Algebraic-Geometry

Let \(M\) be a skew-symmetric matrix (with real entries). Let \(\lambda\) be an eigenvalue of \(M\). This means, there exists vector \(v\) such that \(Mv=\lambda v\). To relate with ``skew-symmetricness'' of \(M\), we apply transpose on both sides of previous equation, to get \(v^TM^T=\lambda v^T\). As \(M\) is skew-symmetric, we see that \(v^TM^T=\lambda v^T\) is equivalent to \(-v^TM=\lambda v^T\). Now, multiply by \(v\) on both sides of the above equation to get \(-v^TMv=\lambda v^Tv\). ...

On a Saturday morning, I was thinking about sylow theorems. The question I asked myself is, do I know how to apply sylow theorems? Only application I was aware about, of sylow theorem, is to assure if a group of finite order is simple or not. As a first step, I thought to check for groups of order less than 100. ...

Once we have a reasonably good notion of an object, we would look at a notion of morphisms. Let \((L,A,\rho,\tau)\) to \((L',A',\rho',\tau')\) be Lie-Rinehart algebras. Our experience suggests that the data of a morphism of Lie-Rinehart algebras from \((L,A,\rho,\tau)\) to \((L',A',\rho',\tau')\) should at least have two morphisms, one a morphism of Lie algebras \(\Phi:L\rightarrow L'\) and a morphism of associative algebras \(\Psi:A\rightarrow A'\) such that the following diagram commute, ...

These are “notes” I have written for myself when reading the book Model Categories by Mark Hovey. This book has some typos, there is an errata by its Author. There might be some more typos. I am assuming some notation and results about topological spaces (fibrations, cofibrations, etc) and homological algebra (chain complexes, etc). Other references for Model categories are : An Introduction to Homotopical categories by Julie Bergner.  

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 \(\mathcal{D}\).

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 \(i^*(a),i^*(b)\in \mathcal{S}(V)\).

