Category Theory on Geometry and some category theory

limit/limsup/liminf of a sequence (by an example)

Fri, 13 Sep 2024 05:29:07 +0000

Let us check for limit/limsup/liminf of the sequence $\frac{n}{10^{\lceil \log_{10}n \rceil}}$, where the notation $\lceil x \rceil$ means the smallest integer greater than or equal to $x$.

For example, $\lceil 0.1 \rceil=1, \lceil 0.9 \rceil=1, \lceil -1.2 \rceil=-1, \lceil -2.5 \rceil=-2$

To compute limit (to have a hope of computing), we need to know it converge (which we can check by checking it is Cauchy sequence).

Model categories : Part 2 (Definitions)

Thu, 21 May 2020 17:59:45 +0000

Before we move to the notion of "a model structure on a category", we need to recall (or introduce) some definitions. Definition : Let $\mathcal{C}$ be a category. An object $X$ of $\mathcal{C}$ is said to be a retract of an object $\mathcal{C}$ if there exists arrows $X\xrightarrow{f} Y\xrightarrow{g} X$ such that, the composition $g\circ f:X\rightarrow X$ is equal to the identity arrow $1_X:X\rightarrow X$. Definition : Let $\mathcal{C}$ be a category. We define the morphism category of $\mathcal{C}$, denoted by $\text{Map}(\mathcal{C})$ whose

objects are the arrows of $\mathcal{C}$,
morphisms are commutative diagrams in $\mathcal{C}$.

Definition : Let $\mathcal{C}$ be a category. A morphism $f$ in $\mathcal{C}$ is said to be a retract of a morphism $g$ in $\mathcal{C}$, if, $f$ is a retract of $g$, when both $f$ and $g$ are seen as objects of $\text{Map}(\mathcal{C})$. Definition : Let $\mathcal{C}$ be a category. Let $i:A\rightarrow B$ and $p:X\rightarrow Y$ be morphisms in $\mathcal{C}$. We say that $i$ has the left lifting property with respect to $p$ or $p$ has the right lifting property with respect to $i$ if, for every commutative diagram there exists an arrow $h:B\rightarrow X$ such that $h\circ i=f$ and $p\circ h=g$. Definition : Let $\mathcal{C}$ be a category. A model structure on $\mathcal{C}$ consists of the following data :

a subcategory of $\mathcal{C}$ called “weak equivalences”,
a subcategory of $\mathcal{C}$ called “fibrations”,
a subcategory of $\mathcal{C}$ called “cofibrations”,

satisfying certain conditions:

If $f,g$ are morphisms of $\mathcal{C}$ such that $gf$ is defined and two of $f,g,gf$ are "weak equivalences" then so is the third.
A retract of a "weak equivalece" is a "weak equivalence".
A retract of a "fibration" is a "fibration".
A retract of a "cofibration" is a "cofibration".
factorizing property (part $1$) of arrows of $\mathcal{C}$ : for each $f$ in $\text{Mor}(\mathcal{C})$, there exists a cofibration $i$ and a traivial fibration $q$ such that $f=qi$.
factorizing property (part $2$) of arrows of $\mathcal{C}$ : for each $f$ in $\text{Mor}(\mathcal{C})$, there exists a fibration $p$ and a trivial cofibration $j$ such that $f=pj$.
Any commutative diagram of the type has lifting property if either $i$ or $p$ is a "weak equivalence".

Definition : A model category is defined to be a category that has

all small limits,
all small colimits,
a model structure in $\mathcal{C}$.

Construction of new model categories from old model categories:

Let $\mathcal{C}$ and $\mathcal{D}$ be model categories. Defining the collection of fibrations (cofibrations, weak equivalences) as pairs $(f,g)$ where both $f$ and $g$ are fibrations (cofibrations, weak equivalences) defined a model structure on the product category $\mathcal{C}\times \mathcal{D}$. This model category is called the product model category produced from model categories $\mathcal{C}$ and $\mathcal{D}$.

