# Tagged Questions

Homological algebra studies homology in a general algebraic setting. The purpose is extraction of information about structures involved in terms of tangible objects like rings groups and modules.

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### Projective resolution of tensor product

Let $M,N$ are $R$ modules and $P^\bullet, Q^\bullet$ are their projective resolutions. Can we obtain projective resolution $M\otimes N$ using $P^\bullet, Q^\bullet$. If i understand correctly homology ...
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I read here the formula for cohomology of a free group with trivial action: $$H^q(G, M) = \begin{cases} M &\text{for } q = 0\\ M^n &\text{for } q = 1\\ 0 &\text{for } q \geq 2\\ \end{cases}... 0answers 14 views ### Minus signs in internal hom Consider the internal hom of chain complexes A^\bullet and B^\bullet$$\operatorname{Hom}^\bullet(A^\bullet, B^\bullet):=\{\text{degree $n$ maps}\} \qquad df:= d^B\circ f - (-1)^{|f|}f \circ d^A$$... 1answer 20 views ### Choosing an injective resolution of a short exact sequence of complexes Lemma: Given a short exact sequence of cochain complexes in an abelian category \mathcal{C} with enough injectives,$$0 \to P^\bullet \xrightarrow{f} Q^\bullet \xrightarrow{g} R^\bullet \to 0,$$... 0answers 7 views ### On selfinjectivity of Hopf algebras Any group algebra kG is selfinjective. More generally Gentile proves that for a group ring RG with R commutative and torsion free as a \Bbb Z-module, RG is selfinjective if and only if G ... 3answers 92 views ### Finitely generated projective modules over polynomial rings with integral coefficients There is famous Quillen-Suslin theorem which states that every finitely generated projective module over a ring of polynomials k[x_1,...,x_n], where k is a field, is free. I have never carefully ... 3answers 263 views ### What is the product and coproduct of Morphism category (Arrow category)? Given a category C, its morphism category D means a category that has 1) "morphisms of C" as its objects 2) "pairs (f,g) s.t. the diagram (square) commutes" as its morphisms. The above definition ... 1answer 22 views ### Finite product exists implies finite coproduce exist. Let C be a category such that the law composition of morphisms is bilinear, and there exists a zero object 0, and the products exists for arbitrary finite sets of objects of C. Then the ... 0answers 52 views ### Chain Homotopy in abelian category When dealing with complexes of modules or groups, the following lemma is pretty easy: If f,g :E\rightarrow E' are homotopic, i.e. f-g=d'h+hd for some h, then f,g induce the same homomorphism ... 0answers 55 views ### Homotopy Colimit of Truncations Let \mathcal{A} be an additive category with countable coproducts. I am just starting to learn about homotopy colimits and I am struggling with the following example that I am very interested in ... 1answer 51 views ### Meaning of functorial It's known that for a short exact sequence of complexes, 0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0, it associates a homology sequences ...\rightarrow H(E')\rightarrow H(E)\rightarrow ... 0answers 23 views ### What properties are preserved by direct limits? [on hold] We know that direct limit of a directed family of flat R-modules is also flat (R is a commutative ring with 1 and all modules are unital). I am looking for other properties of modules which ... 1answer 16 views ### Seifert matrix, linking numbers, generators I have been asked to compute the seifert form of a knot, the twist knot. I know how to compute the seifert surface, and then the seifert matrix seems to be defined accordingly (according to all the ... 1answer 33 views ### Why solving linear equations is taking a quotient by some subspace? Linear equation can be represented by a linear form, and its solution space is the same thing as kernel of this form. The same is true for system of linear equations. But this lecture notes suggest ... 1answer 42 views ### A left exact functor preserves quasi-isomorphisms between acyclic complexes A homological algebra theorem states Theorem: Let T: \mathscr{A} \to \mathscr{B} be a left exact functor between abelian categories, and let X^\bullet \xrightarrow{f} Y^\bullet be a quasi-... 0answers 30 views ### The relation between Weyl character formula and Frobenius characteristic map Let \mathfrak{gl}(n) be the general linear Lie algebra of rank n, and \mathfrak{S}_d be the symmetric group of rank d. It is well-known that the Schur-Weyl duality provide a equivalence ... 0answers 28 views ### someone desbribe what Homology theory is in 1hour lecture? [on hold] Am self studying abstract algebra.I feel the concept of ideals,cosets is what exactily is what is called homology group a fundamental principles of homology theory?so,I need someone out there to ... 2answers 35 views ### If \text{Ext}_R^1(A,I) = 0 for all A\in ob(_R\text{Mod}), then I\in ob(_R\text{Mod}) is injective. [duplicate] Let \text{Ext}^1(A,I)=0 for all A\in ob(_R\text{Mod}), then I\in ob(_R\text{Mod}) is injective. I got stuck by this problem. Any ideas? 1answer 47 views ### How do you find the free resolution of the module M and of F/M where F=(K[x,y])^3? M is a module generated by$$f_1=(xy,y,x), f_2=(x^2+x,y+x^2,y), f_3=(-y,x,y),f_4=(x^2,x,y).$$We're to use the lex ordering with x<y and e_1>e_2>e_3, where terms are given preference ... 1answer 70 views ### Splitting a short exact sequence of complexes of vector spaces It's well-known that any complex of vector spaces is isomorphic to a direct sum of two types of indecomposable complexes (a one-dimensional space concentrated in one degree, or two one dimensional ... 0answers 31 views ### Ext group of bundles on moduli space of curves Let \mathcal{M}_{g} be the moduli space of curves of genus g. Let's suppose g \geq 2. Let T be the tangent bundle of \mathcal{M}_{g}. Is the Ext group \text{Ext}^1(\bigwedge^2T, T) trivial?... 1answer 152 views ### If two chain maps over a PID induce the same homomorphism, then they are homotopic If two chain maps f,g:\mathcal{X} \rightarrow \mathcal{Y}, where \mathcal{X},\mathcal{Y} are chain complexes with free modules X_p and Y_p over a PID, R, induce the same homomorphism in the ... 0answers 71 views ### Homology Groups of Tangent 2-Spheres I have been trying to compute the Homology Groups  H_n  of two tangent 2-Spheres (we will call this space, X). By previous results, I am able to easily determine that  H_0(X)  is isomorphic to the ... 1answer 224 views ### Functor preserves kernels iff it's left exact I'm trying to understand the proof to a statement in Rotman's 'Introduction to Homological Algebra': Proposition 5.25, p. 240: Let F :_R\text{Mod} \to \text{Ab} be a covariant functor. Then F ... 0answers 16 views ### Mapping cylinder of chain complexes via -\otimes \Delta An instructor gave me a homework set where the mapping cylinder of a chain map C_\bullet \xrightarrow{f} D_\bullet is defined as (\Delta^1_\bullet \otimes C_\bullet) \oplus_{C_\bullet} D_\bullet, ... 0answers 35 views ### projective resolution for an I-torsion R-module Let R be a commutative Noetherian ring with non-zero identity, I be an ideal of R and M be an I-torsion R-module. We know that there exists an injective resolution of M in which each ... 2answers 570 views ### Vanishing of a certain Tor I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ... 1answer 90 views ### Splitness of quotient sequence Let A, B, C be holomorphic vector bundles over some complex manifold X. Let A', B', C' be sub bundles, respectively. Suppose that we have short exact sequences:$$0 \rightarrow A \rightarrow B \...
While trying to understand the Hochschild-Kostant-Rosenberg theorem, I learned that $Ext_{R \otimes R}^1(R, R) = Der_K(R)$, where $R$ is an regular affine (commutative) $\mathbb{C}$-algebra. I am ...