A morphism of stacks \(f:\mathcal{D}\rightarrow \mathcal{C}\) is called a representable submersion if, for every morphism \(\underline{M}\rightarrow \mathcal{C}\), the fibred product \(\mathcal{D}\times_{\mathcal{C}}\underline{M}\) is representable i.e., isomorphic to a stack coming from a manifold and that the map of manifolds \(\mathcal{D}\times_{\mathcal{C}}\underline{M}\rightarrow M\) is a submersion. Following is a criterion for a map of stacks to be representable submersion. The result says it is enough to check for epimorphisms \(\underline{M}\rightarrow \mathcal{C}\). Precise statement is as follows. Let \(f:\mathcal{D}\rightarrow \mathcal{C}\) be a morphism of stacks. Suppose given a manifold \(U\) and a morphism of stacks \(\underline{U}\rightarrow \mathcal{C}\) which is an epimorphism. If the fibered product \(\mathcal{D}\times_{\mathcal{C}}\underline{U}\) is representable i.e., isomorphic to a stack coming from a manifold and that the map of manifolds \(\mathcal{D}\times_{\mathcal{C}}\underline{U}\rightarrow U\) is a submersion, then \(f\) is representable submersion. Let us see what this means in the set up of manifolds. Let \(f:\mathcal{D}\rightarrow \mathcal{C}\) be a morphism of stacks. Let \(f:M\rightarrow N\) be a morphism of manifolds (which gives a morphism of stacks \(\underline{M}\rightarrow \underline{N}\)). Suppose given a manifold \(U\) and a morphism of stacks \(\underline{U}\rightarrow \mathcal{C}\) which is an epimorphism. A representable surjective submersion is an epimorphism. So, we consider a surjective submersion \(g:U\rightarrow N\) (which gives an epimorphism \(\underline{U}\rightarrow \underline{N}\) being a representable surjective submersion). If the fibered product \(\mathcal{D}\times_{\mathcal{C}}\underline{U}\) is representable i.e., isomorphic to a stack coming from a manifold and that the map of manifolds \(\mathcal{D}\times_{\mathcal{C}}\underline{U}\rightarrow U\) is a submersion. As \(U\rightarrow N\) is submersion, it is anyways true that \(M\times_N U\) is a smooth manifold. What is extra that we have here is that \(M\times_NU\rightarrow U\) is a submersion. It is anyways true that \(M\times_NU\rightarrow M\) is a submersion being a pullback of submersion. But it is not true in general that \(M\times_NU\rightarrow U\) is a submersion. Here, we are given that \(M\times_NU\rightarrow U\) is a submersion. So, Let \(f:\mathcal{D}\rightarrow \mathcal{C}\) be a morphism of stacks. Suppose given a manifold \(U\) and a morphism of stacks \(\underline{U}\rightarrow \mathcal{C}\) which is an epimorphism. If the fibered product \(\mathcal{D}\times_{\mathcal{C}}\underline{U}\) is representable i.e., isomorphic to a stack coming from a manifold and that the map of manifolds \(\mathcal{D}\times_{\mathcal{C}}\underline{U}\rightarrow U\) is a submersion, then \(f\) is representable submersion. turns to Let \(f:M\rightarrow N\) be a morphism of manifolds be a morphism of manifolds. Suppose given a manifold \(U\) and a sujective submersion \(\underline{U}\rightarrow N\). If the fibered product \(M\times_{N}U\) is a manifold and that the map of manifolds \(M\times_{N}U\rightarrow U\) is a submersion, then \(f\) is submersion. This is more or less obvious. We have following commutative diagram As \(G:U\rightarrow N\) is a submersion (we have started with this) and \(p_2: M\times_N U\rightarrow U\) is a submersion (we are given this), the composition \(G\circ p_2=F\circ p_1\) is a submersion which then imply that \(F:M\rightarrow N\) is a submersion. Let \(m\in M\). As \(G:U\rightarrow N\) is surjective, so is \(p_1\) (pullback of surjective is surjective) i.e., there exists \((m,u)\in M\times_N U\) such that \(p_1(m,u)=m\). As \(F\circ p_1\) is submersion, \((F\circ p_1)_{*,(m,u)}(T_{m,u}(M\times_N U))=T_{F(m)}N\). Applying chain rule, we have \(F_{*,m}((p_1)_{*,(m,u)}(T_{m,u}(M\times_N U)))=T_{F(m)}N\), in particular, \(F_{*,m}(T_mM)=T_{F(m)}N\). Thus, \(F\) is submersion. So, we need both surjectivity and submersion of \(U\rightarrow N\). Now, let us look at more general case. Now, \(U\rightarrow N\) is not a surjective submersion but induces an epimorphism \(U\rightarrow N\). Suppose that the pullback \(M\times_N U\) is a manifold and that the map \(M\times_N U\rightarrow U\) is a submerson. Let \(W\rightarrow N\) be a map. We need to prove that \(M\times_N W\) is a manifold. We have following diagram As \(M\times_N U\rightarrow U\) is a submersion, the pullback \((M\times_N U)\times_U W_i=M\times_N W_i\) is a manifold. So, we have an open cover \(\{W_i\rightarrow W\}\) of \(W\) such that the pullbacks \(M\times_N W_i\) are manifolds. I think this should confirm that \(M\times_N W\) is a manifold and just because \(M\times_N W\rightarrow W_i\) are submersions, so is the map \(M\times_N W\rightarrow W\). Thus, \(f:M\rightarrow N\) is a representable submersion. The same idea works for an arbitrary map of stacks \(\mathcal{D}\rightarrow \mathcal{C}\). Let \(\underline{W}\rightarrow \mathcal{C}\) be a map of stacks. We have to prove that \(\mathcal{D}\times_{\mathcal{C}}\underline{W}\) is representable and that the map of manifolds \(\mathcal{D}\times_{\mathcal{W}}\underline{W}\rightarrow W\) is a submersion. As \(\underline{U}\rightarrow \mathcal{C}\) is epimorphism, for \(\underline{W}\rightarrow \mathcal{C}\) there exists an open cover \(\{W_i\rightarrow W\}\) with commutative diagram as shown below. We have following diagram As \(\mathcal{D}\times_{\mathcal{C}} \underline{U}\rightarrow U\) is a submersion, the pullback \((\mathcal{D}\times_{\mathcal{C}} \underline{U})\times_U W_i=\mathcal{D}\times_{\mathcal{C}} W_i\) is a manifold. So, we have an open cover \(\{W_i\rightarrow W\}\) of $ W$ such that the \((\mathcal{D}\times_{\mathcal{C}}W)\times_W W_i=\mathcal{D}\times_{\mathcal{C}} W_i\) ...

