3
votes
0answers
41 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
153 views

A proof using Yoneda lemma

Some clever geezer Mister 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
75 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
52 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
65 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
64 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
189 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
196 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 ...
17
votes
1answer
357 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
80 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
186 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
36 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
66 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
56 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
66 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
59 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
23 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
67 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
71 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
169 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
45 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
45 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
67 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
64 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
73 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
54 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
91 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
77 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
74 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
86 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
182 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
122 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
127 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
147 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
53 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
184 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
147 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 ...
4
votes
2answers
158 views

explicitly represent a representable functor

Assume you were given a functor $$F : k\text{-}\mathbf{alg}\to\mathbf{set},$$ with the additional information that it is representable. Is there then a procedure to find an object $A$ that represents ...
3
votes
1answer
95 views

Does the direct image functor on sheaves reflect epimorphisms?

Let $f:X\to Y$ be a morphism of schemes, and let $f_*:\mathbf{Sh}(X)\to\mathbf{Sh}(Y)$ be the direct image functor. Does $f_*$ reflect epimorphisms? That is, suppose ...
2
votes
0answers
44 views

Categories of sheaves of finite homological dimension revisited

This is the question When is the category of (quasi-coherent) sheaves of finite homological dimension? revisited. I made some mistakes: I posted the question before I sign in, I tried to make it more ...
3
votes
1answer
74 views

When is the category of (quasi-coherent) sheaves of finite homological dimension?

Let say from the beginning that my background is category of modules over a ring. So I know that if we take a given (nice) scheme $X$, then category of sheaves on $X$ is Grothendieck, so it must have ...
3
votes
0answers
50 views

Covariant functors represented by schemes

Let $S$ be a nice base scheme and let $F : \mathsf{Sch}/S \to \mathsf{Set}$ be a functor. Are there necessary and sufficient conditions that $F$ is represented by some scheme $X$, i.e. $F \cong ...
4
votes
0answers
66 views

Products of sites

Does the category of sites (i.e. small categories equipped with a Grothendieck topology) has products? Is there a connection to the product of locales (as discussed in Johnstone's Stone spaces, ...
3
votes
3answers
110 views

Does the inclusion from affine schemes into schemes preserve pushouts?

Let $K$ be a field. What is an example of two $K$-algebra morphisms $R\to T$ and $S\to T$ such that $\operatorname{Spec}(R\times_T S)$ is not the pushout of the diagram $$ ...
6
votes
1answer
92 views

A new(?) partial order on the set of continuous maps

Let $X,Y$ be topological spaces. Define a partial order on $\hom(Y,X)$ as follows: $f \leq g$ if $f^{-1}(U) \subseteq g^{-1}(U)$ for all open subsets $U \subseteq X$. Equivalently, $f(y)$ is a ...
2
votes
1answer
111 views

Is this square diagram cocartesian for every regular local ring?

Let $K$ be a field and $R=\{f\in K[X]\mid f(0)=f(1)\}$ the $K$-algebra obtained by pulling back $K[X]\to K\times K$, $X\mapsto (0,1)$ along the diagonal. Is the induced square \begin{eqnarray} ...
10
votes
1answer
160 views

What are the “correct” modules over locally ringed spaces?

$$\begin{array}{ccccc} \text{schemes} & \longrightarrow & \text{locally ringed spaces} & \longrightarrow & \text{ringed spaces} \\ | && | && | \\ \text{quasi-coherent ...
3
votes
1answer
72 views

Property kept under base change and composition is preserved by products

The following is true? Why? Let $P$ be a property of morphisms preserved under base change and composition. Let $X\to Y$ and $X'\to Y'$ be morphisms of $S$-schemes with property $P$. Then the unique ...