Stackification of fibred categories

Sat, 02 Feb 2019 13:50:32 +0000

I understood most of this from Introduction to the language of stacks and gerbes (section 2) by Ieke Moerdijk and from stacks project Stackification of fibred categories. It is necessary to know what is the sheafification of a presheaf to understand what is the stackification. I studied sheafification from Hartshorne's Algebraic geometry book. You can choose what you are comfortable with. I will mention the result first as in Lemma $8.8.1$.

Lemma : Let $\mathcal{C}$ be a site. Let $p:\mathcal{S}\rightarrow \mathcal{C}$ be a fibred category over $\mathcal{C}$. There exists a stack $p':\mathcal{S}'\rightarrow \mathcal{C}$ and a morphisms $G:\mathcal{S}\rightarrow \mathcal{S}'$ of fibred categories over $\mathcal{C}$ such that

for every $U\in \text{Ob}(\mathcal{C})$ and $x,y\in \mathcal{S}(U)$, the map $\text{Mor}(x,y)\rightarrow \text{Mor}(G(x),G(y))$ induced by $G$ identifies the right hand side with the sheafification of the left hand side.
For $U\in \mathcal{C}_0$ and $x'\in \mathcal{S}'(U)$ there exists a covering $\{U_i\rightarrow U\}$ such that each $x'|_{U_i}$ is in the essential image of the functor $G:\mathcal{S}(U)\rightarrow \mathcal{S}'(U)$.

We recall what is $\text{Mor}(a,b)$. This is a presheaf on $U$ defined as follows. Given an inclusion $i : V\hookrightarrow U$ we have $i^*(a),i^*(b)\in \mathcal{S}(V)$.

Category theory

Thu, 24 Jan 2019 23:24:08 +0000

In this page, I will give links of Category theory posts that I have made here. I learned some category theory from

Hilton and Stammbach's book A Course in Homological Algebra.
Angelo Vistoli's Descent theory notes.

Limit of a diagram/functor preserved by Hom functor

Wed, 23 Jan 2019 16:48:25 +0000

Let $F:\mathcal{I}\rightarrow \mathcal{C}$ is a functor. This is also called as diagram indexed by $\mathcal{I}$. By the Limit of this diagram, we mean an object (universal) $L$ of $\mathcal{C}$ and a collection of arrows (universal again) $\pi_i:L\rightarrow F(i)$ such that, for each arrow $m:i\rightarrow j$ in $\mathcal{I}$ the following diagram is commutative. This is usually denoted by $\varprojlim_{\mathcal{I}}F(i)$ or simply by $\varprojlim_{\mathcal{I}}F$. Fixing an object $X$ in $\mathcal{C}$, I want to prove that

$\varprojlim_{\mathcal{I}}(\text{Hom}_{\mathcal{C}}(X,F(i))) =\text{Hom}_{\mathcal{C}}(X,\varprojlim_{\mathcal{I}}F(i))$

Morphism of Lie groups giving a functor

Fri, 18 Jan 2019 20:46:30 +0000

Given a morphism of Lie groups $\theta:G\rightarrow H$ and a principal $G$ bundle $\pi:P\rightarrow M$ there are (at least) two ways to assign a principal $H$ bundle.

