concentrated in degrees above some $m\in \mathbb Z$ . Then $P$ is actually a free $A$-module.

Under the same hypothesis over $A$, show that if $M$ is a graded module and $M\otimes_A k=0$ where $k$ is the quotient of $A$ by the ideal of positively graded elements, then $M=0$.

Using the above, show that if $M$ is graded and $\mathrm{Tor}_A^1(M,k)=0$, then $M$ is free. To illustrate, it follows projective graded modules over polynomial algebras are free.

# Wish I had known this in 2012

Let $A$ be an integral domain, $I,J\subseteq A$ ideals such that $I+J=A$. If $I\cap J$ is principal, then $I$ and $J$ are projective.

# The Eilenberg swindle

If $M$ is a projective $A$-module, there exists a free $A$-module $F$ such that $F \simeq M \bigoplus F$.

# Module structures

Consider the group ring $\mathbb Z C_p$. How many module structures, up to isomorphism, does $C_{p^2}$ admit? Do the same for $C_p^2$. With this information, calculate $H^2(C_p,-)$ in such cases.

# Duality in polynomial rings

Let $A$ be a polynomial ring over a commutative ring (!) $k$ in $n$ variables. Show that for any $A$-module $M$ and $0\leqslant q\leqslant n$, there are natural isomorphisms

$\mathrm{Tor}_q^A(k,M) \simeq \mathrm{Ext}_A^{n-q}(k,M)$

# Links of singular points in complex hypersurfaces

Consider the canonical embedding of a sphere of radius $\varepsilon$, say $S_\varepsilon^3$ inside $\mathbb C^2$ as the set of pairs $(z,w)$ with $|z|^2+|w|^2=\varepsilon^2$, and the curve $X= \{(z,w):z^k + w^l =0\}$. Calculate the homology of the intersection $X\cap S_\varepsilon^3$ for as many values of $(k,l)$ as possible where $k,l>1$, with $\varepsilon$ of your liking.