5
votes
1answer
52 views

Does pullback of schemes by monomorphism produce topological pullback?

Suppose I have scheme maps $X\to Z,Y\to Z$. It is not in general true that the fibre product $X\times_{Z}Y$ has the same underlying topological space (or even the same underlying set) as the fiber ...
1
vote
2answers
92 views

Soft sheaves adapted to $f_!$

I'm reading Gelfand-Manin, Homological Algebra. I understand that the class of soft sheaves is sufficiently large, because every injective sheaf is soft. Now to see that this class is adapted to ...
3
votes
0answers
63 views

Weil restriction - from abstract nonsense to a practical procedure

Let $L/K$ be a finite field extension. Let $X$ be a $L$-scheme, that is $\exists$ a morphism of schemes $\pi : X \to \operatorname{Spec} L$. The Weil restriction of $X$ is a $K$-scheme $W_{L/K}X$ ...
2
votes
1answer
55 views

Coarse moduli space and rational points

Let $K$ be a field (not necessarily algraically closed). Let $\mathcal{F}$ be a contravariant functor from the category of schemes over $K$ to sets and $M$ be a coase moduli space for the functor. So, ...
3
votes
0answers
54 views

Topological characterisations of freeness and separability?

According to wikipedia, a spectral space is defined to be homeomorphic to the spectrum of some commutative ring. They form a category $Spec$ where we take morphisms to be those whose preimage of open ...
2
votes
1answer
43 views

Algebraic Compact manifold originates from a proper scheme?

If $M$ is a compact complex manifold, which is the analytification of some scheme $X$ of finite type over $\operatorname{Spec}(\mathbb{C})$, then must $X$ be proper over ...
2
votes
0answers
74 views

What are the prerequisites for reading SGA 1?

My question concerns, basically, scheme theory. If there is someone who has actually read SGA 1, I would really like to hear what their opinion is on that. For example, is EGA in its entirety a ...
8
votes
2answers
323 views

Trying to understand the use of the “word” pullback/pushforward.

Essentially, my question is the following : Is everything we call "pullback" or "pushforward" an actual categorical pullback/pushout? I have seen tons of pullbacks in differential geometry but we ...
3
votes
0answers
49 views

For which categories do injections induce surjections in cohomology?

I'll ask a specific question first, but I believe my question might have a rather immediate abstraction, with which I'll finish. Let $H,G$ be finite affine group schemes over an algebraically closed ...
6
votes
4answers
206 views

Learning about Grothendieck's Galois Theory.

I have background in category theory and I am familiar with the very basics of algebraic geometry - Chapters I and II of Hartshorne. What would be a recommended (self-contained, maybe?) text for ...
2
votes
0answers
86 views

How can I make peace with contravariance?

My question is a bit vague, but I hope it can be answered in a good way. Various arguments involving contravariance sometimes trip me up when coming up with proofs in algebraic geometry and related ...
3
votes
0answers
43 views

Prime Ideals as Ring theoretic Ultrafilters

I am confused by the following statement in Awodey's Category Theory p. 35: Ring homomorphisms $A\to \mathbb Z$ into the initial ring $\mathbb Z$ play an analogous and equally important role [to ...
8
votes
2answers
271 views

A proof using Yoneda lemma

Martin Brandenburg pointed out elsewhere in the comments that he could give a one line proof, using the Yoneda lemma, of $$\frac{\mathbf{C}[x_1,\ldots,x_{n+m}]}{I(X)^e+I(Y)^e} \cong ...
4
votes
2answers
77 views

Hom of algebras

For any $R$-algebras $A$ and $B$, doea their set of R-algebra morphisms $\mathrm{Hom}_{R_{\mathrm{Alg}}}(A,B)$ necessarily again have the strucutre of an $R$-algebra?
2
votes
1answer
55 views

graph of the compostion of morphisms category-theoretically

My question is about a certain category-theoretic statement really but since I came to it trying to prove something about non-reduced schemes, I'll state it in this language. Let $M$ be a scheme ...
0
votes
2answers
79 views

Sufficient condition for a function to be a bijection

We want to prove two sets $A$, and $B$ have the same cardinality. Assume we have found a function $f:A\to B$, and a function $g:B\to A$, with $f\circ g=id$. Can we conclude that $f$ is bijective? ...
2
votes
0answers
77 views

Criteria for a contravariant functor from Schemes to Sets to be representable by a scheme

I am trying to prove that a functor $\mathcal{F}:(Sch/S)^{op}\rightarrow (Sets)$ is representable by an $S$-scheme $F$. My intuition says that it is indeed representable. I have been reading Fulton's ...
8
votes
1answer
224 views

Advice: Algebra and category theory for geometry?