See that the morphism of Lie groups $\theta:G\rightarrow H$ gives an action of $G$ on $H$ by $g.h=\theta(g).h$. Given an action of $G$ on manifold (Lie group in this case) $H$ there is an associated fibre bundle $P\times_G H\rightarrow M$ with fibre $H$. This gives a principal $H$ bundle.
For principal bundle $\pi:P\rightarrow M$, we can find an open cover $\{U_\alpha\}$ of $M$ and (transition) maps $g_\alpha g_\beta:U_{\alpha\beta}\rightarrow G$ satifsying the cocycle condition $g_{\alpha\beta}g_{\beta\gamma}=g_{\alpha\gamma}$ on $U_\alpha\cap U_\beta\cap U_\gamma$. Then the compositions $\tau_{\alpha\beta}=\theta\circ g_{\alpha\beta}:U_{\alpha\beta}\rightarrow G\rightarrow H$ also satifies the cocycle condition $\tau_{\alpha\beta}\tau_{\beta\gamma}=\tau_{\alpha\gamma}$ on $U_\alpha\cap U_\beta\cap U_\gamma$. One can then produce a principal $H$ bundle over $M$ given this open cover $\{U_\alpha\}$ of $M$ and smooth maps $\tau_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow H$ satisfying the cocycle condition. This gives a principal $H$ bundle.

It is a good exercise (that I have not tried) to check that principal $H$ bundles obtained from above two methods are (naturally) isomorphic i.e., one and the same. Given a Lie group $G$, let $BG$ denote the category of principal $G$ bundles. Objects are principal $G$ bundles and morphisms are $G$-equivariant morphisms. Given a morphism of Lie groups $\theta:G\rightarrow H$, above construction gives a functor (at the level of objects) $B\theta:BG\rightarrow BH$. It is not difficult to see that, a $G$-equivarint map induce a $H$-equivariant map. This gives a functor $BG\rightarrow BH$.

What is a Stack?

Sat, 12 Jan 2019 18:18:08 +0000

Given a manifold $M$ we have the concept of open cover of $M$. We usually write an open cover of a manifold $M$ as a collection of open subsets $\{U_i\}$ (such that $\bigcup U_i=M$). In this note we see an open cover of $M$ as a collection of maps (inclusions) $\{U_i\rightarrow M\}$. Some properties of "open cover" are.

(Pull back exists and gives an open cover) Suppose $\{U_i\rightarrow M\}$ is an open cover for $M$ and $\pi:V\rightarrow M$ is a smooth map. Then, $\{\pi^{-1}(U_i) \rightarrow V\}$ is a cover for $V$.
(Diffeomorphisms gives open cover) For any manifold $M$, $M$ itself is considered as an open cover $\{M\rightarrow M\}$. More generally, for any diffeomorphism $M'\rightarrow M$, $\{M'\rightarrow M\}$ is considered as an open cover.
(Open cover of open cover is an open cover) Let $\{U_\alpha\rightarrow U\}$ be an open cover for $U$ i.e., $\bigcup_{\alpha} U_\alpha=U$. Suppose $\{V_{\alpha\beta}\rightarrow U_\alpha\}$ is an open cover for $U_\alpha$ for each $\alpha$ i.e., $\bigcup_{\beta}V_{\alpha\beta}=U_\alpha$. Then, $\bigcup_{\alpha\beta}V_{\alpha\beta}=U$ i.e., $\{V_{\alpha\beta}\rightarrow U\}$ is an open cover for $U$.

For a category $\mathcal{C}$ and an object $U$ of $\mathcal{C}$, a collection of arrows $\{U_i\rightarrow U\}$ is said to be a cover for $U$. Definition : Let $\mathcal{C}$ be a category. A Grothendieck topology on $\mathcal{C}$ is given by a collection of covers $\mathcal{W}=\{\{U_i\rightarrow U\}: U\in \mathcal{C}_0\}$ satisfying following conditions.

(Pullbacks exists and gives a cover) Suppose $\{U_i\rightarrow U\}\in \mathcal{W}$ and $\pi:V\rightarrow U$ be an arrow. Then, the pull back $U_i\times_UV$ exists (as an object in $\mathcal{C}$) and $\{U_i\times_UV \rightarrow V\}$ is a cover for $V$.
(Isomorphisms gives an open cover) Suppose $V\in \mathcal{C}_0$ and $V\rightarrow U$ is an isomorphism in $\mathcal{C}$ then, $\{V\rightarrow U\}\in \mathcal{W}$.
(cover of a cover is a cover) Suppose $\{U_\alpha\rightarrow U\}\in \mathcal{W}$ and $\{U_{\alpha\beta}\rightarrow U_\alpha\}\in \mathcal{W}$ for each $\alpha$. Then, the collection of compositions $\{U_{\alpha\beta}\rightarrow U_\alpha\rightarrow U\}\in \mathcal{W}$.

