$f_\ast\mathcal{O}_X$ is locally free of rank $r=\deg f$ for a finite $r$-to-one morphism $f$.
47143
Push forward of the structure sheaf along covering
86977
If $\cup \mathcal{F}=A$ then $A \in \mathcal{F}$. Prove that $A$ has exactly one element.
37041
Why is $\text{Aut}(F)$ of the forgetful functor $F$ on $G$-sets isomorphic to $G$?
111129
Show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.
101986
Prob. 2, Sec. 20 in Munkres' TOPOLOGY, 2nd ed: The dictionary order topology on $\mathbb{R} \times \mathbb{R}$ is metrizable.
40996
Proving that free modules are flat (without appealing projective modules)
73825
Every $R$-module is a quotient of a free module
69398
Rank-nullity theorem for free $\mathbb Z$-modules
140883
If $R$ is a local ring, is $R[[x]]$ (the ring of formal power series) also a local ring?
42652
Must $k$-subalgebra of $k[x]$ be finitely generated?
156005
Pullback of locally free sheaves is locally free
127886
Artin's lemma and some morphisms.
114640
Showing that an ideal is not a free $R-$module.
91222
Isomorphism between $\mathfrak o(4,\mathbb R)$ and $\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $
25260
Extension of a morphism $k^*\rightarrow X$ to $\mathbb{P}^1\rightarrow \overline{X}$
63447
Prove that the kernel of a homomorphism is a principal ideal. (Artin, Exercise 9.13)
14830
A transformation of F[x]