I'm interested in learning a bit of geometry. To start I'm (slowly) working my way towards differential geometry via Lee's Introduction to Smooth Manifolds. But, later on, I'd also like to study some ...
9
votes
2answers
216 views

Introduction to sheaves using categorical approach

When I first started to learn about sheaves, it was a very geometric approach. This is nice, but it seems like knowing more abstract categorical approach is very useful. For example, sheafification ...
15
votes
1answer
411 views

Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
3
votes
1answer
84 views

Product in the category of varieties

I am tasked to show that given affine varieties $X$ and $Y$, that $A(X\times Y)\cong A(X)\otimes A(y)$. I think I am very close I just need a little nudge. Proof so far Define the map ...
11
votes
1answer
214 views

What is a concrete example of why one wants to have a *derived category* in algebraic geometry?

My question asks for a concrete (and hopefully easy) example, why one wants to derive things in algebraic geometry. I heard, that a resolution of an object by free ones behaves much better than the ...
0
votes
1answer
39 views

Monics and monomorphisms are the same as kernels in the additive category of R-modules.

How can we show that in an additive category monics and monomorphisms are the same as kernels? Actually, I can show that a kernel is a monic and a monomorphism but I could not show that "a monic is a ...
1
vote
2answers
70 views

When should we take direct limit and when should we take inverse limit?

We know that we can take direct limit for a direct system and inverse limit for an inverse system. For example, when can defined the stalk of a presheaf $\mathcal{F}$ on a topological space $X$ at a ...
2
votes
1answer
62 views

Retrieve affine schemes by adjunction

In an introductory course about schemes, I've seen the adjunction $$ {\mathbf{LocRngSpace}}^{\mathrm{op}} \overset{\mathrm{Spec}}{\underset{\Gamma}{\leftrightarrows}} \mathbf{Ring}, $$ where ...
1
vote
1answer
75 views

what's wrong with this categorical proof that maps between two covering spaces are unique?

Let $\mathcal{C}$ be the category of finite covers of a fixed base space $S$ (say, connected, locally path connected, locally simply connected. Hell, we can even assume $S$ is a manifold). Morphisms ...
4
votes
1answer
63 views

Exact sequences in the Mumford example

This question concerns the proof for the Mumford example of an Hilbert Scheme that has a non-reduced component. I am studying the proof given on R. Hartshone, "Deformation Theory", pp. 91-94. I do not ...
2
votes
1answer
25 views

About the automorphism groups of the objects in a connected groupoid

In the note named Foundation of Algebraic Geometry, the author gives an example: given a topological space $X$, there is a fundamental groupoid which is the category in which the objects are points of ...
3
votes
0answers
73 views

Morphisms in the derived category

I have just started to learn about derived categories, I am now trying to understand what morphisms look like in some easy examples. Let me describe one for you. Let $D(\mathcal A)$ be the derived ...
3
votes
2answers
78 views

The morphism is monic iff the diagonal is an isomorphism

Prop: $f:X\rightarrow Y$ is monic iff $\Delta_{X\mid Y}$ is an isomorphism from $X$ to $X\times_{Y}X$. I read the following argument: $f$ is monic is equivalent to for any $Z$ over $Y$, ...
5
votes
3answers
193 views

What does it mean to have exact derived functors?

Let $F:\mathcal A\to \mathcal B$ be a functor between abelian categories. Suppose $F$ is, say, left exact (plus additive and covariant). We have built its right derived functors $R^iF$. I see no ...
3
votes
1answer
47 views

$GL(-)$ as a k-group functor

My question is essentially may lye simply in a notational obstruction. For a k-algebra M, Jantzen J. defines the k-group functor $GL(M)$ as: $GL(M)(A):=(End_A(M\otimes_{\mathbb{k}} A)^*$. My ...
1
vote
1answer
50 views

Flatness preserved under Cartesian product

Let $X_1, X_2, X_3, Y_1, Y_2$ and $Y_3$ be projective schemes. Let $f_1:X_1 \to Y_1, f_2:X_2 \to Y_2$ and $f_3:X_3 \to Y_3$ be flat morphisms. Suppose there are morphism $g_1:X_1 \to X_2$, $g_2:X_3 ...
1
vote
1answer
55 views

$\operatorname{Eq}(f,g) = X$ implies that $f= g $ as morphisms?