To talk about a stack over category $\mathcal{C}$ we fix a Grothendieck topology $\mathcal{W}$ on $\mathcal{C}$. When we say cover, we mean it belongs to $\mathcal{W}$. Let $\mathcal{D}$ be a category fibered in groupoids over $\mathcal{C}$ i.e., we have a functor $F:\mathcal{D}\rightarrow \mathcal{C}$ satisfying some conditions.

Given an object $U$ of $\mathcal{C}$ we have what is called fibre of $U$ in $\mathcal{D}$ usually denoted by $\mathcal{D}(U)$.
Given an object $U$ of $\mathcal{C}$ and a cover $\{U_i\rightarrow U\}$ (i.e., it belongs to $\mathcal{W}$) we have what is called descent category associated to the cover $\{U_i\rightarrow U\}$, usually denoted by $\mathcal{D}(\{U_i\rightarrow U\})$.

There is an obvious functor $\mathcal{D}(U)\rightarrow \mathcal{D}(\{U_i\rightarrow U\})$.

Definition : Let $\mathcal{C}$ be a category with Grothendieck topology $\mathcal{W}$. A category fibered in groupoids $\mathcal{D}\rightarrow \mathcal{C}$ is said to be a stack over $\mathcal{C}$ if, for every object $U$ of $\mathcal{C}$ and every cover $\{U_i\rightarrow U\}$, the functor $\mathcal{D}(U)\rightarrow \mathcal{D}(\{U_i\rightarrow U\})$ is an equivalence of categories.

The fibre categroy $\mathcal{D}(U)$ is a category whose objects are that of $\mathcal{D}$ which map to $U$ under $F$ i.e.,

$\mathcal{D}(U)_0=\{V\in \mathcal{D}_0:F(V)=U\}$.

Additive categories

Wed, 09 Aug 2017 19:21:42 +0000

Pull back and Push forward of two morphisms

Wed, 09 Aug 2017 19:20:53 +0000

Definition : Let $\mathcal{C}$ be a category and $f:A\rightarrow C$ and $g:B\rightarrow C$ be morphisms. We define pull back of $f,g$ to be a triple $(P,f',g')$ where

$P$ is an object of $\mathcal{C}$ and
$f':P\rightarrow B, g':P\rightarrow A$ with $g\circ f'=f\circ g'$

such that given any object $P'$ of $\mathcal{C}$ and morphisms $g'':P'\rightarrow A$ and $f'':P'\rightarrow B$ with $g\circ f''=f\circ g''$, there exists a unique morphism $\eta:P'\rightarrow P$ such that $g''=g'\circ \eta$ and $f''=f'\circ \eta$ as in the following commutative diagram. Definition : Let $\mathcal{C}$ be a category and $f:A\rightarrow C$ and $g:B\rightarrow C$ be morphisms. We define pull back of $f,g$ to be a triple $(P,f',g')$ where

$P$ is an object of $\mathcal{C}$ and
$f':P\rightarrow B, g':P\rightarrow A$ with $g\circ f'=f\circ g'$

Equalizers and Coequalizers

Wed, 09 Aug 2017 19:19:58 +0000

Monomorphisms and epimorphisms

Wed, 09 Aug 2017 19:19:16 +0000

Left exactness of Global section functor

Fri, 28 Jul 2017 17:11:38 +0000

Adjointness of the global section functor and the Spec functor

Sat, 22 Jul 2017 17:38:23 +0000

Let $A$ be a ring and let $(X,\mathcal{O}_X)$ be a scheme. Given a morphism $f:X\rightarrow \text{Spec}(A)$ we have an associated map on sheaves $f^{\#}:\mathcal{O}_{\text{Spec}(A)}\rightarrow f_* \mathcal{O}_X$. Taking global sections, we obtain a homomorphism $A\rightarrow \mathcal{O}_X(X)$. Thus there is a natural map

