The Question : 108 people think this question is useful Suppose we have two schemes $X, Y$ and a map $f\colon X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where $\mathcal{F}$ is an $\mathcal{O}_X$-module and $\mathcal{G}$ an $\mathcal{O}_Y$-module (and the Homs are in the category of $\mathcal{O}_X$-modules etc). This gives a natural map

The Question : 109 people think this question is useful I was reading Barry Mazur’s biography and come across this part: Grothendieck was exceptionally patient with me, for when we first met I knew next to nothing about algebra. In one of his first conversations with me, he raised the question (asked of him by

The Question : 118 people think this question is useful Many students – myself included – have a lot of problems in learning scheme theory. I don’t think that the obstacle is the extreme abstraction of the subject, on the contrary, this is really the strong point of modern algebraic geometry. I’m reading many books

The Question : 130 people think this question is useful I’m actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the Picard group continues to elude me. One of the

The Question : 130 people think this question is useful In SmalĂ¸: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert’s basis theorem that I don’t understand: Two orders are defined on the set of $d$-dimensional modules over an algebra $\Lambda$ that is finite dimensional over a field.

The Question : 227 people think this question is useful I’m going to start self-stydying algebraic geometry very soon. So, my question is why do mathematicians study algebraic geometry? What are the types of problems in which algebraic geometers are interested in? And what are some of the most beautiful theorems in algebraic geometry? The

The Question : 464 people think this question is useful In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ counted the number of subspaces of an $n$-dimensional vector space over $GF(x)$ (which I was using to determine the