Let $X,Y$ be schemes over $S$ and assumed $Y/S$ is separated and $X$ reduced. Let $\operatorname{Eq}(f,g)$ denote the equalizer of $f$ and $g$. By base change to an affine open subset $U$ of $X$, I ...
2
votes
2answers
68 views

The equivalence of the definitions of stalks

The definition of direct limits is: I'm trying to see how this definition works in the stalks: The index $I$ is the open sets containing $x$ under the inclusion and the restrictions ...
3
votes
2answers
65 views

A generator (or a cogenrerator) for the category of schemes

Does the category of Schemes admit a (single) generator (or a cogenerator)? What if we restrict to the category of schemes of finite type over a field $k$?
5
votes
1answer
80 views

Is a pushout of a closed immersion $f$ again a closed immersion?

Assume $$ \begin{eqnarray} X&\xrightarrow{f}& Y\\ \downarrow && \downarrow\\ Z&\xrightarrow{f'}& W \end{eqnarray} $$ is a pushout in the category of schemes (and in particular ...
3
votes
1answer
62 views

Is a pushout $W$ of schemes along a closed subscheme also a pullback?

Assume $$ \begin{eqnarray} X&\xrightarrow{f}& Y\\ \downarrow && \downarrow\\ Z&\to& W \end{eqnarray} $$ is a pushout in the category of schemes (and in particular $W$ is a ...
6
votes
1answer
96 views

How does indexing work in EGA/ how to search for a result in EGA?

I am interested in a certain result which says that if we have an open cover $F_i$ of a sheaf $F$ with each $F_i$ representable, then $F$ is representable. The reason I am interested in this is ...
2
votes
1answer
84 views

Commutativity of diagram involving two arrows

Hi suppose I have a diagram that looks like this: but where we only have $fe = hf'$ and $ge = hf'$. What would I call the square? I can't say that it commutes yes? Is it true that in general given ...
0
votes
1answer
78 views

Are the hom sets in the category of varieties abelian groups?

This is supposedly (though I know of the proof bud haven't read it) for the Hom sets of noetherian schemes. Since every variety can be thought of as a noetherian scheme then it seems right... when ...
3
votes
1answer
87 views

Definition of equalizer for $\textbf{Sh}(X)$

Let $\textbf{Sh}(X)$ denote the category of all (set - valued) sheaves on a topological space $X$. My question is: Given sheaves $F,G \in \textbf{Sh}(X)$ and morphisms $\varphi : F \to G$ ...
6
votes
1answer
212 views

Gluing sheaves - can we realize $\mathcal{F}(W)$ as some kind of limit?

Let $X$ be a topological space and $\{U_i\}$ and open cover for $X$. Suppose we have sheaves $\mathcal{F}_i$ on $U_i$ and for each $i,j$ an isomorphism $\varphi_{ij} : \mathcal{F}_i|_{U_i \cap U_j} ...
1
vote
1answer
145 views

Definition of sheaf using equalizer

Wikipedia give sheaf property using equalizer diagram by saying sheaf property means for any open cover $\{U_i\}$ of $U$ $$F(U) \rightarrow \prod_{i} F(U_i) {{{} \atop ...
5
votes
1answer
143 views

Hartshorne's weird definition of right derived functors and prop. III 2.6

There is something very weird with the way Hartshorne defines right derived functors. Hartshorne p 204 Let $\mathfrak A$ be an abelian category with enough injectives, and let $F \colon \mathfrak ...
5
votes
2answers
154 views

Sheafication of a sheaf restricted to a open set

Let $X$ be a topological space and $U$ be open in $X$. Let $\mathcal F$ be a presheaf of rings on $X$. Let $\mathcal F_u$ denote the presheaf restricted to the open set $U$. $\mathcal F^+$ denote the ...
4
votes
1answer
54 views

Existence of product in the category of pre-sheaves of abelian categories

Let $X$ be $Top(X)$ be the category of open sets of $X$ with inclusion maps as morphism. Let $\mathcal{C}$ be abelian category and $\mathcal{C}_x$ denote the category of contravariant functors from ...
5
votes
1answer
206 views

Why is the category of coherent sheaves not grothendieck?

Let $(X, \mathscr{O}_X)$ be a ringed space. It is well-known that the category $\mathbf{Mod}(\mathscr{O}_X)$ of $\mathscr{O}_X$-modules is a grothendieck abelian category (see e.g. Grothendieck's ...
4
votes
0answers
155 views

What does it mean for a ring to be unital?

What is the category of unital rings like? I only know that it no more has a terminal object. But what about the products and coproducts? Are they as usual, different or nonexistent? In Gelfand ...
1
vote
1answer
73 views

Some exact sequences of cohomology on picard schemes

I'm looking for the answer of this question. $X$ is a variety over a field $k$, and $Art_k$ is the category of local artinian $k$-algebras whose residue field is $k$. I consider the formal completion ...