$\alpha: \text{Hom}_{\text{Schemes}}(X,\text{Spec}(A))\rightarrow \text{Hom}_{\text{Rings}}(A,\mathcal{O}_X(X)).$

Then, $\alpha$ is bijective. We will try to understand this adjointness of Global section functor and Spec functor. Suppose we are given a scheme $(X,\mathcal{O}_X)$ and a ring homomoprhism $\varphi: A\rightarrow \mathcal{O}_X(X)$. We construct a morphism of schemes $(f,f^{\#}):(X,\mathcal{O}_X)\rightarrow (\text{Spec}(A),\mathcal{O}_{\text{Spec}(A)})$. We first define morphism of topological spaces $f:X\rightarrow \text{Spec}(A)$. Let $x\in X$, we want to assign a prime ideal $P$ in $A$. Let $X=\text{Spec}(B)$ and $x=\mathfrak{P}\in X=\text{Spec}(B)$, as we have a ring homomorphism

$\varphi : A\rightarrow \mathcal{O}_X(X)=O_{\text{Spec}(B)}(\text{Spec}(B))=B$

Group as a category with one object

Wed, 19 Jul 2017 09:30:17 +0000

Let $G$ be a group. We are going to construct a category with one element whose morphisms are elements of this group $G$. Definition : A category $\mathcal{C}$ consists of

a collection of objects, $\text{Ob}(\mathcal{C})$,
for each $A,B\in \text{Ob}(\mathcal{C})$ a collection of maps from $A$ to $B$ denoted by $\mathcal{C}(A,B)$.
for each $A,B,C \in \text{Ob}(\mathcal{C})$ a function $\mathcal{C}(B,C)\times \mathcal{C}(A,B)\rightarrow \mathcal{C}(A,C)$ with $(g,f)\mapsto g\circ f$ called the composition
for each $A\in \text{Ob}(\mathcal{C})$ an element $1_A\in \mathcal{C}(A,A)$ called the identity on $A$

satisfying the following conditions

Associativity : for each $f\in \mathcal{C}(A,B), g\in \mathcal{C}(B,C), h\in \mathcal{C}(C,D)$ we have $(h\circ g)\circ f=h\circ (g\circ f)$.
Identity law : for each $f\in \mathcal{C}(A,B)$ we have $f\circ 1_A=f=1_B\circ f$.

We are not constructing a categroy $\mathcal{C}$ with $\text{Ob}(\mathcal{C})=G$, we are constructing a category $\mathcal{C}$ with $\text{Ob}(\mathcal{C})=\{A\}$(one point set) and $\mathcal{C}(A,A)=G$. We have defined objects and morphisms collections (here there is only one object so there is only one morphisms collection). We have to define composition $\mathcal{C}(A,A)\times \mathcal{C}(A,A)\rightarrow \mathcal{C}(A,A)$ i.e., we have to give a map $G\times G\rightarrow G$. There are two natural ways to give this map $(g,h)\mapsto g.h$ or $(g,h)\mapsto h.g$. We consider the map $(g,h)\mapsto g\circ h:=g.h$ to give composition. Obvious choice of an identity element in $\mathcal{C}$ is $llatex 1_A=e$ identity element of the group. Associativity of group implies associativity of composition

$(g\circ h)\circ f=(g.h)\circ f=(g.h).f=g.(h.f)=g\circ (h\circ f).$

Yoneda Lemma

Sun, 09 Jul 2017 08:06:02 +0000

Yoneda lemma : Let $\mathcal{C}$ be a (locally) small category. Then naturally in $A\in \mathcal{C}$ and $X\in [\mathcal{C}^{\rm{op}},\rm{Set}]$. Terminology :

