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Perhaps an easier approach if you wanted to avoid going into the world of schemes would be to define orthogonal, symplectic, and unitary matrices over $R$ in analogy with the case when $R=\mathbb{C}$. …

I learned about this in a course taught by Mark Hovey, but we didn't use a book and his lecture notes are not available online. As I recall, you first understand how the signature of a quadratic form …

The proof is given on page 3 of Mark Hovey's paper "Cotorsion Pairs and Model Categories" http://homepages.math.uic.edu/~bshipley/hovey.pdf After writing "It is not at all obvious that this is a coto …

Unless I'm misunderstanding your question, the answer is yes. Your second way of defining limits, via "the universal property of a diagram with some arrows in $\mathcal{F}$," is actually a special cas …

Isn't this treated in Claude Schochet's 1992 paper "On Equivariant Kasparov Theory and Spanier-Whitehead Duality"? Also, if you want more references, I'd recommend using Google Scholar to see who cite …

Fernando's answer tells you how to prove the statement directly. Alternatively, if you want a reference, this is proven in Corollary 3.6 of my PhD thesis paper (published in JPAA) Model Structures on …

The comment by skd basically answers your question. I am writing to flesh it out with references, so your question doesn't stay open forever. The derived category of $I$-complete $A$-modules has recen …

Short exact sequences form a bridge of sorts between homological algebra and representation theory. For example, Maschke's theorem is the statement that, if $G$ is a finite group and $k$ is a field wh …

An example of a short exact sequence satisfying your first desiderata, but one which you probably won't fully understand till you are further along in homological algebra, is the Universal Coefficient …

I guess the quintessential example, satisfying your second desiderata, is $$ 0 \rightarrow A \stackrel{f}{\rightarrow} B \rightarrow B/f(A) \rightarrow 0. $$ For example, if $f = \mu_n: \mathbb{Z} \to …

The literature about constructing model structures on abelian categories has grown significantly since Hovey's book came out. In particular, there is now a connection between this process and cotorsio …

I recommend the Handbook of K-theory. It was published in 2005 and Part II seems to contain what you're looking for.

The answer is yes, this theory has been worked out. As a topologist, my favorite reference is Christensen-Hovey "Quillen model structures for relative homological algebra," which works in extreme gene …

Yes. The category of $\mathbb{Z}$-coalgebras is locally presentable, and objects are filtered colimits of finite dimensional subobjects. See the appendix to Coalgebraic models for combinatorial model …

Yes, there is a nice way. The mapping cone $C_i^*$ is the homotopy pushout of $\ast \gets A_i^* \to B_i^*$, and $-\otimes X$ is a monoidal left Quillen functor (symmetric if the underlying ring is com …