Given a principal \(G\) bundle \(P\rightarrow M\) we associate what is called a Weil homomorphism \(I(G)\rightarrow H^*(M,\mathbb{R})\). Given \(f\in I^k(G)\) i.e., \(f:\underbrace{\mathfrak{g}\times\cdots\times\mathfrak{g}}_{k\text{ times}}\rightarrow \mathbb{R}\) we associate an element in \(H^{2k}(M,\mathbb{R})\) as follows. This is only an outline. It is useful if you can fill the gaps by your self. Fix a connection \(\Gamma\) on \(P(M,G)\) and let \(\Omega\) denote the curvature form associated to \(\Gamma\). The element \(f\in I^k(G)\) gives a \(2k\)-form \(f(\Omega):P\rightarrow \Lambda^{2k}T^*P\) on \(P\) as follows. \(f(\Omega)(v_1,\cdots,v_{2k})=\frac{1}{(2k)!}\sum_{\sigma\in S_{2k}} f(\Omega(v_{\sigma(1)}.v_{\sigma(2)}),\cdots\Omega(v_{\sigma(2k-1)},v_{\sigma(2k)}))\) ...

Definition : Let \(X\) be a topological space, \(\mathcal{F},\mathcal{G}\) be presheaves on \(X\). A morphism \(\varphi:\mathcal{F}\rightarrow \mathcal{G}\) is a collection \(\{\varphi(U):\mathcal{F}(U)\rightarrow \mathcal{G}(U)\}\) indexing over all open \(U\subseteq X\) such that the following diagram is commutative for open \(U\subseteq V\subseteq X\). Morphism of sheaves inducing Morphism of stalks : We see that any morphism of sheaves \(\varphi:\mathcal{F}\rightarrow \mathcal{G}\) induces morphism of stalks \(\varphi_p:\mathcal{F}_p\rightarrow \mathcal{G}_p\) for each \(p\in X\). Fix \(p\in X\). Let us define \(\varphi_p:\mathcal{F}_p\rightarrow \mathcal{G}_p\) i.e., for \((U,s)\in \mathcal{F}_p\) we give an open set \(V\) containing \(p\) and a section \(t\in \mathcal{G}(V)\) giving an element \((V,t)\in \mathcal{G}_p\). One obvious choice of an open set containing \(p\) is \(U\). For this \(U\), we have \(\varphi(U):\mathcal{F}(U)\rightarrow \mathcal{G}(U)\) sending \(s\) to \(\varphi(U)(s)\in \mathcal{G}(U)\). Define \(\varphi_p:\mathcal{F}_p\rightarrow \mathcal{G}_p\) as \((U,s)\mapsto (U,\varphi(U)(s))\). Map is well defined : We prove \((U,s) \sim (V,t)\) implies \((U,\varphi(U)(s))\sim (V,\varphi(V)(t))\). As \((U,s) \sim (V,t)\) there exists an open subset \(W\subseteq U\cap V\) containing \(p\) such that \(s|_W=t|_W\). We prove that \(\varphi(U)(s)|_W=\varphi(V)(t)|_W\) which implies \((U,\varphi(U)(s))\sim (V,\varphi(V)(t))\). The commutative diagram gives \(\varphi(U)(s)|_W=\varphi(W)(s|_W)\). Similar diagram in which \(U\) is replaced by \(V\) gives that \(\varphi(V)(t)|_W=\varphi(W)(t|_W)\). As \(s|_W=t|_W\) we have \(\varphi(W)(s|_W)=\varphi(W)(t|_W)\), concluding that \(\varphi(U)(s)|_W=\varphi(V)(t)|_W\). So, given a morphism of sheaves \(\varphi:\mathcal{F}\rightarrow \mathcal{G}\) we have well defined morphism \(\varphi_p:\mathcal{F}_p\rightarrow\mathcal{G}_p\) for each \(p\in X\). Theorem : A morphism of sheaves \(\varphi:\mathcal{F}\rightarrow \mathcal{G}\) is an isomorphism of sheaves iff the induced map \(\varphi_p:\mathcal{F}_p\rightarrow\mathcal{G}_p\) is an isomorphism for each \(p\in X\). Proof : Let \(\varphi:\mathcal{F}\rightarrow \mathcal{G}\) is an isomorphism of sheaves i.e., \(\varphi(U):\mathcal{F}(U)\rightarrow \mathcal{G}(U)\) is an isomorphism of groups for each open \(U\subseteq X\). Fixing \(p\in X\) we prove that \(\varphi_p:\mathcal{F}_p\rightarrow\mathcal{G}_p\) is an isomorphism. Let \((U,s)\in \mathcal{F}_p\) be such hat \((U,\varphi(U)(s))=0\in \mathcal{G}_p\) i.e., \(\varphi(U)(s)|_W=0\) for some open \(W\subseteq U\). We thus have \(\varphi(W)(s|_W)=\varphi(U)(s)|_W=0\). As \(\varphi(W):\mathcal{F}(W)\rightarrow \mathcal{G}(W)\) is injective, this means \(s|_W=0\). Thus, \(\varphi_p:\mathcal{F}_p\rightarrow\mathcal{G}_p\) is an injective map. Let \((V,t)\in \mathcal{G}_p\) i.e., \(p\in V\) and \(t\in \mathcal{F}(V)\). As \(\varphi(V):\mathcal{F}(V)\rightarrow \mathcal{G}(V)\) is surjective, there exists \(s\in \mathcal{F}(V)\) such that \(\varphi(V)(s)=t\). So, \(\varphi_p((V,s))=(V,\varphi(V)(s))=(V,t)\). Thus, \(\varphi_p\) is surjective. So, \(\varphi:\mathcal{F}\rightarrow \mathcal{G}\) is an isomorphism of sheaves implies \(\varphi_p:\mathcal{F}_p\rightarrow\mathcal{G}_p\) is an isomorphism for each \(p\in X\). Conversely, suppose that \(\varphi_p:\mathcal{F}_p\rightarrow\mathcal{G}_p\) is an isomorphism for each \(p\in X\). We prove \(\varphi(U):\mathcal{F}(U)\rightarrow \mathcal{G}(U)\) is an isomorphism for each open \(U\subseteq X\). Fix \(U\subseteq X\) and consider \(\varphi(U):\mathcal{F}(U)\rightarrow\mathcal{G}(U)\). Let \(s\in \mathcal{F}(U)\) be such that \(\varphi(U)(s)=0\). Fix \(p\in U\) and consider \(\varphi_p:\mathcal{F}_p\rightarrow \mathcal{G}_p\). We have \((U,s)\in \mathcal{F}_p\) with \(\varphi_p((U,s))=(U,\varphi(U)(s))=0\). As \(\varphi_p\) is injective, this means that \(s|_{W_p}=0\) for some \(W_p\subseteq U\) containing \(p\). This is true for all \(p\in U\). So, we have an open cover \(\{W_p\}_{p\in U}\) of \(U\) and \(s\in \mathcal{F}(U)\) such that \(s|_{W_p}=0\). Identity axiom of sheaf implies that \(s=0\). So, \(\varphi(U):\mathcal{F}(U)\rightarrow \mathcal{G}(U)\) is injective. Let \(s\in \mathcal{G}(U)\). Fix \(p\in U\) and consider \(\varphi_p:\mathcal{F}_p\rightarrow \mathcal{G}_p\). As \((U,s)\in \mathcal{G}_p\) and \(\varphi_p\) is surjective, there exists \((V,t_p)\in \mathcal{F}_p\) such that \(\varphi_p((V,t_p))=(U,s)\) i.e., \((V,\varphi(V)(t_p))=(U,s)\in \mathcal{F}_p\) i.e., \(s|_{W_p}=\varphi(V)(t_p)|_{W_p}\) for some \(p\in W_p\subseteq U\cap V\). Idea is to glue the sections \(t_p|_{W_p}\in \mathcal{F}(W_p)\) to get a section \(t\in \mathcal{F}(U)\). For that we show that \(t_p|_{W_p\cap W_q}=t_q|_{W_p\cap W_q}\). We have the following commuative diagram,which says that \(\varphi(W_p)(t_p|_{W_p})|_{W_p\cap W_q}=\varphi(W_p\cap W_q)(t_p|_{W_p\cap W_q}).\) ...