$\mathcal{C}$ is a category mentioned in the lemma, $\mathcal{C}^{\rm{op}}$ is the opposite category associated to $\mathcal{C}$.
$\rm{Set}$ is the category with elements as sets and morphisms as functions.
$X:\mathcal{C}^{op}\rightarrow \rm{Set}$ is a functor.
Given $A\in \mathcal{C}$, $H_A$ is the functor $H_A:\mathcal{C}^{\rm{op}}\rightarrow \rm{Set}$ given by $B\mapsto \mathcal{C}(B,A)$.
$[\mathcal{C}^{\rm{op}},\rm{Set}]$ is the category with functors from $\mathcal{C}^{\rm{op}}$ to $\rm{Set}$ as elements and natural transformations between these functors as morphisms.
$[\mathcal{C},\rm{Set}]$ is the category with functors from $\mathcal{C}$ to $rm{Set}$ as elements and natural transformations between these functors as morphisms.

Given $X,A$ as above, $X(A)$ is a set and $[\mathcal{C}^{\rm{op}},\rm{Set}](H_A,X)$ is a set. Yoneda lemma says that there is a bijection between these sets, natural bijection in both $A$ and $X$. Natural transformation : Let $\mathcal{A},\mathcal{B}$ be two categories and $F,G:\mathcal{A}\rightarrow \mathcal{B}$ be both contravariant or both covariant functors. A natural transformation $\eta:F\rightarrow G$ is a family of arrows (morphisms) $F(A)\xrightarrow{\eta(A)}G(A)$ such that for each $A\xrightarrow{f}A'$ in $\mathcal{A}$ the following appropriate diagram commutes. Now that we have defined terminology used in the statement, we will now prove the statement. We will first give a bijectionand then prove that it is natural in $A$ and $X$. Construction of Bijective map : Let $\eta\in [\mathcal{C}^{\rm{op}},\rm{Set}](H_A,X)$ i.e., $\eta:H_A\rightarrow X$ is a natural transformation. We want to assign an element in $X(A)$ with this $\eta$. It is only natural to consider the map $\eta(A):H_A(A)\rightarrow X(A)$. The set $H_A(A)$ has a special element namely $1_A\in H_A(A)$, its image $\eta(A)(1_A)\in X(A)$. Define $\Phi(\eta)=\eta(A)(1_A)$. This give a map

$\Phi:[\mathcal{C}^{\rm{op}},\rm{Set}](H_A,X)\rightarrow X(A)$.

Sheaf associated to a Module over a ring

Sat, 08 Jul 2017 03:46:15 +0000

Let $A$ be a ring and $M$ be an $A$ module. We associate a sheaf of modules $\widetilde{M}$ on $X= \text{Spec(A)}$ with this module $M$. These modules are our models for quasi-coherent sheaves. For an open subset $U\subseteq \text{Spec(A)}$ we definewhere the condition $\dagger$ says that given $p\in U$ we have $s(p)\in M_p$ and that $s$ is locally a fraction i.e., given $p\in U$ there exists an open set $U(p)\subseteq U$ and $m\in M, f\in A$ such that $s(q)=\frac{m}{f}\in M_q$ for all $q\in U(p)$. With obvious restriction maps this defines a sheaf $\widetilde{M}$ on $X= \text{Spec(A)}$ called the sheaf associated with $M$. This should remind you something similar we have done before. We have defined structure sheaf on $X=\text{Spec(A)}$ in exactly same way where $M$ in this definition is replaced by ring $A$. Just to confirm, we havewhere the condition $\dagger$ says that given $p\in U$ we have $s(p)\in A_p$ and that $s$ is locally a fraction i.e., given $p\in U$ there exists an open set $U(p)\subseteq U$ and $a\in A, f\in A$ such that $s(q)=\frac{a}{f}\in A_q$ for all $q\in U(p)$. This should suggest some relation between $\widetilde{M}(U)$ and $\mathcal{O}_X(U)$. It turns out that $\widetilde{M}(U)$ is an $\mathcal{O}_X(U)$ module for every open $U\subseteq \text{Spec(A)}$. So, $\widetilde{M}$ is a sheaf of $\mathcal{O}_X$ modules. In case of structure sheaf $\mathcal{O}_X$ on $X=\text{Spec (A)}$ we have following proposition. Proposition : Let $A$ be a ring, and $(\text{Spec A},\mathcal{O})$ its spectrum.

For any $\mathfrak{p}\in \text{Spec A}$, the stalk $\mathcal{O}_{\mathfrak{p}}$ of the sheaf $\mathcal{O}_{}$ is isomorphic to the local ring $A_{\mathfrak{p}}$ i.e., $\mathcal{O}_{\mathfrak{p}}\cong A_{\mathfrak{p}}$.
For any element $f\in A$, the ring $\mathcal{O}(D(f))$ is isomorphic to the localized ring $A_f$ i.e., $\mathcal{O}(D(f))\cong A_f$.
In particular, $\Gamma(\text{Spec A}, \mathcal{O})\cong A$.

In case of associated sheaf $\widetilde{M}$ on $X=\text{Spec A}$ we have same results with $A$ replaced by $M$ and $\mathcal{O}$ replaced by $\widetilde{M}$. Precisely, we have following proposition. Proposition : Let $A$ be a ring, $M$ be an $A$ module and and $\widetilde{M}$ be the associated sheaf on $X=\text{Spec A}$.

For any $\mathfrak{p}\in \text{Spec A}$, the stalk $\widetilde{M}_{\mathfrak{p}}$ of the sheaf $\widetilde{<}$ is isomorphic to the localization $M_{\mathfrak{p}}$ i.e., $\widetilde{M}_{\mathfrak{p}}\cong M_{\mathfrak{p}}$.
For any element $f\in A$, the ring $\widetilde{M}(D(f))$ is isomorphic to the localization $M_f$ i.e., $\widetilde{M}(D(f))\cong M_f$.
In particular, $\Gamma(\text{Spec A}, \widetilde{M})\cong M$.

Proposition : Let $A$ be a ring and let $X=\text{Spec}(A)$. Also let $A\rightarrow B$ be a ring homomorphism, and let $f:\text{Spec}(B)\rightarrow \text{Spec}(A)$ be the correpsonding morphism of spectra. Then :

the map $M\mapsto \widetilde{M}$ gives an exact, fully faithful functor from category of $A$ modules to the category of $\mathcal{O}_X$ modules.
$\widetilde{M\otimes_A N}=\widetilde{M}\otimes_{\mathcal{O}_X}\widetilde{N}$.
$\widetilde{\bigoplus M_i}=\bigoplus \widetilde{M_i}$.
For a $B$ module $N$, we have $f_*(\widetilde{N})=\widetilde{~_A N}$ where $~_A N$ is $N$ considered as an $A$ module.
For a $A$ module $M$, we have $f^*(\widetilde{M})=\widetilde{M\otimes_A B}$.

About

Tue, 02 Jun 2015 08:24:56 +0000

I am Praphulla Koushik, faculty member in department of Mathematics, NIT Calicut. I am interested broadly in Differential geometry and Category theory. I want to share my thoughts as I read the following books.

Hartshorne's Algebraic geometry.
...
...

I wish to add some articles that seems to be interesting for me on different topics. Most of the times, I start writing a post and leave it in the middle.. I usually write in usual latex before copying it here. It is time consuming to put latex in between dollars. I have to take screen shot of figures, crop into image and then add here which is really time consuming. So, If I feel I am wasting time, I will leave it like that. If you find any of my posts interesting and incomplete, do leave a message. I will continue it.